**Directions:** The
Mathematics subtest will test your ability to solve quantitative problems
quickly and accurately. Analyze each problem and select the best answer from
the multiple choices.

The practice tests are organized around five fundamental math concepts that will be included in the actual UPCAT: arithmetic and number sense, algebra, geometry, trigonometry, and statistics.

To mimic the actual test conditions, you’re not allowed to use calculators. Because UPCAT is a time-pressured exam, we also recommend using a timer when taking the practice test.

1. The average of 7 numbers is 24. The smallest of the numbers is 2 and the largest of the numbers is 31. What is the average of the middle 5 numbers?

A. 25

B. 26

C. 27

D. 28

2. Two side lengths of a triangle are 5 and 6, which of the following CANNOT be the length of the third side?

A. 3

B. 6

C. 9

D. 12

3. Eric wishes
to shift the graph of the function ƒ(*x*) = *x*² four
places to the right and six places down from the origin (0,0). Which equation
represents this translation?

A. ƒ(x) = x – 4^{2} + 6

B. ƒ(x) = (x − 4)^{2} − 6

C. ƒ(x) =
(x^{2} − 4)^{2} − 6

D. ƒ(x) = x + 4^{2} – 6

4. What is the
slope intercept form of (13*x* − 5*x*) + 12 − 2*y* =
6?

A. y = 3x – 9

B. y = 4x + 3

C. y = 6x + 2

D. y = -2x + 5

5. The coordinates (−3, 5) and (3, 5) designate the diameter of a circle, what is its circumference?

A. 2π

B. 4π

C. 6π

D. 8π

6. What is the solution set of | 2(x – 1) – 15 | = 7?

A. { 5 }

B. { 12 }

C. { 5,12 }

D. { -5,12 }

7. *A* and *B* are
reciprocals (when multiplied together their product is 1). If *A* <
−1, then *B* must be which of the following?

A. 0 < B < 1

B. -1 < B < 0

C. B < -1

D. B < 0

8. Which of
these is true for all real numbers *x* and *y* such
that *x* > 0 and *y* < 0?

A. xy > 0

B. x + y > 0

C. x^{2} + y^{2 }> 0

D. x – y < 0

9. At what point on the Cartesian plane do the following two lines intersect?

y/3 – 7/3 = x and y/3 + 4/3 = x

A. (4, 3)

B. (6, 2)

C. The lines do not intersect.

D. (5, -2)

10. The slope
of the line represented by the equation 2(*x* − 3) − 6*y* =
10 is equal to what?

A. – 1/2

B. 1/3

C. 1/2

D. 2

11. Jenny’s house is 2 km away from her school. One day when going to school, Jenny runs for 20 minutes and arrived 5 minutes late. How many minutes earlier would Jenny be if she would use her bicycle at a rate of one-third kilometer per minute?

A. 5 minutes

B. 6 minutes

C. 8 minutes

D. 9 minutes

12. The chart below shows the monthly profits of 3 companies. What is the total profit generated by Store X and Store Z in the month of March?

A. 120,000

B. 140,000

C. 180,000

D. 200,000

13. Which of the following statements is true?

A. The diagonals of a rhombus are congruent.

B. All rectangles are similar.

C. All rectangles are rhombuses.

D. Some rhombuses are rectangles.

14. In order
for the two triangles shown to be similar, what is one possible value for *x*?

A. 8 in.

B. 10 in.

C. 16 in.

D. 20 in.

15. Which
coordinates satisfy the inequality: *y*+ 3 > −3(*x* − 2)

A. (0,2)

B. (1,0)

C. (-3, 15)

D. (2, -2)

16. If *h*(*x*)
= 3*x* + 4, what is *h*(2*x* − 3)?

A. -6X + 5

B. 6X – 5

C. 5X – 2

D. 5X + 2

17. If -1 < a < b < 0, then which of the following has the greatest value?

A. b – a

B. a + b

C. a – b

D. 2b – a

18. In the
figure below, what is the length of altitude *y*?

A. 7 cm.

B. 8 cm.

C. 9 cm.

D. 10 cm.

19. Which of the following sets of interior angle measures would describe an acute isosceles triangle?

A. 90°, 45°, 45°

B. 80°, 60°, 60°

C. 60°, 60°, 60°

D. 60°, 50°, 50°

20. Which trigonometric function can equal or be greater than 1.000?

A. Sine

B. Cosine

C. Tangent

D. none of the above

1. Answer: C.

Begin by setting up an equation representing the average. (2 + x + 31) ÷ 7 = 24. Solve for x to find 135 and recognize that this x represents the sum of the remaining 5 scores. To find the average, divide 135 by 5 to find 27.

2. Answer: D.

If a triangle has side lengths a, b, and c, the sum of the lengths of any 2 sides must be larger than the length of the 3rd side. So in this case, 5 + 6 = 11 must be larger than side length c. From the answer choices, 12 is the only length greater than 11, so it cannot be the length of the third side.

3. Answer: B.

Recall that vertex- form of a parabola is:

a(x − h)^{2} + k, where
(h, k) represents the vertex.

We wish to translate our vertex from (0,0) to (4,−6) so h = 4 and k = −6.

ƒ(x) = (x − 4)^{2} – 6

4. Answer: B.

Recall that slope-intercept form is y = mx + b where m is the slope and b is the y-intercept. Solve for y:

8x − 2y = −6

2y = 8x + 6

Divide everything by 2:

y = 4x + 3

5. Answer: C.

The circumference of a circle is the distance around defined by π * diameter. The diameter, in this case, can be found through the difference between the x values:

3 − (−3) = 6, so π * 6 is the circumference.

6. Answer: C.

| 2(x – 1) – 15 | = 7

| 2x – 2 – 15 | = 7

| 2x – 17 | = 7

2x – 17 = – 7 and 2x – 17 = 7

2x = 10 and 2x = 24

x = 5 and x = 12

The solution set: { 5,12 }

7. Answer: B.

If the product of two numbers is positive, the two numbers must have the same sign. That is, if ab > 0, then either a > 0 and b > 0, or a < 0 and b < 0.

We are told that A < −1 (which implies that A < 0).

So we know that B < 0.

We also know that AB = 1, so A = 1/B

Since A = 1/B, and A < -1, we can infer that 1/B < -1

If we take reciprocals on both sides of the last inequality, we must flip the inequality sign. Hence: B > −1

So we know that B < 0, and B > −1. We can represent this as a compound inequality: −1 < B < 0

8. Answer: C.

For this question, you have to examine all the answer options individually in order to eliminate all those that cannot be true. First, if x is positive and y is negative, their product must be negative, so (A) is incorrect.

Next, the sum of a positive and a negative number could be either positive or negative, depending on which number has the greater absolute value; this rules out (B) because it’s not always true.

Similar reasoning applies to choice (D) as well. However, both positive and negative real numbers have positive squares, and adding those positive squares will always yield a positive number, so (C) is correct.

9. Answer: C.

Because both of these equations are already solved for the variable x, we can set them equal to each other to find the value of y. Begin by multiplying both sides by 3 to remove the denominator.

y − 7 = y + 4

Notice that this equation will never be true. Since there is no solution, so we can conclude that the lines do not intersect.

10. Answer: B.

Recall the slope-intercept form of a line:

y = mx + b where m is the slope.

Solve the given equation for y to find the slope:

2x − 6 − 6y = 10

−6y = −2x + 16

y = 1/3x − 16/6

Slope is equal to 1/3.

11. Answer: D.

She runs for 20 minutes and arrived 5 minutes late → She needs to be exactly there in 15 minutes.

Using a bike with a speed of 1/3 km per minutes → t = d/r → t = 2/1/3 → t = 6

15 minutes – 6 minutes = 9 minutes earlier.

12. Answer: C.

From the chart, we can see that Store X had 80 (thousand) in profits and Store Z had 100 (thousand) in profits. Combining these two, we arrive at 180 (thousand) in profits.

13. Answer: D.

A square is a rhombus and a rectangle. Therefore, some rhombuses are rectangles.

14. Answer: C.

Problems involving similar figures can be solved using proportions. The issue with this problem is that we are given a similarity across inches to feet with the answer choices containing only inches. First, we must convert the feet measurement into inches:

2 1/3 ft. x 12 in./ft. = 28 in.

We can now set up our proportions:

4/7 = x/28

x = 16

15. Answer: C.

Solve the inequality for one variable:

y + 3 > −3x + 6

y > −3x + 3

This states that the y-coordinate must be larger than −3 times the x-coordinate plus 3. Test the points provided to see which one satisfies the given inequality (this can also be done graphically). Only (−3, 15) satisfies the inequality.

16. Answer: B.

We solve this problem by replacing every x in h(x) with 2x − 3 and evaluating the expression:

h(2x − 3) = 3(2x − 3) + 4

= 6x − 9 + 4

= 6x − 5

17. Answer: A

Let a = -0.9 and b = -0.1

A. b – a = (-0.1) – (- 0.9) = 0.8

B. a + b = (-0.1) + (-0.9) = -1.0

C. a – b = (-0.9) – (-0.1) = -0.8

D. 2b – a = 2(-0.1) – (-0.9) = 0.7

18. Answer: B.

Sometimes questions will provide unnecessary information. In this case, the angle measurement of the top right angle. We can focus exclusively on the right triangle shown and use the Pythagorean Theorem, or the recognition of a Pythagorean triple to see that the length of y is 8 cm.

19. Answer: C.

Choice A is not an acute triangle because it has one right angle. In choice B, the sum of interior angle measures exceeds 180°. Choice D suffers the reverse problem; its sum does not make 180°. Though choice C describes an equilateral triangle; it also describes an isosceles triangle.

20. Answer: C.

The trigonometric ratios sine and cosine never equal or exceed 1.000 because the hypotenuse, the longest side of a right triangle, is always their denominator. The trigonometric ratio Tangent can equal and exceed the value 1.000 because the hypotenuse is never its denominator.