ตัวอย่างข้อสอบ gmat GMAT EXAM - GMAT TEST MATH SECTION

ตัวอย่างข้อสอบ gmat
GMAT EXAM - GMAT TEST MATH SECTION

FORMAT OF THE GMAT MATH TEST SECTION

The Math section consists of 37 multiple-choice questions. The questions come in two formats: the standard multiple-choice question which we will study in this section and the Data Sufficiency question which we will study in the next section. The math section is designed to test your ability to solve problems, not to test your mathematical knowledge.
GMAT TEST VS. SAT TEST

GMAT math is very similar to SAT math, though slightly harder. The mathematical skills tested are very basic: only first year high school algebra and geometry (no proofs). However, this does not mean that the math section is easy. The medium of basic mathematics is chosen so that everyone taking the test will be on a fairly even playing field. Although the questions require only basic mathematics and all have simple solutions, it can require considerable ingenuity to find the simple solution. If you have taken a course in calculus or another advanced math topic, don't assume that you will find the math section easy. Other than increasing your mathematical maturity, little you learned in calculus will help on the GMAT.

As mentioned above, every GMAT math problem has a simple solution, but finding that simple solution may not be easy. The intent of the math section is to test how skilled you are at finding the simple solutions. The premise is that if you spend a lot of time working out long solutions you will not finish as much of the test as students who spot the short, simple solutions. So if you find yourself performing long calculations or applying advanced mathematics--stop. You're heading in the wrong direction.

Don't worry if you fail to reach the last few questions. It's better to work accurately than quickly.
SUBSTITUTION

Substitution is a very useful technique for solving GMAT math problems. It often reduces hard problems to routine ones. In the substitution method, we choose numbers that have the properties given in the problem and plug them into the answer-choices.

Example:

If n is an odd integer, which one of the following is an even integer?

(A) 3n + 2 (B) n/4 (C) 2n + 3 (D) n(n + 3) (E) nn

We are told that n is an odd integer. So choose an odd integer for n, say, 1 and substitute it into each answer-choice. In Choice (A), 3(1) + 2 = 5, which is not an even integer. So eliminate (A). Next, n/4 = 1/4 is not an even integer--eliminate (B). Next, 2n + 3 = 2(1) + 3 = 5 is not an even integer--eliminate (C). Next, n(n + 3) = 1(1 + 3) = 4 is even and hence the answer is possibly (D). Finally, in Choice (E), the nn = 1(1) = 1, which is not even--eliminate (E). The answer is (D).

When using the substitution method, be sure to check every answer-choice because the number you choose may work for more than one answer-choice. If this does occur, then choose another number and plug it in, and so on, until you have eliminated all but the answer. This may sound like a lot of computing, but the calculations can usually be done in a few seconds.

When substituting in quantitative comparison problems, don't rely on only positive whole numbers. You must also check negative numbers, fractions, 0, and 1 because they often give results different from those of positive whole numbers. Plug in the numbers 0, 1, 2, -2, and 1/2, in that order.

Example : If n is an integer, which of the following CANNOT be an even integer?
(A) 2n + 2 (B) n - 5 (C) 2n (D) 2n + 3 (E) 5n + 2

Choose n to be 1. Then 2n + 2 = 2(1) + 2 = 4, which is even. So eliminate (A). Next, n - 5 = 1 - 5 = -4. Eliminate (B). Next, 2n = 2(1) = 2. Eliminate (C). Next, 2n + 3 = 2(1) + 3 = 5 is not even--it may be our answer. However, 5n + 2 = 5(1) + 2 = 7 is not even as well. So we choose another number, say, 2. Then 5n + 2 = 5(2) + 2 = 12 is even, which eliminates (E). Thus, choice (D), 2n + 3, is the answer.

Sometimes instead of making up numbers to substitute into the problem, we can use the actual answer-choices. This is called Plugging In. It is a very effective technique but not as common as Substitution.

Example: The digits of a three-digit number add up to 18. If the ten's digit is twice the hundred's digit and the hundred's digit is 1/3 the unit's digit, what is the number?
(A) 246 (B) 369 (C) 531 (D) 855 (E) 893

First, check to see which of the answer-choices has a sum of digits equal to 18. For choice (A), 2 + 4 + 6 = 12. Eliminate. For choice (B), 3 + 6 + 9 = 18. This may be the answer. For choice (C), 5 + 3 + 1 = 9. Eliminate. For choice (D), 8 + 5 + 5 = 18. This too may be the answer. For choice (E), 8 + 9 + 3 = 20. Eliminate. Now, in choice (D), the ten's digit is not twice the hundred's digit, 5 does not equal 2(8). Eliminate. Hence, by process of elimination, the answer is (B). Note that we did not need the fact that the hundred's digit is 1/3 the unit's digit.

DEFINED FUNCTIONS

Defined functions are very common on the GMAT, and most students struggle with them. Yet once you get used to them, defined functions can be some of the easiest problems on the test. In this type of problem, you will be given a symbol and a property that defines the symbol.

Example: Define x # y by the equation x # y = xy - y. Then 2 # 3 =

(A) 1 (B) 3 (C) 12 (D) 15 (E) 18

From the above definition, we know that x # y = xy - y. So all we have to do is replace x with 2 and y with 3 in the definition: 2 # 3 = 2(3) - 3 = 3. Hence, the answer is (B).
GMAT NUMBER THEORY

This broad category is a popular source for GMAT questions. At first, students often struggle with these problems since they have forgotten many of the basic properties of arithmetic. So before we begin solving these problems, let's review some of these basic properties.

• "The remainder is r when p is divided by q" means p = qz + r; the integer z is called the quotient. For instance, "The remainder is 1 when 7 is divided by 3" means 7 = 3(2) + 1.

Example: When the integer n is divided by 2, the quotient is u and the remainder is 1. When the integer n is divided by 5, the quotient is v and the remainder is 3. Which one of the following must be true?

(A) 2u + 5v = 4
(B) 2u - 5v = 2
(C) 4u + 5v = 2
(D) 4u - 5v = 2
(E) 3u - 5v = 2

Translating "When the integer n is divided by 2, the quotient is u and the remainder is 1" into an equation gives n = 2 u + 1. Translating "When the integer n is divided by 5, the quotient is v and the remainder is 3" into an equation gives n = 5v + 3. Since both expressions equal n, we can set them equal to each other: 2u + 1 = 5v + 3. Rearranging and then combining like terms yields 2u - 5v = 2. The answer is (B).

• A number n is even if the remainder is zero when n is divided by 2: n = 2z + 0, or n = 2z.

• A number n is odd if the remainder is one when n is divided by 2: n = 2z + 1.

• The following properties for odd and even numbers are very useful--you should memorize them:

even x even = even
odd x odd = odd
even x odd = even

even + even = even
odd + odd = even
even + odd = odd

• Consecutive integers are written as x, x + 1, x + 2, . . .

• Consecutive even or odd integers are written as , x + 2, x + 4, . . .

• The integer zero is neither positive nor negative, but it is even: 0 = 2(0).

• A prime number is an integer that is divisible only by itself and 1.

The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, . . .

• A number is divisible by 3 if the sum of its digits is divisible by 3.

For example, 135 is divisible by 3 because the sum of its digits (1 + 3 + 5 = 9) is divisible by 3.

• The absolute value of a number, | |, is always positive. In other words, the absolute value symbol eliminates negative signs.

For example, | -7 | = 7. Caution, the absolute value symbol acts only on what is inside the symbol, | |. For example, -| -7 | = -(+7) = -7. Here, only the negative sign inside the absolute value symbol is eliminated.

Example: If a, b, and c are consecutive integers and a < b < c, which of the following must be true?

I. b - c = 1
II. abc/3 is an integer.
III. a + b + c is even.

(A) I only (B) II only (C) III only (D) I and II only (E) II and III only

Let x, x + 1, x + 2 stand for the consecutive integers a, b, and c, in that order. Plugging this into Statement I yields b - c = (x + 1) -(x + 2) = -1. Hence, Statement I is false.

As to Statement II, since a, b, and c are three consecutive integers, one of them must be divisible by 3. Hence, abc/3 is an integer, and Statement II is true.

As to Statement III, suppose a is even, b is odd, and c is even. Then a + b is odd since even + odd = odd. Hence, a + b + c = (a + b) + c = (odd) + even = odd. Thus, Statement III is not necessarily true. The answer is (B).
GEOMETRY on the GMAT TEST

One-fourth of the math problems on the GMAT involve geometry. (There are no proofs.) Fortunately, except for Data Sufficiency section, the figures on the GMAT are usually drawn to scale. Hence, you can check your work and in some cases even solve a problem by "eyeballing" the drawing.

Following are some of the basic properties of geometry. You probably know many of them. Memorize any that you do not know.

1. There are 180 degrees in a straight angle.

2. Two angles are supplementary if their angle sum is 180 degrees.

3. Two angles are complementary if their angle sum is 90 degrees.

4. Perpendicular lines meet at right angles.

5. A triangle with two equal sides is called isosceles. The angles opposite the equal sides are called the base angles.

6. The altitude to the base of an isosceles or equilateral triangle bisects the base and bisects the vertex angle.

7. The angle sum of a triangle is 180 degrees.

8. In an equilateral triangle all three sides are equal, and each angle is 60 degrees.

9. The area of a triangle is bh/2, where b is the base and h is the height.

10. In a triangle, the longer side is opposite the larger angle, and vice versa.

11. Two triangles are similar (same shape and usually different size) if their corresponding angles are equal. If two triangles are similar, their corresponding sides are proportional.

12. Two triangles are congruent (identical) if they have the same size and shape.

13. In a triangle, an exterior angle is equal to the sum of its remote interior angles and is therefore greater than either of them.

14. Opposite sides of a parallelogram are both parallel and congruent.

15. The diagonals of a parallelogram bisect each other.

16. If w is the width and l is the length of a rectangle, then its area is A = lw and its perimeter is P=2w + 2l.

17. The volume of a rectangular solid (a box) is the product of the length, width, and height. The surface area is the sum of the area of the six faces.

18. If the length, width, and height of a rectangular solid (a box) are the same, it is a cube. Its volume is the cube of one of its sides, and its surface area is the sum of the areas of the six faces.

19. A tangent line to a circle intersects the circle at only one point. The radius of the circle is perpendicular to the tangent line at the point of tangency.

20. An angle inscribed in a semicircle is a right angle.
Example: In the figure to the right, what is the value of x?

(A) 30
(B) 32
(C) 35
(D) 40
(E) 47


Since 2x + 60 is an exterior angle, it is equal to the sum of the remote interior angles. That is, 2x + 60 = x + 90. Solving for x gives x = 30. The answer is (A).

Most geometry problems on the GMAT require straightforward calculations. However, some problems measure your insight into the basic rules of geometry. For this type of problem, you should step back and take a "birds-eye" view of the problem. The following example will illustrate.
Example: In the figure to the right, O is both the center of the circle with radius 2 and a vertex of the square OPRS. What is the length of diagonal PS?

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


The diagonals of a square are equal. Hence, line segment OR (not shown) is equal to SP. Now, OR is a radius of the circle and therefore OR = 2. Hence, SP = 2 as well, and the answer is (D).
COORDINATE GEOMETRY

Distance Formula:

The distance between points (x, y) and (a, b) is given by the following formula:


Example: In the figure to the right, the circle is centered at the origin and passes through point P. Which of the following points does it also pass through?

(A) (3, 3)
(B)
(C) (2, 6)
(D) (1.5, 1.3)
(E) (-3, 4)


Since the circle is centered at the origin and passes through the point (0,-3), the radius of the circle is 3. Now, if any other point is on the circle, the distance from that point to the center of the circle (the radius) must also be 3. Look at choice (B). Using the distance formula to calculate the distance between and (0, 0) (the origin) yields


Hence, is on the circle, and the answer is (B).
Midpoint Formula:

The midpoint M between points (x, y) and (a, b) is given by
M = ([x + a]/2, [y + b]/2)

In other words, to find the midpoint, simply average the corresponding coordinates of the two points.
Example: In the figure to the right, polygon PQRO is a square and T is the midpoint of side QR. What are the coordinates of T ?

(A) (1, 1)
(B) (1, 2)
(C) (1.5, 1.5)
(D) (2, 1)
(E) (2, 3)


Since point R is on the x-axis, its y-coordinate is 0. Further, since PQRO is a square and the x-coordinate of Q is 2, the x-coordinate of R is also 2. Since T is the midpoint of side QR, the midpoint formula yields T = ([2 + 2]/2, [2 + 0]/2) = (4/2, 2/2) = (2, 1). The answer is (D).
Slope Formula:

The slope of a line measures the inclination of the line. By definition, it is the ratio of the vertical change to the horizontal change. The vertical change is called the rise, and the horizontal change is called the run. Thus, the slope is the rise over the run. Given the two points (x, y) and (a, b) the slope is
M = (y - b)/(x - a)


Example: In the figure to the right, what is the slope of line passing through the two points?

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

The slope formula yields m = (4 - 2)/(5 - 1) = 2/4 = 1/2. The answer is (C).
Slope-Intercept Form:

Multiplying both sides of the equation m = (y -b)/(x - a) by x-a yields

y - b = m(x - a)

Now, if the line passes through the y-axis at (0, b), then the equation becomes

y - b = m(x - 0)
y - b = mx
y = mx + b

This is called the slope-intercept form of the equation of a line, where m is the slope and b is the y-intercept. This form is convenient because it displays the two most important bits of information about a line: its slope and its y-intercept.

Example: If The equation of the line above is y = 9x/10 + k, which line segment is longer AO or BO?
Since y = 9x/10 + k is in slope-intercept form, we know the slope of the line is 9/10. Now, the ratio of BO to AO is the slope of the line (rise over run). Hence, BO/AO = 9/10. Multiplying both sides of this equation by AO yields BO = 9AC/10. In other words, BO is 9/10 the length of AO. Hence, AO is longer.

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