SAT Diagnostic Test - Math

Test Questions

1. If x and y are both prime and greater than 2, then which of the following CANNOT be a divisor of xy?
(A) 2
(B) 3
(C) 11
(D) 15
(E) 17

Correct Answer: (A)
Solution: Since x and y are prime and greater than 2, xy is the product of two odd numbers and is therefore odd. Hence, 2 cannot be a divisor of xy. The answer is (A).

2. Cars X and Y leave City A at the same time and travel the same route to City B. Car X takes 30 minutes to complete the trip and car Y takes 20 minutes. Which of the following must be true?

I. The average miles per hour at which car X traveled was greater than the average miles per hour at which car Y traveled.
II. The distance between the cities is 30 miles.
III. The average miles per hour at which car Y traveled was greater than the average miles per hour at which car X traveled.

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

Correct Answer: (C)
Solution:

The average speed at which car X traveled is (Total Distance)/30.

The average speed at which car Y traveled is (Total Distance)/20.

The two fractions have the same numerators, and the denominator for car Y is smaller. Hence, the average miles per hour at which car Y traveled is greater than the average miles per hour at which car X traveled. Thus, Statement I is false and Statement III is true. As to Statement II, we do not have enough information to calculate the distance between the cities. Hence, Statement II need not be true. The answer is (C).

3. In the figure above, y =
(A) 75
(B) 76
(C) 77
(D) 78
(E) 79

Correct Answer: (D)
Solution: OS and OT are equal since they are radii of the circle. Hence, Triangle SOT is isosceles. Therefore, S = T = 51. Recalling that the angle sum of a triangle is 180 degrees, we get S + T + y = 51 + 51 + y = 180. Solving for y gives y = 78. The answer is (D).

4. If x = 3y = 4z, which of the following must equal 6x?

I. 18y
II. 3y + 20z
III. (4y + 10z)/3

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

Correct Answer: (D)
Solution: The equation x = 3y = 4z contains three equations:

Multiplying both sides of the equation x = 3y by 6 gives 6x = 18y. Hence, Statement I is true. This eliminates (B) and (C). Next, 3y + 20z = 3y + 5(4z) . Substituting x for 3y and for 4z in this equation gives 3y + 20z = 3y + 5(4z) = x + 5x = 6x. Hence, Statement II is true. This eliminates (A) and (E). Hence, by process of elimination, the answer is (D).

5. The average of four numbers is 20. If one of the numbers is removed, the average of the remaining numbers is 15. What number was removed?
(A) 10
(B) 15
(C) 30
(D) 35
(E) 45

Correct Answer: (D)
Solution: Let the four numbers be a, b, c, and d. Since their average is 20, we get

(a + b + c + d)/4 = 20

Let d be the number that is removed. Since the average of the remaining numbers is 15, we get

(a + b + c)/3 = 15

Solving for a + b + c yields

a + b + c = 45

Substituting this into the first equation yields

(45 + d)/4 = 20

Multiplying both sides of this equation by 4 yields

45 + d = 80

Subtracting 45 from both sides of this equation yields

d = 35

The answer is (D).

6. The ratio of two numbers is 10 and their difference is 18. What is the value of the smaller number?
(A) 2
(B) 5
(C) 10
(D) 21
(E) 27

Correct Answer: (A)
Solution: Let x and y denote the numbers. Then x/y = 10 and x - y = 18. Solving the first equation for x and plugging it into the second equation yields

Plugging this into the equation x - y = 18 yields x = 20. Hence, y is the smaller number. The answer is (A).

7. If 3y + 5 = 7x, then 21y - 49x =
(A) -40
(B) -35
(C) -10
(D) 0
(E) 15

Correct Answer: (B)
Solution: First, interchanging 5 and 7x in the expression 3y + 5 = 7x yields 3y - 7x = -5. Next, factoring 21y - 49x yields

21y - 49x =

7(3y) - 7(7x) =

7(3y - 7x) =

7(-5) = since 3y - 7x = -5

-35

The answer is (B).

8. Seven years ago, Scott was 3 times as old as Kathy was at that time. If Scott is now 5 years older than Kathy, how old is Scott?
(A) 12 1/2
(B) 13
(C) 13 1/2
(D) 14
(E) 14 1/2

Correct Answer: (E)
Solution: Let S be Scott's age and K be Kathy's age. Then translating the sentence "If Scott is now 5 years older than Kathy, how old is Scott" into an equation yields

Now, Scott's age 7 years ago can be represented as S = -7, and Kathy's age can be represented as K = -7. Then translating the sentence "Seven years ago, Scott was 3 times as old as Kathy was at that time" into an equation yields

Combining this equation with S = K + 5 yields the system:

Solving this system gives 14 1/2. The answer is (E).

DIAGNOSIS

If you missed Question 1:
Your score indicates that you need to study Number Theory. Click Number Theory for a review of this type of problem.

If you missed Question 2:
Your score indicates that you need to study Averages . Click Averages for a review of this type of problem.

If you missed Question 3:
Your score indicates that you need to study Geometry. Click Geometry for a review of this type of problem.

If you missed Question 4:
Your score indicates that you need to study Equations. Click Equations for a review of this type of problem.

If you missed Question 5:
Your score indicates that you need to study Averages . Click Averages for a review of this type of problem.

If you missed Question 6:
Your score indicates that you need to study Ratios and Proportions. Click Ratios and Proportions for a review of this type of problem.

If you missed Question 7:
Your score indicates that you need to study Factoring. Click Factoring for a review of this type of problem.

If you missed Question 8:
Your score indicates that you need to study Word Problems. Click Word Problems for a review of this type of problem.

PERFORMANCE

If you perform at this level on a complete SAT math section, your score would be as follows:

Conversion Chart:

Raw Score	Scaled Score
1	280
2	350
3	420
4	480
5	560
6	620
7	710
8	800