ค้นหาข้อมูลเพิ่มเติมเกี่ยวกับ " gmat "

ข้อสอบ gmat

ข้อสอบ gmat
ข้อสอบ gmat

รายละเอียดของข้อสอบ GMAT

รายละเอียดของข้อสอบ GMAT
วันที่ 17 มีนาคม 2549
เรียบเรียงโดย: ดร.สิระ สุทธิคำ, สถาบัน Kendall Square
www.ToeflThailand.com

Graduate Management Admission Test หรือข้อสอบ GMAT เป็นข้อสอบที่ใช้วัดความสามารถของผู้ที่ต้องการเข้าศึกษาในระดับปริญญาโทและปริญญาเอกสาขาบริหารธุรกิจซึ่งรวมทั้ง MBA, M.S. Marketing, M.S. Finance, MIS (สำหรับสาขา MIS ของบางมหาวิทยาลัยอาจต้องใช้คะแนน GRE แทน), DBA, และ Ph.D. ด้านบริหารธุรกิจส่วนใหญ่ต้องใช้คะแนน GMAT ในการพิจารณารับนักศึกษา

ภาพรวมของข้อสอบ
เป็นการวัดความรู้ในการสื่อสารซึ่งรวมทั้งการอ่านและการเขียน, ทักษะการวิเคราะห์, และ ทักษะในการคำนวณ ที่จะสามารถใช้เป็นเครื่องชี้วัดความสำเร็จในการเรียนต่อทางด้านบริหารธุรกิจ
จาก website ของผู้ออกข้อสอบ
www.mba.com ระบุไว้ชัดเจนว่าข้อสอบ GMAT ไม่สามารถใช้วัดความสามารถในเรื่องต่อไปนี้

• ความรู้เฉพาะทางของธุรกิจเช่น มาตรฐานของบัญชี หรือ กฎหมายธุรกิจ
• ทักษะเฉพาะด้านของงาน (specific job skill) หรือเนื้อหาจากบทเรียนระดับปริญญาตรีด้านบัญชีและบริหาร
• คุณสมบัติอื่นของการเป็นนักธุรกิจที่ดีเช่นความมุ่งมั่น, ความคิดริเริ่มสร้างสรรค์, และความสามารถในการทำงานร่วมกับผู้อื่น

รูปแบบของข้อสอบและเวลา
ข้อสอบ GMAT ประกอบด้วยข้อสอบ 3 ส่วนคือ 1. การเขียน (Analytical Writing Assessment) 2. คณิตศาสตร์ (Quantitative) และ 3. ภาษาอังกฤษ (Verbal)

ข้อสอบการเขียน (AWA)
ข้อสอบ GMATเริ่มจากการทำข้อสอบเขียนก่อนเสมอ โดยจะมีรูปแบบของข้อสอบ 2 ลักษณะได้แก่ 1.การเขียนเพื่อแสดงความคิดเห็น (Issue) และ 2.การเขียนเพื่อแสดงวิจารณ์บทความ (Argument) ผู้เข้าสอบจะมีเวลา 30 นาทีต่อหนึ่ง essay

ข้อสอบคณิตศาสตร์ (Quantitative)
หลังจากการพักจากข้อสอบการเขียน 10 นาที ผู้เข้าสอบจะต้องทำโจทย์เลขแบบ multiple-choice จำนวน 37 ข้อโดยมีรูปแบบของข้อสอบสองลักษณะนั้นคือ 1. Problem Solving ~24 ข้อ และ 2. Data Sufficiency ~13 ข้อ โดยมีเวลาทำข้อสอบทั้งสิ้น 75 นาที คอมพิวเตอร์จะหยุดการทำงานทันทีเมื่อเวลาหมด

ข้อสอบภาษาอังกฤษ (Verbal)
หลังจากการพักจากข้อสอบคณิตศาสตร์แล้ว ผู้เข้าสอบสามารถพักหรือทำข้อสอบภาษาอังกฤษ (Verbal) ต่อเนื่องเลยได้ โดยข้อสอบส่วนนี้จะเป็นโจทย์แบบ multiple-choice จำนวน 41 ข้อโดยมีรูปแบบของข้อสอบสามลักษณะนั้นคือ 1. การอ่าน (Reading Comprehension) ~14 ข้อ 2. การวิเคราะห์ (Critical Reasoning) ~14 ข้อและ 3. ไวยกรณ์และการเขียน (Sentence Correction) ~13 ข้อ โดยมีเวลาทำข้อสอบทั้งสิ้น 75 นาที คอมพิวเตอร์จะหยุดการทำงานทันทีเมื่อเวลาหมด

http://toefl121.blogspot.com/

http://toeic1.blogspot.com/

ตัวอย่างข้อสอบGmat

Example:

The rise in negative attitudes toward foreigners indicate that the country is becoming less tolerant, and therefore that the opportunities are ripe for extremist groups to exploit the illegal immigration problem.

(A) indicate that the country is becoming less tolerant, and therefore that
(B) indicates that the country is becoming less tolerant, and therefore
(C) indicates that the country is becoming less tolerant, and therefore that
(D) indicates that the country is being less tolerant, and therefore
(E) indicates that the country is becoming less tolerant of and therefore that

Choice (A) has two flaws. First, the subject of the sentence the rise is singular, and therefore the verb indicate should not be plural. Second, the comma indicates that the sentence is made up of two independent clauses, but the relative pronoun that immediately following therefore forms a subordinate clause.

Choice (C) corrects the number of the verb, but retains the subordinating relative pronoun that.

Choice (D) corrects the number of the verb and eliminates the subordinating relative pronoun that. However, the verb being is less descriptive than the verb becoming: As negative attitudes toward foreigners increase, the country becomes correspondingly less tolerant. Being does not capture this notion of change.

Choice (E) corrects the verb's number, and by dropping the comma makes the subordination allowable. However, it introduces the preposition of which does not have an object: less tolerant of what?

Choice (B) both corrects the verb's number and removes the subordinating relative pronoun that. The answer is (B).

ศึกษาข้อมูลก่อนสอบ gmat

Saturday, April 26, 2008

ดาวน์โหลดดิกชันนารีไว้ใช้งาน ฟรี ! download dictionary free


Oxford Dictionary of English

ดาวน์โหลดดิกชันนารีไว้ใช้งาน ฟรี ! download dictionary free
[1] ประเภทดิกชันนารี อังกฤษ – ไทย, ไทย - อังกฤษ

โปรแกรมที่ 1: LEXiTRON (ดิก อังกฤษ – ไทย & ไทย - อังกฤษ) ประมาณ 20 MB
http://home.dsd.go.th/freeenglish/LEXiTRON.exe

โปรแกรมที่ 2: My Buddy Dictionary ดิก อังกฤษ – ไทย รวมดิกจาก 3 ฐานข้อมูลไว้ในโปรแกรมเดียวกัน ประมาณ 25 MB
http://www.thaibuddy.com/

ดาวน์โหลดดิก: (หมดอายุ 1 มกราคม 2552)
http://home.dsd.go.th/freeenglish/MyBuddy2.1beta.zip

โปรแกรมที่ 3: Loy Dictionary อังกฤษ – ไทย ประมาณ 25 MB
http://home.dsd.go.th/freeenglish/_LoyDictSetupZipFiles.exe
(เป็นไฟล์ Setup และรายละเอียดในการติดตั้งทั้งหมด)

โปรแกรมที่ 4: HighLight Dictionary (อังกฤษ ->ไทย และ ไทย-อังกฤษ ) ขนาด 10.5 MB
http://www.thaiware.com/main/info.php?id=3342
มีคลังข้อมูลบรรจุคำศัพท์จากพจนานุกรมอิเล็คทรอนิกส์ ถึง 3 เล่ม รวมกันกว่า 180,000 รายการ สมบูรณ์แบบด้วย ความหมาย , ประเภทของคำศัพท์ , คำย่อ , คำพ้องเสียง , คำเหมือน , คำไกล้เคียง , คำตรงข้าม, ตัวอย่างประโยค เป็นต้น สามารถใช้ร่วมกับโปรแกรมต่าง ๆ ได้ โดยจะแปลศัพท์ที่ คุณต้องการรู้ความหมายจริง ๆ โดยไม่ต้องพิมพ์คำศัพท์ใหม่ เพียงแค่คุณลาก Highlight คำศัพท์ที่ต้องการแล้วกดปุ่ม Windows Key + X โปรแกรมก็จะทำการแปลและแสดงผล ...


ดาวน์โหลดดิกชันนารีไว้ใช้งาน ฟรี ! download dictionary free
[2] ประเภท ดิกชันนารี อังกฤษ – อังกฤษ
โปรแกรมที่ 1: WordWeb 5 เป็นดิกอังกฤษ – อังกฤษ ใช้ฟรีที่ดีที่สุด [ขนาดประมาณ7 MB]
ดาวน์โหลด:
http://www.download.com/WordWeb/3000-2279_4-1000320

โปรแกรมที่ 2: Merriam Webster´s Concise Dictionary 2.1 [ขนาด 1.59 MB]
ดาวน์โหลด: http://www.download.com/Merriam-Webster-s-Concise-Dictionary/3000-2279_4-10059666.html?tag=lst-0-1

โปรแกรมที่ 3: TheSage´s English Dictionary and Thesaurus 1.1.2 [ขนาด 8.06 MB]
ดาวน์โหลด:
http://www.tucows.com/preview/412035
http://www.snapfiles.com/get/thesage.html

ดาวน์โหลดดิกชันนารีไว้ใช้งาน ฟรี ! download dictionary free

Friday, April 25, 2008

ลิงค์เพื่อความสุดยอดเก่งภาษาอังกฤษ

เรื่อง
Web site
British Council Organization Thailand http://www.britishcouncil.or.th/th/index.asp
Advance English Lessons http://www.englishpage.com
Tower of English Lobby http://www.towerofenglish.com
Nation Junior Newspaper http://www.njconnex.com/
NLL Free Resource http://www.nll.co.uk/free/exercises.shtml
American University Alumni (AUA) http://www.auathailand.org/
English Exercise Online http://www.smic.be/smic5022
Dictionary.com http://dictionary.reference.com
Acronym Dictionary (ค้นหาตัวย่อ) http://www.yindii.com/dictionary/acronym.htm
American Slang (คำศัพท์แสลง) http://www.yindii.com/ref/slang.htm
Forms of Address (การเขียนจดหมาย) http://www.yindii.com/ref/address.htm
Idioms http://www.yindii.com/ref/idioms/idiom1.htm
Online Dictionaries http://www.yindii.com/dictionary/rhyme.htm
Online Grammar http://www.yindii.com/ref/grammar/index.htm
Proverbs (สุภาษิต) http://www.yindii.com/ref/proverbs/proverb1.htm
SuperKids Vocabulary Builders http://www.superkids.com/aweb/tools/words
World English test, learn, and study online http://www.world-english.org
Learn English - British Council http://www.learnenglish.org.uk
Learn English Free With EnglishClub.com http://www.englishclub.com
The Internet Picture Dictionary http://www.pdictionary.com
Oxford Teachers Club Oxford Teachers' Club http://www.oup.com/elt/teachersclub/?cc=th
ESL-Pro.com - ESL PRO Products http://www.esl-pro.com/store.aspx
เกมอักษรไขว้ se-ed http://ww2.se-ed.net/acc2003/cross/cross1.htm
Crossword Puzzle Games http://www.crosswordpuzzlegames.com/create.html
Teach with movies http://www.teachwithmovies.net/index.html
Fun English Study http://www.manythings.org/
English-4U http://www.english-4u.com/
Vacabulary Can be Fun http://www.vocabulary.co.il/
Better English http://www.better-english.com/grammar.htm
คำศัพท์อังกฤษ http://www.vocabulary.com/
Crossword Puzzle ต http://www.mrtawat.com/xword.htm
Teaching English Organization http://www.teachingenglish.org.uk/
Webster Dictionary Online http://www.websters-online-dictionary.org/
Lyrics For All http://www.lyricsforall.com
English Puzzle http://www.puzzles.bambamscorner.nl/
englishpuzzles.html
Grammar Songs http://gardenofpraise.com/mugram.htm
Basic guide to essey writing http://members.tripod.com/~lklivingston/
essay/index.html

ตัวอย่างข้อสอบ gmat CHECKING EXTREME CASES

ตัวอย่างข้อสอบ gmat


CHECKING EXTREME CASES

• When drawing a geometric figure or checking a given one, be sure to include drawings of extreme cases as well as ordinary ones.

Example 1: In the figure to the right, AC is a chord and B is a point on the circle. What is the measure of angle x?

Although in the drawing AC looks to be a diameter, that cannot be assumed. All we know is that AC is a chord. Hence, numerous cases are possible, three of which are illustrated below:


In Case I, x is greater than 45 degrees; in Case II, x equals 45 degrees; in Case III, x is less than 45 degrees. Hence, the given information is not sufficient to answer the question.

Example 2: Three rays emanate from a common point and form three angles with measures p, q, and r. What is the measure of q + r ?

It is natural to make the drawing symmetric as follows:


In this case, p = q = r = 120, so q + r = 240. However, there are other drawings possible. For example:


In this case, q + r = 180. Hence, the given information is not sufficient to answer the question.

Problems:

1. Suppose 3p + 4q = 11. Then what is the value of q?

(1) p is prime.
(2) q = -2p

Solution: (1) is insufficient. For example, if p = 3 and q = 1/2, then 3p + 4q = 3(3) + 4(1/2) = 11. However, if p = 5 and q = -1, then 3p + 4q = 3(5) + 4(-1) = 11. Since the value of q is not unique, (1) is insufficient.

Turning to (2), we now have a system of two equations in two unknowns. Hence, the system can be solved to determine the value of q. Thus, (2) is sufficient, and the answer is B.


2. What is the perimeter of triangle ABC above?

(1) The ratio of DE to BF is 1: 3.
(2) D and E are midpoints of sides AB and CB, respectively.

Solution: Since we do not even know whether BF is an altitude, nothing can be determined from (1). More importantly, there is no information telling us the absolute size of the triangle.

As to (2), although from geometry we know that DE = AC/2, this relationship holds for any size triangle. Hence, (2) is also insufficient.

Together, (1) and (2) are also insufficient since we still don't have information about the size of the triangle, so we can't determine the perimeter. The answer is E.

3. A dress was initially listed at a price that would have given the store a profit of 20 percent of the wholesale cost. What was the wholesale cost of the dress?

(1) After reducing the asking price by 10 percent, the dress sold for a net profit of 10 dollars.
(2) The dress sold for 50 dollars.

Solution: Consider just the question setup. Since the store would have made a profit of 20 percent on the wholesale cost, the original price P of the dress was 120 percent of the cost: P = 1.2C. Now, translating (1) into an equation yields:

P - .1P = C + 10

Simplifying gives

.9P = C + 10

Solving for P yields

P = (C + 10)/.9

Plugging this expression for P into P = 1.2C gives

(C + 10)/.9 = 1.2C

Since we now have only one equation involving the cost, we can determine the cost by solving for C. Hence, the answer is A or D.

(2) is insufficient since it does not relate the selling price to any other information. Note, the phrase "initially listed" implies that there was more than one asking price. If it wasn't for that phrase, (2) would be sufficient. The answer is A.

4. What is the value of the two-digit number x?

(1) The sum of its digits is 4.
(2) The difference of its digits is 4.

Solution: Considering (1) only, x must be 13, 22, 31, or 40. Hence, (1) is not sufficient to determine the value of x.

Considering (2) only, x must be 40, 51, 15, 62, 26, 73, 37, 84, 48, 95, or 59. Hence, (2) is not sufficient to determine the value of x.

Considering (1) and (2) together, we see that 40 and only 40 is common to the two sets of choices for x. Hence, x must be 40. Thus, together (1) and (2) are sufficient to uniquely determine the value of x. The answer is C.

5. If x and y do not equal 0, is x/y an integer?

(1) x is prime.
(2) y is even.

Solution: (1) is not sufficient since we don't know the value of y. Similarly, (2) is not sufficient. Furthermore, (1) and (2) together are still insufficient since there is an even prime number--2. For example, let x be the prime number 2, and let y be the even number 2 (don't forget that different variables can stand for the same number). Then x/y = 2/2 = 1, which is an integer. For all other values of x and y, x/y is not an integer. (Plug in a few values to verify this.) The answer is E.

6. Is 500 the average (arithmetic mean) score on the GMAT?

(1) Half of the people who take the GMAT score above 500 and half of the people score below 500.
(2) The highest GMAT score is 800 and the lowest score is 200.

Solution: Many students mistakenly think that (1) implies the average is 500. Suppose just 2 people take the test and one scores 700 (above 500) and the other scores 400 (below 500). Clearly, the average score for the two test-takers is not 500. (2) is less tempting. Knowing the highest and lowest scores tells us nothing about the other scores. Finally, (1) and (2) together do not determine the average since together they still don't tell us the distribution of most of the scores. The answer is E.

7. The set S of numbers has the following properties:

I) If x is in S, then 1/x is in S.
II) If both x and y are in S, then so is x + y.

Is 3 in S?

(1) 1/3 is in S.
(2) 1 is in S.

Solution: Consider (1) alone. Since 1/3 is in S, we know from Property I that 1/(1/3) = 3 is in S. Hence, (1) is sufficient.

Consider (2) alone. Since 1 is in S, we know from Property II that 1 + 1 = 2 (Note, nothing in Property II prevents x and y from standing for the same number. In this case both stand for 1.) is in S. Applying Property II again shows that 1 + 2 = 3 is in S. Hence, (2) is also sufficient. The answer is D.

8. What is the area of the triangle above?

(1) a = x, b = 2x, and c = 3x.
(2) The side opposite a is 4 and the side opposite b is 3.

Solution: From (1) we can determine the measures of the angles: a + b + c = x + 2x + 3x = 6x = 180

Dividing the last equation by 6 gives: x = 30

Hence, a = 30, b = 60, and c = 90. However, different size triangles can have these angle measures, as the diagram below illustrates:


Hence, (1) is not sufficient to determine the area of the triangle.

Turning to (2), be careful not to assume that c is a right angle. Although from the diagram c appears to be a right angle, it could be 91 degrees or 89 degrees--we can't tell. Hence, (2) is not sufficient to determine the area of the triangle.

However, with both (1) and (2), c is a right angle and the area of the triangle is (1/2)(base)(height) = (1/2)(4)(3). The answer is C.
ตัวอย่างข้อสอบ gmat
ตัวอย่างข้อสอบ gmat
ตัวอย่างข้อสอบ gmat

ตัวอย่างข้อสอบ gmat GMAT EXAM - GMAT TEST Data Sufficiency

ตัวอย่างข้อสอบ gmat

EXAM - GMAT TEST Data Sufficiency

Most people have much more difficulty with the Data Sufficiency problems than with the Standard Math problems. However, the mathematical knowledge and skill required to solve Data Sufficiency problems is no greater than that required to solve standard math problems. What makes Data Sufficiency problems appear harder at first is the complicated directions. But once you become familiar with the directions, you'll find these problems no harder than standard math problems. In fact, people usually become proficient more quickly on Data Sufficiency problems.

THE DIRECTIONS

The directions for Data Sufficiency questions are rather complicated. Before reading any further, take some time to learn the directions cold. Some of the wording in the directions below has been changed from the GMAT to make it clearer. You should never have to look at the instructions during the test.

Directions: Each of the following Data Sufficiency problems contains a question followed by two statements, numbered (1) and (2). You need not solve the problem; rather you must decide whether the information given is sufficient to solve the problem.

The correct answer to a question is

A if statement (1) ALONE is sufficient to answer the question but statement (2) alone is not sufficient;
B if statement (2) ALONE is sufficient to answer the question but statement (1) alone is not sufficient;
C if the two statements TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient;
D if EACH statement ALONE is sufficient to answer the question;
E if the two statements TAKEN TOGETHER are still NOT sufficient to answer the question.

Numbers: Only real numbers are used. That is, there are no complex numbers.

Drawings: The drawings are drawn to scale according to the information given in the question, but may conflict with the information given in statements (1) and (2).

You can assume that a line that appears straight is straight and that angle measures cannot be zero.

You can assume that the relative positions of points, angles, and objects are as shown.

All drawings lie in a plane unless stated otherwise.

Example:

In triangle ABC to the right, what is the value of y?

(1) AB = AC
(2) x = 30

Explanation: By statement (1), triangle ABC is isosceles. Hence, its base angles are equal: y = z. Since the angle sum of a triangle is 180 degrees, we get x + y + z = 180. Replacing z with y in this equation and then simplifying yields x + 2y = 180. Since statement (1) does not give a value for x, we cannot determine the value of y from statement (1) alone. By statement (2), x = 30. Hence, x + y + z = 180 becomes 30 + y + z = 180, or y + z = 150. Since statement (2) does not give a value for z, we cannot determine the value of y from statement (2) alone. However, using both statements in combination, we can find both x and z and therefore y. Hence, the answer is C.

Notice in the above example that the triangle appears to be a right triangle. However, that cannot be assumed: angle A may be 89 degrees or 91 degrees, we can't tell from the drawing. You must be very careful not to assume any more than what is explicitly given in a Data Sufficiency problem.

ELIMINATION

Data Sufficiency questions provide fertile ground for elimination. In fact, it is rare that you won't be able to eliminate some answer-choices. Remember, if you can eliminate at least one answer choice, the odds of gaining points by guessing are in your favor.

The following table summarizes how elimination functions with Data Sufficiency problems.

StatementChoices Eliminated
(1) is sufficientB, C, E
(1) is not sufficientA, D
(2) is sufficientA, C, E
(2) is not sufficientB, D
(1) is not sufficient and (2) is not sufficientA, B, D

Example 1: What is the 1st term in sequence S?

(1) The 3rd term of S is 4.
(2) The 2nd term of S is three times the 1st, and the 3rd term is four times the 2nd.

(1) is no help in finding the first term of S. For example, the following sequences each have 4 as their third term, yet they have different first terms:

0, 2, 4
-4, 0, 4

This eliminates choices A and D. Now, even if we are unable to solve this problem, we have significantly increased our chances of guessing correctly--from 1 in 5 to 1 in 3.

Turning to (2), we completely ignore the information in (1). Although (2) contains a lot of information, it also is not sufficient. For example, the following sequences each satisfy (2), yet they have different first terms:

1, 3, 12
3, 9, 36

This eliminates B, and our chances of guessing correctly have increased to 1 in 2.

Next, we consider (1) and (2) together. From (1), we know "the 3rd term of S is 4." From (2), we know "the 3rd term is four times the 2nd." This is equivalent to saying the 2nd term is 1/4 the 3rd term: (1/4)4 = 1. Further, from (2), we know "the 2nd term is three times the 1st." This is equivalent to saying the 1st term is 1/3 the 2nd term: (1/3)1 = 1/3. Hence, the first term of the sequence is fully determined: 1/3, 1, 4. The answer is C.

Example 2: In the figure to the right, what is the area of the triangle?

(1)
(2) x = 90

Recall that a triangle is a right triangle if and only if the square of the longest side is equal to the sum of the squares of the shorter sides (Pythagorean Theorem). Hence, (1) implies that the triangle is a right triangle. So the area of the triangle is (6)(8)/2. Note, there is no need to calculate the area--we just need to know that the area can be calculated. Hence, the answer is either A or D.

Turning to (2), we see immediately that we have a right triangle. Hence, again the area can be calculated. The answer is D.

Example 3: Is p <>

(1) p/3 < q/3
(2) -p + x > -q + x

Multiplying both sides of p/3 < q/3 by 3 yields p < q.

Hence, (1) is sufficient. As to (2), subtract x from both sides of -p + x > -q + x, which yields -p > -q.

Multiplying both sides of this inequality by -1, and recalling that multiplying both sides of an inequality by a negative number reverses the inequality, yields p < q.

Hence, (2) is also sufficient. The answer is D.

Example 4: If x is both the cube of an integer and between 2 and 200, what is the value of x?

(1) x is odd.
(2) x is the square of an integer.

Since x is both a cube and between 2 and 200, we are looking at the integers:


which reduce to

8, 27, 64, 125

Since there are two odd integers in this set, (1) is not sufficient to uniquely determine the value of x. This eliminates choices A and D.

Next, there is only one perfect square, 64, in the set. Hence, (2) is sufficient to determine the value of x. The answer is B.

Example 5: Is CAB a code word in language Q?

(1) ABC is the base word.
(2) If C immediately follows B, then C can be moved to the front of the code word to generate another word.

From (1), we cannot determine whether CAB is a code word since (1) gives no rule for generating another word from the base word. This eliminates A and D.

Turning to (2), we still cannot determine whether CAB is a code word since now we have no word to apply this rule to. This eliminates B.

However, if we consider (1) and (2) together, then we can determine whether CAB is a code word:

From (1), ABC is a code word.

From (2), the C in the code word ABC can be moved to the front of the word: CAB.

Hence, CAB is a code word and the answer is C.

UNWARRANTED ASSUMPTIONS

Be extra careful not to read any more into a statement than what is given.

• The main purpose of some difficult problems is to lure you into making an unwarranted assumption.

If you avoid the temptation, these problems can become routine.

Example 6: Did Incumbent I get over 50% of the vote?

(1) Challenger C got 49% of the vote.
(2) Incumbent I got 25,000 of the 100,000 votes cast.

If you did not make any unwarranted assumptions, you probably did not find this to be a hard problem. What makes a problem difficult is not necessarily its underlying complexity; rather a problem is classified as difficult if many people miss it. A problem may be simple yet contain a psychological trap that causes people to answer it incorrectly.

The above problem is difficult because many people subconsciously assume that there are only two candidates. They then figure that since the challenger received 49% of the vote the incumbent received 51% of the vote. This would be a valid deduction if C were the only challenger (You might ask, "What if some people voted for none-of-the-above?" But don't get carried away with finding exceptions. The writers of the GMAT would not set a trap that subtle). But we cannot assume that. There may be two or more challengers. Hence, (1) is insufficient.

Now, consider (2) alone. Since Incumbent I received 25,000 of the 100,000 votes cast, I necessarily received 25% of the vote. Hence, the answer to the question is "No, the incumbent did not receive over 50% of the vote." Therefore, (2) is sufficient to answer the question. The answer is B.

Note, some people have trouble with (2) because they feel that the question asks for a "yes" answer. But on Data Sufficiency questions, a "no" answer is just as valid as a "yes" answer. What we're looking for is a definite answer.

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ตัวอย่างข้อสอบ gmat GMAT EXAM VERBAL and GMAT READING TEST SECTION

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GMAT EXAM VERBAL and GMAT READING TEST SECTION

The verbal portion of the test consists of three types of questions: Reading Comprehension, Arguments, and Sentence Correction. They are designed to test your ability to reason using the written word. There is roughly the same number of each type of question, for a total of 41 questions.

READING COMPREHENSION

GMAT READING and VERBAL FORMAT

The GMAT reading comprehension section passages are about 200 to 400 words long. The subject matter of a passage can be almost anything, but the most common themes are politics, history, culture, science, and business.

GMAT READING METHODS

Some books recommend speed-reading the passages. This is a mistake. Speed reading is designed for ordinary, nontechnical material. Because this material is filled with "fluff," you can skim over the nonessential parts and still get the gist--and often more--of the passage. However, GMAT passages are dense. Some are actual quoted articles. Most often, however, they are based on articles that have been condensed to about one-third their original length. During this process no essential information is lost, just the "fluff" is cut. This is why speed reading will not work here--the passages contain too much information. You should, however, read somewhat faster than you normally do, but not to the point that your comprehension suffers. You will have to experiment to find your optimum pace.

Many books recommend reading the questions before the passage. But there are two big problems with this method. First, some of the questions are a paragraph long, and reading a question twice can use up precious time. Second, there are up to seven questions per passage, and psychologists have shown that we can hold in our minds a maximum of about three thoughts at any one time (some of us have trouble simply remembering phone numbers). After reading all seven questions, the student will turn to the passage with his mind clouded by half-remembered thoughts. This will at best waste his time and distract him. More likely it will turn the passage into a disjointed mass of information.

However, one technique that you may find helpful is to preview the passage by reading the first sentence of each paragraph. Generally, the topic of a paragraph is contained in the first sentence. Reading the first sentence of each paragraph will give an overview of the passage. The topic sentences act in essence as a summary of the passage. Furthermore, since each passage is only three or four paragraphs long, previewing the topic sentences will not use up an inordinate amount of time.

The passages presented depend on how well you are performing on the test. However, unlike other parts of the test, the questions presented do not depend on your performance. The longer passages will require you to scroll through the passage.

THE SIX QUESTIONS

The key to performing well on the passages is not the particular reading technique you use (so long as it's neither speed reading nor pre-reading the questions). Rather the key is to become completely familiar with the question types--there are only six--so that you can anticipate the questions that might be asked as you read the passage and answer those that are asked more quickly and efficiently. As you become familiar with the six question types, you will gain an intuitive sense for the places from which questions are likely to be drawn. This will give you the same advantage as that claimed by the "pre-reading-the-questions" technique, without the confusion and waste of time. Note, the order in which the questions are asked roughly corresponds to the order in which the main issues are presented in the passage. Early questions should correspond to information given early in the passage, and so on.

The following passage and accompanying questions illustrate the six question types.

There are two major systems of criminal procedure in the modern world--the adversarial and the inquisitorial. The former is associated with common law tradition and the latter with civil law tradition. Both systems were historically preceded by the system of private vengeance in which the victim of a crime fashioned his own remedy and administered it privately, either personally or through an agent. The vengeance system was a system of self-help, the essence of which was captured in the slogan "an eye for an eye, a tooth for a tooth." The modern adversarial system is only one historical step removed from the private vengeance system and still retains some of its characteristic features. Thus, for example, even though the right to institute criminal action has now been extended to all members of society and even though the police department has taken over the pretrial investigative functions on behalf of the prosecution, the adversarial system still leaves the defendant to conduct his own pretrial investigation. The trial is still viewed as a duel between two adversaries, refereed by a judge who, at the beginning of the trial has no knowledge of the investigative background of the case. In the final analysis the adversarial system of criminal procedure symbolizes and regularizes the punitive combat.

By contrast, the inquisitorial system begins historically where the adversarial system stopped its development. It is two historical steps removed from the system of private vengeance. Therefore, from the standpoint of legal anthropology, it is historically superior to the adversarial system. Under the inquisitorial system the public investigator has the duty to investigate not just on behalf of the prosecutor but also on behalf of the defendant. Additionally, the public prosecutor has the duty to present to the court not only evidence that may lead to the conviction of the defendant but also evidence that may lead to his exoneration. This system mandates that both parties permit full pretrial discovery of the evidence in their possession. Finally, in an effort to make the trial less like a duel between two adversaries, the inquisitorial system mandates that the judge take an active part in the conduct of the trial, with a role that is both directive and protective.

Fact-finding is at the heart of the inquisitorial system. This system operates on the philosophical premise that in a criminal case the crucial factor is not the legal rule but the facts of the case and that the goal of the entire procedure is to experimentally recreate for the court the commission of the alleged crime.

MAIN IDEA QUESTIONS

The main idea is usually stated in the last--occasionally the first--sentence of the first paragraph. If it's not there, it will probably be the last sentence of the entire passage.

Because main idea questions are relatively easy, the GMAT writers try to obscure the correct answer by surrounding it with close answer-choices ("detractors") that either overstate or understate the author's main point. Answer-choices that stress specifics tend to understate the main idea; choices that go beyond the scope of the passage tend to overstate the main idea.

The answer to a main idea question will summarize the author's argument, yet be neither too specific nor too broad.

Example: (Refer to the first passage.)

The primary purpose of the passage is to

(A) explain why the inquisitorial system is the best system of criminal justice
(B) explain how the adversarial and the inquisitorial systems of criminal justice both evolved from the system of private vengeance
(C) show how the adversarial and inquisitorial systems of criminal justice can both complement and hinder each other's development
(D) show how the adversarial and inquisitorial systems of criminal justice are being combined into a new and better system
(E) analyze two systems of criminal justice and deduce which one is better

The answer to a main idea question will summarize the passage without going beyond it. (A) violates these criteria by overstating the scope of the passage. The comparison in the passage is between two specific systems, not between all systems. (A) would be a good answer if "best" were replaced with "better." Beware of extreme words. (B) violates the criteria by understating the scope of the passage. Although the evolution of both the adversarial and the inquisitorial systems is discussed in the passage, it is done to show why one is superior to the other. As to (C) and (D), both can be quickly dismissed since neither is mentioned in the passage. Finally, the passage does two things: it presents two systems of criminal justice and shows why one is better than the other. (E) aptly summarizes this, so it is the best answer.

Description Questions

Description questions, as with main idea questions, refer to a point made by the author. However, description questions refer to a minor point or to incidental information, not to the author's main point.

The answer to a description question must refer directly to a statement in the passage, not to something implied by it. However, the correct answer will paraphrase a statement in the passage, not give an exact quote. In fact, exact quotes ("Same language" traps) are often used to bait wrong answers.

Caution: When answering a description question, you must find the point in the passage from which the question is drawn. Don't rely on memory--too many obfuscating tactics are used with these questions.

Not only must the correct answer refer directly to a statement in the passage, it must refer to the relevant statement. The correct answer will be surrounded by wrong choices which refer directly to the passage but don't address the question. These choices can be tempting because they tend to be quite close to the actual answer.

Once you spot the sentence to which the question refers, you still must read a few sentences before and after it, to put the question in context. If a question refers to line 20, the information needed to answer it can occur anywhere from line 15 to 25. Even if you have spotted the answer in line 20, you should still read a couple more lines to make certain you have the proper perspective.

Example: (Refer to the first passage.)

According to the passage, the inquisitorial system differs from the adversarial system in that

(A) it does not make the defendant solely responsible for gathering evidence for his case
(B) it does not require the police department to work on behalf of the prosecution
(C) it does not allow the victim the satisfaction of private vengeance
(D) it requires the prosecution to drop a weak case
(E) a defendant who is innocent would prefer to be tried under the inquisitorial system

This is a description question, so the information needed to answer it must be stated in the passage--though not in the same language as in the answer. The needed information is contained in the fourth sentence of Paragraph 3, which states that the public prosecutor has to investigate on behalf of both society and the defendant. Thus, the defendant is not solely responsible for investigating his case. Furthermore, the paragraph's opening implies that this feature is not found in the adversarial system. This illustrates why you must determine the context of the situation before you can safely answer the question. The answer is (A).

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ตัวอย่างข้อสอบ GMAT EXAM - GMAT TEST GRAMMAR SECTION

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GMAT EXAM - GMAT TEST GRAMMAR SECTION

FORMAT OF THE GMAT GRAMMAR TEST SECTION

The field of grammar is huge and complex--tomes have been written on the subject. This complexity should be no surprise since grammar deals with the process of communication.

GMAT grammar tests only a small part of standard written English. Grammar can be divided into two parts: Mechanics and Usage.

Mechanics concerns punctuation, capitalization, etc. It is not tested on the GMAT nearly as often as is usage. So don't spend too much time worrying whether the comma is in the right place or whether a particular word should be capitalized.

Usage concerns how we choose our words and how we express our thoughts: in other words, are the connections between the words in a sentence logically sound, and are they expressed in a way that conforms to standard idiom? This is the part of grammar that the GMAT concentrates on. Six major categories of usage are tested:

Pronoun Errors
Subject-Verb Agreement
Misplaced Modifiers
Faulty Parallelism
Faulty Verb Tense
Faulty Idiom

PRONOUN ERRORS

A pronoun is a word that stands for a noun, known as the antecedent of the pronoun. The key point for the use of pronouns is this: pronouns must agree with their antecedents in both number (singular or plural) and person (first, second, or third).

Example:
Steve has yet to receive his degree.

Here, the pronoun his refers to the noun Steve.

Following is a list of the most common pronouns:
PRONOUNS
Singular Plural Both Singular and Plural
I, me we, us any
she, her they none
he, him them all
it these most
anyone those more
either some who
each that which
many a both what
nothing ourselves you
one any
another many
everything few
mine several
his, hers others
this
that
Reference

• A pronoun should be plural when it refers to two nouns joined by and.

Example:
Jane and Katarina believe they passed the final exam.

The plural pronoun they refers to the compound subject Jane and Katarina.

• A pronoun should be singular when it refers to two nouns joined by or or nor.
Faulty Usage

Neither Jane nor Katarina believes they passed the final.

Correct: Neither Jane nor Katarina believes she passed the final.

• A pronoun should refer to one and only one noun or compound noun.

This is probably the most common error on the GMAT. If a pronoun follows two nouns, it is often unclear which of the nouns the pronoun refers to.
Faulty Usage

The breakup of the Soviet Union has left nuclear weapons in the hands of unstable, nascent countries. It is imperative to world security that they be destroyed.

Although one is unlikely to take the sentence to mean that the countries must be destroyed, that interpretation is possible from the structure of the sentence. It is easily corrected:

The breakup of the Soviet Union has left nuclear weapons in the hands of unstable, nascent countries. It is imperative to world security that these weapons be destroyed.
Faulty Usage

In Somalia, they have become jaded by the constant warfare.

This construction is faulty because they does not have an antecedent. The sentence can be corrected by replacing they with people:

In Somalia, people have become jaded by the constant warfare.

Better: The people of Somalia have become jaded by the constant warfare.

• In addition to agreeing with its antecedent in number, a pronoun must agree with its antecedent in person.
Faulty Usage

One enters this world with no responsibilities. Then comes school, then work, then marriage and family. No wonder, you l ook longingly to retirement.

In this sentence, the subject has changed from one (third person) to you (second person). To correct the sentence either replace one with you or vice versa:

You enter this world with no responsibilities. Then comes school, then work, then marriage and family. No wonder, you look longingly to retirement.

One enters this world with no responsibilities. Then comes school, then work, then marriage and family. No wonder, one looks longingly to retirement.

Example:

In the following sentence, part or all of the sentence is underlined. The answer-choices offer five ways of phrasing the underlined part. If you think the sentence as written is better than the alternatives, choose A, which merely repeats the underlined part; otherwise choose one of the alternatives.

Had the President's Administration not lost the vote on the budget reduction package, his first year in office would have been rated an A.

(A) Had the President's Administration not lost the vote on the budget reduction package, his first year in office would have been rated an A.
(B) If the Administration had not lost the vote on the budget reduction package, his first year in office would have been rated an A.
(C) Had the President's Administration not lost the vote on the budget reduction package, it would have been rated an A.
(D) Had the President's Administration not lost the vote on its budget reduction package, his first year in office would have been rated an A.
(E) If the President had not lost the vote on the budget reduction package, the Administration's first year in office would have been rated an A.

Choice (A) is incorrect because his appears to refer to the President, but the subject of the subordinate clause is the President's Administration, not the President.

Choice (B) changes the structure of the sentence, but retains the same flawed reference.

In choice (C), it can refer to either the President's Administration or the budget reduction package. Thus, the reference is ambiguous.

Choice (D) adds another pronoun, its, but still retains the same flawed reference.

Choice (E) corrects the flawed reference by removing all pronouns. The answer is (E).

SUBJECT-VERB AGREEMENT

Within a sentence there are certain requirements for the relationship between the subject and the verb.

• The subject and verb must agree both in number and person.

Example:
We have surpassed our sales goal of one million dollars.

Here, the first person plural verb have agrees with its first person plural subject we.

Note, ironically, third person singular verbs often end in s or es:

He seems to be fair.

• Intervening phrases and clauses have no effect on subject-verb agreement.

Example:
Only one of the President's nominees was confirmed.

Here, the singular verb was agrees with its singular subject one. The intervening prepositional phrase of the President's nominees has no effect on the number or person of the verb.

• When the subject and verb are reversed, they still must agree in both number and person.

Example:
Attached are copies of the contract.

Here, the plural verb are attached agrees with its plural subject copies. The sentence could be rewritten as

Copies of the contract are attached.

Example:
The rise in negative attitudes toward foreigners indicate that the country is becoming less tolerant, and therefore that the opportunities are ripe for extremist groups to exploit the illegal immigration problem.

(A) indicate that the country is becoming less tolerant, and therefore that
(B) indicates that the country is becoming less tolerant, and therefore
(C) indicates that the country is becoming less tolerant, and therefore that
(D) indicates that the country is being less tolerant, and therefore (E) indicates that the country is becoming less tolerant of and therefore that

Choice (A) has two flaws. First, the subject of the sentence the rise is singular, and therefore the verb indicate should not be plural. Second, the comma indicates that the sentence is made up of two independent clauses, but the relative pronoun that immediately following therefore forms a subordinate clause.

Choice (C) corrects the number of the verb, but retains the subordinating relative pronoun that.

Choice (D) corrects the number of the verb and eliminates the subordinating relative pronoun that. However, the verb being is less descriptive than the verb becoming: As negative attitudes toward foreigners increase, the country becomes correspondingly less tolerant. Being does not capture this notion of change.

Choice (E) corrects the verb's number, and by dropping the comma makes the subordination allowable. However, it introduces the preposition of which does not have an object: less tolerant of what?

Choice (B) both corrects the verb's number and removes the subordinating relative pronoun that. The answer is (B).

MISPLACED MODIFIERS

• As a general rule, a modifier should be placed as close as possible to what it modifies.

Example:
Following are some useful tips for protecting your person and property from the FBI.

As written, the sentence implies that the FBI is a threat to your person and property. To correct the sentence put the modifier from the FBI next to the word it modifies, tips:

Following are some useful tips from the FBI for protecting your person and property.

• When a phrase begins a sentence, make sure that it modifies the subject of the sentence.

Example:
Coming around the corner, a few moments passed before I could recognize my old home.

As worded, the sentence implies that the moments were coming around the corner. The sentence can be corrected as follows:

As I came around the corner, a few moments passed before I could recognize my old home.

or

Coming around the corner, I paused a few moments before I could recognize my old home.

Example:
By focusing on poverty, the other causes of crime--such as the breakup of the nuclear family, changing morals, the loss of community, etc.--have been overlooked by sociologists.

(A) the other causes of crime--such as the breakup of the nuclear family, changing morals, the loss of community, etc.--have been overlooked by sociologists.
(B) the other causes of crime have been overlooked by sociologists--such as the breakup of the nuclear family, changing morals, the loss of community, etc.
(C) there are other causes of crime that have been overlooked by sociologists--such as the breakup of the nuclear family, changing morals, the loss of community, etc.
(D) crimes--such as the breakup of the nuclear family, changing morals, the loss of community, etc.--have been overlooked by sociologists.
(E) sociologists have overlooked the other causes of crime--such as the breakup of the nuclear family, changing morals, the loss of community, etc.

Choice (A) is incorrect since it implies that the other causes of crime are doing the focusing.

Choice (B) has the same flaw.

Choice (C) is incorrect. The phrase by focusing on poverty must modify the subject of the sentence, but there cannot be the subject since the construction there are is used to introduce a subject.

Choice (D) implies that crimes are focusing on poverty.

Choice (E) puts the subject of the sentence sociologists immediately next to its modifying phrase by focusing on poverty. The answer is (E).

FAULTY PARALLELISM

• For a sentence to be parallel, similar elements must be expressed in similar form.

• When two adjectives modify the same noun, they should have similar forms.

Example:
The topology course was both rigorous and a challenge.

Since both rigorous and a challenge are modifying course, they should have the same form:

The topology course was both rigorous and challenging.

• When a series of clauses is listed, the verbs in each clause must have the same form.

Example:
During his trip to Europe, the President will discuss ways to stimulate trade, offer economic aid, and trying to forge a new coalition with moderate forces in Russia.

In this example, the first two verbs, discuss and offer, are active. But the third verb in the series, trying, is passive. The form of the verb should be active:

During his trip to Europe, the President will discuss ways to stimulate trade, offer economic aid, and try to forge a new coalition with moderate forces in Russia.

• When the first half of a sentence has a certain structure, the second half should preserve that structure.

Example:
To acknowledge that one is an alcoholic is taking the first and hardest step to recovery.

The first half of the above sentence has an infinitive structure, to acknowledge, so the second half must have a similar structure:

To acknowledge that one is an alcoholic is to take the first and hardest step to recovery.

Example:
This century began with war brewing in Europe, the industrial revolution well-established, and a nascent communication age.

(A) war brewing in Europe, the industrial revolution well-established, and a nascent communication age.
(B) war brewing in Europe, the industrial revolution surging, and a nascent communication age.
(C) war in Europe, the industrial revolution well-established, and a nascent communication age.
(D) war brewing in Europe, the industrial revolution well-established, and the communication age beginning.
(E) war brewing in Europe, the industrial revolution well-established, and saw the birth of the communication age.

Choice (A) is incorrect. Although the first two phrases, war brewing in Europe and the industrial revolution well-established, have different structures, the thoughts are parallel. However, the third phrase, and a nascent communication age, is not parallel to the first two.

Choice (B) does not make the third phrase parallel to the first two.

Choice (C) changes the meaning of the sentence: the new formulation states that war already existed in Europe while the original sentence states that war was only developing.

Choice (E) is not parallel since the first two phrases in the series are noun phrases, but saw the birth of the communication age is a verb phrase. When a word introduces a series, each element of the series must agree with the introductory word. You can test the correctness of a phrase in a series by dropping the other phrases and checking whether the remaining phrase agrees with the introductory word. In this series, each phrase must be the object of the preposition with:

This century began with war brewing in Europe

This century began with the industrial revolution well-established

This century began with saw the birth of the communication age

In this form, it is clear the verb saw cannot be the object of the preposition with.

Choice (D) offers three phrases in parallel form. The answer is (D).

FAULTY VERB TENSE

A verb has four principal parts:
1. Present Tense

a. Used to express present tense.

He studies hard.

b. Used to express general truths.

During a recession, people are cautious about taking on more debt.

c. Used with will or shall to express future time.

He will take the GMAT next year.
2. Past Tense

a. Used to express past tense.

He took the GMAT last year.
3. Past Participle

a. Used to form the present perfect tense, which indicates that an action was started in the past and its effects are continuing in the present. It is formed using have or has and the past participle of the verb.

He has prepared thoroughly for the GMAT.

b. Used to form the past perfect tense, which indicates that an action was completed before another past action. It is formed using had and the past participle of the verb.

He had prepared thoroughly before taking the GMAT.

c. Used to form the future perfect tense, which indicates that an action will be completed before another future action. It is formed using will have or shall have and the past participle of the verb.

He will have prepared thoroughly before taking the GMAT.
4. Present Participle (-ing form of the verb)

a. Used to form the present progressive tense, which indicates that an action is ongoing. It is formed using is, am, or are and the present participle of the verb.

He is preparing thoroughly for the GMAT.

b. Used to form the past progressive tense, which indicates that an action was in progress in the past. It is formed using was or were and the present participle of the verb.

He was preparing for the GMAT.

c. Used to form the future progressive tense, which indicates that an action will be in progress in the future. It is formed using will be or shall be and the present participle of the verb.
He will be preparing thoroughly for the GMAT.


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PASSIVE VOICE

The passive voice removes the subject from the sentence. It is formed with the verb to be and the past participle of the main verb.

Passive: The bill was resubmitted.

Active: The Senator has resubmitted the bill.

Unless you want to de-emphasize the doer of an action, you should favor the active voice.

Example:
In the past few years and to this day, many teachers of math and science had chosen to return to the private sector.

(A) had chosen to return to the private sector.
(B) having chosen to return to the private sector.
(C) chose to return to the private sector.
(D) have chosen to return to the private sector.
(E) have chosen returning to the private sector.

Choice (A) is incorrect because it uses the past perfect had chosen, which describes an event that has been completed before another event. But the sentence implies that teachers have and are continuing to return to the private sector. Hence, the present perfect tense should be used.

Choice (B) is incorrect because it uses the present progressive tense having chosen, which describes an ongoing event. Although this is the case, it does not capture the fact that the event began in the past.

Choice (C) is incorrect because it uses the simple past chose, which describes a past event. But again, the sentence implies that the teachers are continuing to opt for the private sector.

Choice (D) is the correct answer because it uses the present perfect have chosen to describe an event that occurred in the past and is continuing into the present.

Choice (E) is incorrect because it leaves the thought in the sentence uncompleted.

IDIOM & USAGE

Accept/Except:

Accept means "to agree to" or "to receive." Except means "to object to" or "to leave out."

We will accept (receive) your manuscript for review.

No parking is allowed, except (leave out) on holidays.

Account for:

When explaining something, the correct idiom is account for:

We had to account for all the missing money.

When receiving blame or credit, the correct idiom is account to:

You will have to account to the state for your crimes.

Adapted to/for/from

Adapted to means "naturally suited for." Adapted for means "created to be suited for." Adapted from means "changed to be suited for."

The polar bear is adapted to the subzero temperatures.

For any "New Order" to be successful, it must be adapted for the continually changing world power structure.

Lucas' latest release is adapted from the 1950 B-movie "Attack of the Amazons."

Affect/Effect:

Effect is a noun meaning "a result."

Increased fighting will be the effect of the failed peace conference.

Affect is a verb meaning "to influence."

The rain affected their plans for a picnic.

All ready vs. Already

All ready means "everything is ready."

Already means "earlier."

Alot vs. A lot

Alot is nonstandard; a lot is the correct form.

Among/Between:

Between should be used when referring to two things, and among should be used when referring to more than two things.

The young lady must choose between two suitors.

The fault is spread evenly among the three defendants.

Being that vs. Since:

Being that is nonstandard and should be replaced by since.

(Faulty) Being that darkness was fast approaching, we had to abandon the search.

(Better) Since darkness was fast approaching, we had to abandon the search.

Beside/Besides:

Adding an s to beside completely changes its meaning: Beside means "next to." Besides means "in addition."

We sat beside (next to) the host.

Besides (in addition), money was not even an issue in the contract negotiations.

Center on vs. Center around

Center around is colloquial. It should not be used in formal writing.

(Faulty) The dispute centers around the effects of undocumented workers.

(Correct) The dispute centers on the effects of undocumented workers.
Conform to (not with):

Stewart's writing does not conform to standard literary conventions.
Consensus of opinion

Consensus of opinion is redundant: consensus means "general agreement."
Correspond to/with:

Correspond to means "in agreement with":

The penalty does not correspond to the severity of the crime.

Correspond with means "to exchange letters":

He corresponded with many of the top European leaders of his time.
Different from/Different than:

The preferred form is different from. Only in rare cases is different than acceptable.

The new Cadillacs are very different from the imported luxury cars.
Double negatives:

(Faulty) Scarcely nothing was learned during the seminar.

(Better) Scarcely anything was learned during the seminar.
Doubt that vs. Doubt whether

Doubt whether is nonstandard.

(Faulty) I doubt whether his new business will succeed.

(Correct) I doubt that his new business will succeed.
Farther/Further:

Use farther when referring to distance, and use further when referring to degree.

They went no further (degree) than necking.

He threw the discs farther (distance) than the top seated competitor.
Fewer/Less:

Use fewer when referring to a number of items. Use less when referring to a continuous quantity.

In the past, we had fewer options.

The impact was less than what was expected.
Identical with (not to):

This bid is identical with the one submitted by you.
In contrast to (not of):

In contrast to the conservative attitudes of her time, Mae West was quite provocative.
Independent of (not from):

The judiciary is independent of the other branches of government.
Not only . . . but also:

In this construction, but cannot be replaced with and.

(Faulty) Peterson is not only the top salesman in the department and also the most proficient.

(Correct) Peterson is not only the top salesman in the department but also the most proficient.
On account of vs. Because:

Because is always better than the circumlocution on account of.

(Poor) On account of his poor behavior, he was expelled.

(Better) Because he behaved poorly, he was expelled.
One another/Each other:

Each other should be used when referring to two things, and one another should be used when referring to more than two things.

The members of the basketball team (more than two) congratulated one another on their victory.

The business partners (two) congratulated each other on their successful first year.
Plus vs. And:

Do not use plus as a conjunction meaning and.

(Faulty) His contributions to this community are considerable, plus his character is beyond reproach.

(Correct) His contributions to this community are considerable, and his character is beyond reproach.

Note: Plus can be used to mean and so long as it is not being used as a conjunction.

(Acceptable) His generous financial contribution plus his donated time has made this project a success.

In this sentence, plus is being used as a preposition. Note, the verb has is singular because an intervening prepositional phrase (plus his donated time) does not affect subject verb agreement.
Regard vs. Regards:

Unless you are giving best wishes to someone, you should use regard.

(Faulty) In regards to your letter, we would be interested in distributing your product.

(Correct) In regard to your letter, we would be interested in distributing your product.
Regardless vs. Irregardless

Regardless means "not withstanding." Hence, the "ir" in irregardless is redundant. Regardless is the correct form.
Retroactive to (not from):

The correct idiom is retroactive to:

The tax increase is retroactive to February.
Speak to/with:

To speak to someone is to tell them something:

We spoke to Jennings about the alleged embezzlement.

To speak with someone is to discuss something with them:

Steve spoke with his friend Dave for hours yesterday.
The reason is because:

This structure is redundant. Equally common and doubly redundant is the structure the reason why is because.

(Poor) The reason why I could not attend the party is because I had to work.

(Better) I could not attend the party because I had to work.
Whether vs. As to whether

The circumlocution as to whether should be replaced by whether.

(Poor) The United Nations has not decided as to whether to authorize a trade embargo.

(Better) The United Nations has not decided whether to authorize a trade embargo.
Whether vs. If

Whether introduces a choice; if introduces a condition. A common mistake is to use if to present a choice.

(Faulty) He inquired if we had decided to keep the gift.

(Correct) He inquired whether we had decided to keep the gift.

Example:
The studio's retrospective art exhibit refers back to a simpler time in American history.

(A) The studio's retrospective art exhibit refers back to
(B) The studio's retrospective art exhibit harkens back to
(C) The studio's retrospective art exhibit refers to
(D) The studio's retrospective art exhibit refers from
(E) The studio's retrospective art exhibit looks back to

Choice (A) is incorrect. Retrospective means looking back on the past. Hence, in the phrase refers back, the word back is redundant.

Choice (B) is incorrect because harkens back is also redundant.

Choice (C) is correct. Dropping the word back eliminates the redundancy.

Choice (D) is incorrect because the preposition from is non-idiomatic.

Choice (E) is incorrect because looks back is also redundant.

Note: One could argue that the phrase American history also makes the sentence redundant. However, it is not underlined in the sentence. It is not at all uncommon to find questionable structures in parts of the sentence that are not underlined. In fact, you may even find questionable structures in the underlined part of the sentence that are not corrected by any of the answer choices because the writers are testing a different mistake. Concern yourself with correcting only the underlined part of the sentence.

ตัวอย่างข้อสอบ GMAT

ตัวอย่างข้อสอบ gmat

ตัวอย่างข้อสอบ gmat
INEQUALITIES

Inequalities are manipulated algebraically the same way as equations with one exception:

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality. That is, if x > y and c <>

Example: For which values of x is 4x + 3 > 6x - 8?

As with equations, our goal is to isolate x on one side:

Subtracting 6x from both sides yields -2x + 3 > -8

Subtracting 3 from both sides yields -2x > -11

Dividing both sides by -2 and reversing the inequality yields x < 11/2

Positive & Negative Numbers

A number greater than 0 is positive. On the number line, positive numbers are to the right of 0. A number less than 0 is negative. On the number line, negative numbers are to the left of 0. Zero is the only number that is neither positive nor negative; it divides the two sets of numbers. On the number line, numbers increase to the right and decrease to the left.

The expression x > y means that x is greater than y. In other words, x is to the right of y on the number line.

We usually have no trouble determining which of two numbers is larger when both are positive or one is positive and the other negative (e.g., 5 > 2 and 3.1 > -2). However, we sometimes hesitate when both numbers are negative (e.g., -2 > -4.5). When in doubt, think of the number line: if one number is to the right of the number, then it is larger.

Miscellaneous Properties of Positive and Negative Numbers

1. The product (quotient) of positive numbers is positive.
2. The product (quotient) of a positive number and a negative number is negative.
3. The product (quotient) of an even number of negative numbers is positive.
4. The product (quotient) of an odd number of negative numbers is negative.
5. The sum of negative numbers is negative.
6. A number raised to an even exponent is greater than or equal to zero.

Absolute Value

The absolute value of a number is its distance on the number line from 0. Since distance is a positive number, absolute value of a number is positive. Two vertical bars denote the absolute value of a number: | x |. For example, | 3 | = 3 and | -3 | = 3.

Students rarely struggle with the absolute value of numbers: if the number is negative, simply make it positive; and if it is already positive, leave it as is. For example, since -2.4 is negative, | -2.4 | = 2.4 and since 5.01 is positive | 5.01 | = 5.01.

Further, students rarely struggle with the absolute value of positive variables: if the variable is positive, simply drop the absolute value symbol. For example, if x > 0, then | x | = x.

However, negative variables can cause students much consternation. If x is negative, then | x | = -x. This often confuses students because the absolute value is positive but the -x appears to be negative. It is actually positive--it is the negative of a negative number, which is positive. To see this more clearly let x = -k, where k is a positive number. Then x is a negative number. So | x | = -x = -(-k) = k. Since k is positive so is -x. Another way to view this is | x | = -x = (-1)x = (-1)(a negative number) = a positive number.

Transitive Property

If x <>

Example: If 1/Q > 1, is 1 > QQ ?

Since 1/Q > 1 and 1 > 0, we know from the transitive property that 1/Q is positive. Hence, Q is positive. Therefore, we can multiply both sides of 1/Q > 1 by Q without reversing the inequality:

Q(1/Q) > 1(Q)
Reducing yields 1 > Q
Multiplying both sides again by Q yields Q > QQ
Using the transitive property to combine the last two inequalities yields 1 > QQ

FRACTIONS

I. To compare two fractions, cross-multiply. The larger number will be on the same side as the larger fraction.

Example: Example: Which fraction is greater 9/10 or 10/11 ?

Cross-multiplying gives (9)(11) versus (10)(10), which reduces to 99 versus 100. Now, 100 is greater than 99. Hence, 10/11 is greater than 9/10.

III. To solve a fractional equation, multiply both sides by the LCD (lowest common denominator) to clear fractions.

Example: If (x + 3)/(x - 3) = y, what is the value of x in terms of y?

(A) 3 - y (B) 3/y (C) (2 + y)/(y - 2) (D) (-3y -3)/(1 - y) (E) 3y/2

First, multiply both sides of the equation by x - 3: (x - 3)(x + 3)/(x - 3) = (x - 3)y

Cancel the (x - 3's) on the left side of the equation: x + 3 = (x - 3)y

Distribute the y: x + 3 = xy - 3y

Subtract xy and 3 from both sides: x - xy = -3y - 3

Factor out the x on the left side of the equation: x(1 - y) = -3y - 3

Finally, divide both sides of the equation by 1 - y: x = (-3y -3)/(1 - y)

Hence, the answer is (D).

IV. When dividing a fraction by a whole number (or vice versa), you must keep track of the main division bar.

Example: a/(b/c) = a(c/b) = ac/b. But (a/b)/c = (a/b)(1/c) = a/(bc).

V. Two fractions can be added quickly by cross-multiplying: a/b + c/d = (ad + bc)/bd

Example: 1/2 - 3/4 =

(A) -5/4 (B) -2/3 (C) -1/4 (D) 1/2 (E) 2/3

Cross multiplying the expression 1/2 - 3/4 yields [1(4) - 2(3)]/2(4) = (4 - 6)/8 = -2/8 = -1/4. Hence, the answer is (C).

VI. To find a common denominator of a set of fractions, simply double the largest denominator until all the other denominators divide into it evenly.

VII. Fractions often behave in unusual ways: Squaring a fraction makes it smaller, and taking the square root of a fraction makes it larger. (Caution: This is true only for proper fractions, that is, fractions between 0 and 1.)

Example: 1/3 squared equals 1/9 and 1/9 is less than 1/3. Also the square root of 1/4 is 1/2 and 1/2 is greater than 1/4.

AVERAGES

Problems involving averages are very common on the GMAT. They can be classified into four major categories as follows.

I. The average of N numbers is their sum divided by N, that is, average = sum/N.

Example: The average of x, 2x, and 6 is (x + 2x + 6)/3 = (3x + 6)/3 = 3(x + 2)/3 = x + 2.

II. Weighted average: The average between two sets of numbers is closer to the set with more numbers.

Example: If on a test three people answered 90% of the questions correctly and two people answered 80% correctly, then the average for the group is not 85% but rather [3(90) + 2(80)]/5 = 430/5 = 86. Here, 90 has a weight of 3--it occurs 3 times. Whereas 80 has a weight of 2--it occurs 2 times. So the average is closer to 90 than to 80 as we have just calculated.

III. Using an average to find a number.

Sometimes you will be asked to find a number by using a given average. An example will illustrate.

Example: If the average of five numbers is -10, and the sum of three of the numbers is 16, then what is the average of the other two numbers?

(A) -33 (B) -1 (C) 5 (D) 20 (E) 25

Let the five numbers be a, b, c, d, e. Then their average is (a + b + c + d + e)/5 = -10. Now three of the numbers have a sum of 16, say, a + b + c = 16. So substitute 16 for a + b + c in the average above: (16 + d + e)/5 = -10. Solving this equation for d + e gives d + e = -66. Finally, dividing by 2 (to form the average) gives (d + e)/2 = -33. Hence, the answer is (A).

IV. Average Speed = Total Distance/Total Time

Although the formula for average speed is simple, few people solve these problems correctly because most fail to find both the total distance and the total time.

Example: In traveling from city A to city B, John drove for 1 hour at 50 mph and for 3 hours at 60 mph. What was his average speed for the whole trip?

(A) 50 (B) 53 1/2 (C) 55 (D) 56 (E) 57 1/2

The total distance is 1(50) + 3(60) = 230. And the total time is 4 hours. Hence, Average Speed = Total Distance/Total Time = 230/4 = 57 1/2. The answer is (E). Note, the answer is not the mere average of 50 and 60. Rather the average is closer to 60 because he traveled longer at 60 mph (3 hrs) than at 50 mph (1 hr).

RATIO & PROPORTION
Ratio

A ratio is simply a fraction. Both of the following notations express the ratio of x to y: x:y, x/y. A ratio compares two numbers. Just as you cannot compare apples and oranges, so too must the numbers you are comparing have the same units. For example, you cannot form the ratio of 2 feet to 4 yards because the two numbers are expressed in different units--feet vs. yards. It is quite common for the GMAT to ask for the ratio of two numbers with different units. Before you form any ratio, make sure the two numbers are expressed in the same units.

Proportion

A proportion is simply an equality between two ratios (fractions). For example, the ratio of x to y is equal to the ratio of 3 to 2 is translated as x/y = 3/2. Two variables are directly proportional if one is a constant multiple of the other:

y = kx, where k is a constant.

The above equation shows that as x increases (or decreases) so does y. This simple concept has numerous applications in mathematics. For example, in constant velocity problems, distance is directly proportional to time: d = vt, where v is a constant. Note, sometimes the word directly is suppressed.

Example: If the ratio of y to x is equal to 3 and the sum of y and x is 80, what is the value of y?

(A) -10 (B) -2 (C) 5 (D) 20 (E) 60

Translating "the ratio of y to x is equal to 3" into an equation yields: y/x = 3

Translating "the sum of y and x is 80" into an equation yields: y + x = 80

Solving the first equation for y gives: y = 3x.

Substituting this into the second equation yields

3x + x = 80
4x = 80
x = 20

Hence, y = 3x = 3(20) = 60. The answer is (E).

In many word problems, as one quantity increases (decreases), another quantity also increases (decreases). This type of problem can be solved by setting up a direct proportion.

Example: If Biff can shape 3 surfboards in 50 minutes, how many surfboards can he shape in 5 hours?

(A) 16 (B) 17 (C) 18 (D) 19 (E) 20

As time increases so does the number of shaped surfboards. Hence, we set up a direct proportion. First, convert 5 hours into minutes: 5 hours = 5 x 60 minutes = 300 minutes. Next, let x be the number of surfboards shaped in 5 hours. Finally, forming the proportion yields

3/50 = x/300
3(300)/50 = x
18 =x

The answer is (C).

If one quantity increases (or decreases) while another quantity decreases (or increases), the quantities are said to be inversely proportional. The statement "y is inversely proportional to x" is written as

y = k/x, where k is a constant.

Multiplying both sides of y = k/x by x yields

yx = k

Hence, in an inverse proportion, the product of the two quantities is constant. Therefore, instead of setting ratios equal, we set products equal.

In many word problems, as one quantity increases (decreases), another quantity decreases (increases). This type of problem can be solved by setting up a product of terms.

Example: If 7 workers can assemble a car in 8 hours, how long would it take 12 workers to assemble the same car?

(A) 3hrs (B) 3 1/2hrs (C) 4 2/3hrs (D) 5hrs (E) 6 1/3hrs

As the number of workers increases, the amount time required to assemble the car decreases. Hence, we set the products of the terms equal. Let x be the time it takes the 12 workers to assemble the car. Forming the equation yields

7(8) = 12x
56/12 = x
4 2/3 = x

The answer is (C).

To summarize: if one quantity increases (decreases) as another quantity also increases (decreases), set ratios equal. If one quantity increases (decreases) as another quantity decreases (increases), set products equal.

EXPONENTS & ROOTS
Exponents

There are five rules that govern the behavior of exponents:


Problems involving these five rules are common on the GMAT, and they are often listed as hard problems. However, the process of solving these problems is quite mechanical: simply apply the five rules until they can no longer be applied.

Roots

There are only two rules for roots that you need to know for the GMAT:

FACTORING

To factor an algebraic expression is to rewrite it as a product of two or more expressions, called factors. In general, any expression on the GMAT that can be factored should be factored, and any expression that can be unfactored (multiplied out) should be unfactored.

Distributive Rule

The most basic type of factoring involves the distributive rule:
ax + ay = a(x + y)

For example, 3h + 3k = 3(h + k), and 5xy + 45x = 5xy + 9(5x) = 5x(y + 9). The distributive rule can be generalized to any number of terms. For three terms, it looks like ax + ay + az = a(x + y + z). For example, 2x + 4y + 8 = 2x + 2(2y) + 2(4) = 2(x + 2y + 4).

Example: If x - y = 9, then (x - y/3) - (y - x/3) =

(A) -4 (B) -3 (C) 0 (D) 12 (E) 27

(x - y/3) - (y - x/3) =
x - y/3 - y + x/3 =
4x/3 - 4y/3 =
4(x - y)/3 =
4(9)/3 =
12

The answer is (D).

Difference of Squares

One of the most important formulas on the GMAT is the difference of squares:

Example: If x does not equal -2, then

(A) 2(x - 2) (B) 2(x - 4) (C) 8(x + 2) (D) x - 2 (E) x + 4

In most algebraic expressions involving multiplication or division, you won't actually multiply or divide, rather you will factor and cancel, as in this problem.


2(x - 2)

The answer is (A).

Perfect Square Trinomials

Like the difference of squares formula, perfect square trinomial formulas are very common on the GMAT.

For example,.

ALGEBRAIC EXPRESSIONS

A mathematical expression that contains a variable is called an algebraic expression. Some examples of algebraic expressions are 3x - 2y, 2z/y. Two algebraic expressions are called like terms if both the variable parts and the exponents are identical. That is, the only parts of the expressions that can differ are the coefficients. For example, x + y and -7(x + y) are like terms. However, x - y and 2 - y are not like terms.

Adding & Subtracting Algebraic Expressions

Only like terms may be added or subtracted. To add or subtract like terms, merely add or subtract their coefficients:


You may add or multiply algebraic expressions in any order. This is called the commutative property:
x + y = y + x xy = yx

For example, -2x + 5x = 5x + (-2x) = (5 - 2)x = -3x and (x - y)(-3) = (-3)(x - y) = (-3)x - (-3)y = -3x + 3y.

Caution: the commutative property does not apply to division or subtraction.

When adding or multiplying algebraic expressions, you may regroup the terms. This is called the associative property:
x + (y + z) = (x + y) + z x(yz) = (xy)z

Notice in these formulas that the variables have not been moved, only the way they are grouped has changed: on the left side of the formulas the last two variables are grouped together, and on the right side of the formulas the first two variables are grouped together.

For example, (x -2x) + 5x = (x + [-2x]) + 5x = x + (-2x + 5x) = x + 3x = 4x and 24x = 2x(12x) = 2x(3x4x) = (2x3x)4x = 6x4x = 24x

Caution: the associative property doesn't apply to division or subtraction.

Notice in the first example that we changed the subtraction into negative addition: (x - 2x) = (x + [- 2x]). This allowed us to apply the associative property over addition.

Parentheses

When simplifying expressions with nested parentheses, work from the inner most parentheses out:

5x + (y - (2x - 3x)) = 5x + (y - (-x)) = 5x + (y + x) = 6x + y

Sometimes when an expression involves several pairs of parentheses, one or more pairs are written as brackets. This makes the expression easier to read:

2x(1 -[y + 2(3 - y)]) =
2x(1 -[y + 6 - 2y]) =
2x(1 -[-y + 6]) =
2x(1 + y - 6) =
2x(y - 5) =
2xy - 10x

Order of Operations: (PEMDAS)

When simplifying algebraic expressions, perform operations within parentheses first and then exponents and then multiplication and then division and then addition and then subtraction. This can be remembered by the mnemonic:
PEMDAS Please Excuse My Dear Aunt Sally

GRAPHS

Questions involving graphs rarely involve any significant calculating. Usually, the solution is merely a matter of interpreting the graph.

1. During which year was the company's earnings 10 percent of its sales?

(A) 85 (B) 86 (C) 87 (D) 88 (E) 90

Reading from the graph, we see that in 1985 the company's earnings were $8 million and its sales were $80 million. This gives 8/80 = 1/10 = 10/100 = 10%. The answer is (A).

2. During what two-year period did the company's earnings increase the greatest?

(A) 85-87 (B) 86-87 (C) 86-88 (D) 87-89 (E) 88-90

Reading from the graph, we see that the company's earnings increased from $5 million in 1986 to $10 million in 1987, and then to $12 million in 1988. The two-year increase from '86 to '88 was $7 million--clearly the largest on the graph. The answer is (C).

3. During the years 1986 through 1988, what were the average earnings per year?

(A) 6 million (B) 7.5 million (C) 9 million (D) 10 million (E) 27 million

The graph yields the following information:

YearEarnings
1986$5 million
1987$10 million
1988$12 million

Forming the average yields (5 + 10 + 12)/3 = 27/3 = 9. The answer is (C).

4. If Consolidated Conglomerate's earnings are less than or equal to 10 percent of sales during a year, then the stockholders must take a dividend cut at the end of the year. In how many years did the stockholders of Consolidated Conglomerate suffer a dividend cut?

(A) None (B) One (C) Two (D) Three (E) Four

Calculating 10 percent of the sales for each year yields

Year10% of Sales (millions)Earnings (millions)
85.10 x 80 = 88
86.10 x 70 = 75
87.10 x 50 = 510
8810 x 80 = 812
89.10 x 90 = 911
90.10 x 100 = 108

Comparing the right columns shows that earnings were 10 percent or less of sales in 1985, 1986, and 1990. The answer is (D).

WORD PROBLEMS

Although exact steps for solving word problems cannot be given, the following guidelines will help:

(1) First, choose a variable to stand for the least unknown quantity, and then try to write the other unknown quantities in terms of that variable.

For example, suppose we are given that Sue's age is 5 years less than twice Jane's and the sum of their ages is 16. Then Jane's age would be the least unknown, and we let x = Jane's age. Expressing Sue's age in terms of x gives Sue's age = 2x - 5.

(2) Second, write an equation that involves the expressions in Step 1. Most (though not all) word problems pivot on the fact that two quantities in the problem are equal. Deciding which two quantities should be set equal is usually the hardest part in solving a word problem since it can require considerable ingenuity to discover which expressions are equal.

For the example above, we would get (2x - 5) + x = 16.

(3) Third, solve the equation in Step 2 and interpret the result.

For the example above, we would get by adding the x's: 3x - 5 = 16. Then adding 5 to both sides gives 3x = 21. Finally, dividing by 3 gives x = 7. Hence, Jane is 7 years old and Sue is 2x - 5 = 2(7) - 5 = 9 years old.

Motion Problems

Virtually, all motion problems involve the formula Distance = Rate x Time, or

D = R x T

Example: Scott starts jogging from point X to point Y. A half-hour later his friend Garrett who jogs 1 mile per hour slower than twice Scott's rate starts from the same point and follows the same path. If Garrett overtakes Scott in 2 hours, how many miles will Garrett have covered?

(A) 2 1/5 (B) 3 1/3 (C) 4 (D) 6 (E) 6 2/3

Following Guideline 1, we let r = Scott's rate. Then 2r - 1 = Garrett's rate. Turning to Guideline 2, we look for two quantities that are equal to each other. When Garrett overtakes Scott, they will have traveled the same distance. Now, from the formula D = R x T, Scott's distance is D = r x 2 1/2 and Garrett's distance is D = (2r - 1)2 = 4r - 2. Setting these expressions equal to each other gives 4r - 2 = r x 2 1/2. Solving this equation for r gives r = 4/3. Hence, Garrett will have traveled D = 4r - 2 = 4(4/3) - 2 = 3 1/3 miles. The answer is (B).

Work Problems

The formula for work problems is Work = Rate x Time, or W = R x T. The amount of work done is usually 1 unit. Hence, the formula becomes 1 = R x T. Solving this for R gives R = 1/T.

Example : If Johnny can mow the lawn in 30 minutes and with the help of his brother, Bobby, they can mow the lawn 20 minutes, how long would take Bobby working alone to mow the lawn?

(A) 1/2 hour (B) 3/4 hour (C) 1 hour (D) 3/2 hours (E) 2 hours

Let r = 1/t be Bobby's rate. Now, the rate at which they work together is merely the sum of their rates:

Total Rate = Johnny's Rate + Bobby's Rate
1/20 = 1/30 + 1/t
1/20 - 1/30 = 1/t
(30 - 20)/(30)(20) = 1/t
1/60 = 1/t
t = 60

Hence, working alone, Bobby can do the job in 1 hour. The answer is (C).

Mixture Problems

The key to these problems is that the combined total of the concentrations in the two parts must be the same as the whole mixture.

Example : How many ounces of a solution that is 30 percent salt must be added to a 50-ounce solution that is 10 percent salt so that the resulting solution is 20 percent salt?

(A) 20 (B) 30 (C) 40 (D) 50 (E) 60

Let x be the ounces of the 30 percent solution. Then 30%x is the amount of salt in that solution. The final solution will be 50 + x ounces, and its concentration of salt will be 20%(50 + x). The original amount of salt in the solution is 10%(50). Now, the concentration of salt in the original solution plus the concentration of salt in the added solution must equal the concentration of salt in the resulting solution: 10%(50) + 30%x = 20%(50 + x). Multiply this equation by 100 to clear the percent symbol and then solving for x yields x = 50. The answer is (D).

Coin Problems

The key to these problems is to keep the quantity of coins distinct from the value of the coins. An example will illustrate.

Example : Laura has 20 coins consisting of quarters and dimes. If she has a total of $3.05, how many dimes does she have?

(A) 3 (B) 7 (C) 10 (D) 13 (E) 16

Let D stand for the number of dimes, and let Q stand for the number of quarters. Since the total number of coins in 20, we get D + Q = 20, or Q = 20 - D. Now, each dime is worth 10 cents, so the value of the dimes is 10D. Similarly, the value of the quarters is 25Q = 25(20 - D). Summarizing this information in a table yields

DimesQuartersTotal
NumberD20 - D20
Value10D25(20 - D)305

Notice that the total value entry in the table was converted from $3.05 to 305 cents. Adding up the value of the dimes and the quarters yields the following equation:

10D + 25(20 - D) = 305
10D + 500 - 25D = 305
-15D = -195
D = 13

Hence, there are 13 dimes, and the answer is (D).

Age Problems

Typically, in these problems, we start by letting x be a person's current age and then the person's age a years ago will be x - a and the person's age a years in future will be x + a. An example will illustrate.

Example : John is 20 years older than Steve. In 10 years, Steve's age will be half that of John's. What is Steve's age?

(A) 2 (B) 8 (C) 10 (D) 20 (E) 25

Steve's age is the most unknown quantity. So we let x = Steve's age and then x + 20 is John's age. Ten years from now, Steve and John's ages will be x + 10 and x + 30, respectively. Summarizing this information in a table yields

Age nowAge in 10 years
Stevexx + 10
Johnx + 20x + 30

Since "in 10 years, Steve's age will be half that of John's," we get

(x + 30)/2 = x + 10
x + 30 = 2(x + 10)
x + 30 = 2x + 20
x = 10

Hence, Steve is 10 years old, and the answer is (C).


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