Praxis Mathematics: Number and Quantity and Algebra

Question 1

Which of these whole numbers is prime?

Accepted Answer

47, which has no whole-number factors other than 1 and itself.

Question 2

What is the greatest common factor of 36 and 48?

Accepted Answer

12, because 36 = 12 × 3 and 48 = 12 × 4, and no larger common factor works.

Question 3

What is the least common multiple of 8 and 12?

Accepted Answer

24, the smallest positive common multiple.

Question 4

A student says −3 − (−8) equals −11, because "two minuses make a bigger minus." Which move best answers that idea?

Accepted Answer

Rewrite the expression as −3 + 8 and compute that sum.

Question 5

What is |−7| + |2|?

Accepted Answer

9, because the absolute values are 7 and 2.

Question 6

Which number is an integer?

Accepted Answer

8/2, which equals the whole number 4.

Question 7

Which expression is 4,500 written in scientific notation?

Accepted Answer

4.5 × 10^3, the form with one nonzero digit before the decimal.

Question 8

A student claims 2^5 = 10, because "the exponent tells you to multiply the base by 5." What should the teacher do next?

Accepted Answer

Have the student write 2 × 2 × 2 × 2 × 2 and compare that product with 10.

Question 9

What is the principal square root of 81?

Accepted Answer

9, because 9 × 9 = 81 and the principal root is nonnegative.

Question 10

What is 5^0?

Accepted Answer

1, because a nonzero number to the power 0 equals 1.

Question 11

What is 3/4 + 1/8?

Accepted Answer

7/8, because 3/4 = 6/8 and 6/8 + 1/8 = 7/8. Common denominator eighths.

Question 12

A student computes (1/2) ÷ (1/4) and gets 1/8. Which question best targets the error?

Accepted Answer

How many fourths are in one half? Count them.

Question 13

Which fraction, in lowest terms, equals 0.6?

Accepted Answer

3/5, because 0.6 = 6/10 and that fraction reduces by 2.

Question 14

What is 15% of 80?

Accepted Answer

12, because 0.15 × 80 = 12. Ten percent is 8 and five percent is 4.

Question 15

A $40 price rises by 10% and then falls by 10% of the new price. What is the final price?

Accepted Answer

$39.60, because 44 × 0.90 = 39.60.

Question 16

A student simplifies (2 + x)/2 by canceling the 2s and leaving x. Which check best shows the mistake?

Accepted Answer

Substitute x = 4. The original expression equals 3, but canceling the 2s leaves 4.

Question 17

The larger part of a 2:3 ratio is how much of a total of 20?

Accepted Answer

12, because 20 splits into 5 parts of 4 and the larger share is 3 parts.

Question 18

Four pencils cost $2. At the same price per pencil, what do 10 pencils cost?

Accepted Answer

$5, because each pencil costs $0.50.

Question 19

How many meters are in 3 kilometers?

Accepted Answer

3,000, because a kilometer is 1,000 meters.

Question 20

A lab group says 2.5 hours is 25 minutes, because they moved the decimal one place. What is the useful correction?

Accepted Answer

Have them use 60 minutes in an hour: 2 hours is 120 minutes, and half an hour is 30 more, so 2.5 hours is 150 minutes.

Question 21

A sample has a mass of 12 grams and a volume of 4 cubic centimeters. What is its density?

Accepted Answer

3 grams per cubic centimeter, from 12 divided by 4. Mass over volume.

Question 22

Which number is irrational?

Accepted Answer

The square root of 2, which is not a ratio of integers.

Question 23

Which list is in order from least to greatest?

Accepted Answer

−2, −0.5, 0, 1/2

Question 24

What is 5/6 − 1/3?

Accepted Answer

1/2, because 5/6 − 2/6 = 3/6. That reduces to one half.

Question 25

What is (3/4) ÷ (1/2)?

Accepted Answer

3/2, because dividing by 1/2 doubles 3/4.

Question 26

12 is 30% of what number?

Accepted Answer

40, because 12 ÷ 0.30 = 40.

Question 27

$200 is invested at 5% simple interest per year. What is the interest after 2 years?

Accepted Answer

$20, because 200 × 0.05 × 2 = 20.

Question 28

A student says the square root of 9 + 16 is 3 + 4 = 7. Which comparison should the teacher ask for?

Accepted Answer

Compare the square root of 25 with 3 + 4. One result is 5 and the other is 7.

Question 29

A car travels 150 miles in 3 hours at a steady speed. What is that speed?

Accepted Answer

50 miles per hour, from 150 divided by 3. Total distance over total time.

Question 30

Which decimal equals 3/8?

Accepted Answer

0.375, because 3 ÷ 8 = 0.375.

Question 31

Between which two consecutive whole numbers is the square root of 50?

Accepted Answer

7 and 8, because 49 < 50 < 64.

Question 32

A student solves x^2 = 9 and reports only x = 3. What should the teacher ask?

Accepted Answer

What other number has a square of 9?

Question 33

What is 4!, that is, 4 factorial?

Accepted Answer

24, because 4 × 3 × 2 × 1 = 24.

Question 34

Which product is a prime factorization of 60?

Accepted Answer

2^2 × 3 × 5, the factorization into primes.

Question 35

What is the remainder when 23 is divided by 5?

Accepted Answer

3, because 23 = 5 × 4 + 3. The quotient is 4.

Question 36

The sum of an even integer and an odd integer is which kind of integer?

Accepted Answer

Odd, as in 4 + 3 = 7. Even plus odd stays odd.

Question 37

What is the product of two negative numbers?

Accepted Answer

A positive number, as in (−2) × (−3) = 6.

Question 38

What is 10 to the power −2?

Accepted Answer

0.01, because 10^(−2) = 1/100.

Question 39

What is (2^3) × (2^2)?

Accepted Answer

2^5, which equals 32, from adding the exponents.

Question 40

A student says a negative times a negative is negative, because "two negatives make it worse." Which pattern challenges that sentence?

Accepted Answer

List (−2) × 2 = −4, (−2) × 1 = −2, (−2) × 0 = 0, and ask what (−2) × (−1) must be to continue the pattern.

Question 41

What is the solution of 2x + 6 = 14?

Accepted Answer

x = 4, because 2x = 8. Then divide by 2.

Question 42

What is the solution of 3(x − 2) = 9?

Accepted Answer

x = 5, because x − 2 = 3. Then add 2.

Question 43

What is the solution of 2x + 3x = 15?

Accepted Answer

x = 3, because 5x = 15. The coefficients add to 5.

Question 44

A student solves 2x + 4 = 10 by dividing only the 2x and the 10 by 2, leaving x + 4 = 5, then x = 1. Which next step uses the student's own result?

Accepted Answer

Ask the student to substitute x = 1 into 2x + 4 and compare the result with 10.

Question 45

Which pair solves |x − 1| = 3?

Accepted Answer

x = 4 and x = −2, the two numbers 3 units from 1.

Question 46

The system x + y = 10 and x − y = 2 has which solution?

Accepted Answer

x = 6 and y = 4, from adding the equations.

Question 47

What is the slope of the line through (1, 2) and (3, 6)?

Accepted Answer

2, because the rise over the run is 4/2.

Question 48

A student draws a line parallel to y = 2x + 5 and gives it slope 5, using the intercept. What should the teacher point to?

Accepted Answer

Parallel lines share the coefficient of x, so the new line also has slope 2.

Question 49

Which product equals (x + 3)(x − 3)?

Accepted Answer

x^2 − 9, the difference of squares.

Question 50

Which expression equals (x + 2)^2?

Accepted Answer

x^2 + 4x + 4, including the middle term 4x.

Question 51

Which factorization is correct for x^2 − 5x + 6?

Accepted Answer

(x − 2)(x − 3), since −2 and −3 add to −5 and multiply to 6.

Question 52

A student writes (x + 3)^2 = x^2 + 9. Which request best tests that claim?

Accepted Answer

Ask the student to multiply (x + 3)(x + 3), or to compare both sides at x = 1.

Question 53

Which values solve x^2 = 16?

Accepted Answer

x = 4 and x = −4

Question 54

Which values solve x^2 − 4x + 3 = 0?

Accepted Answer

x = 1 and x = 3

Question 55

For x ≠ 1, which expression equals (x^2 − 1)/(x − 1)?

Accepted Answer

x + 1, for every x other than 1.

Question 56

The sequence 3, 7, 11 continues with a constant difference. What is the next term?

Accepted Answer

15, because the common difference is 4.

Question 57

The sequence 2, 6, 18 continues by multiplying by the same number. What is the next term?

Accepted Answer

54, because each term is multiplied by 3.

Question 58

What is the sum of the first four positive integers?

Accepted Answer

10, the total of 1 + 2 + 3 + 4.

Question 59

Which expression equals 2(a + 3) − 4?

Accepted Answer

2a + 2, after combining 6 − 4.

Question 60

A student factors 4x + 8 as 4(x + 8). Which substitution shows the error?

Accepted Answer

Use x = 1: the original value is 12, while 4(1 + 8) = 36.

Question 61

What is x if 2/x = 6/9?

Accepted Answer

x = 3, because 2/x = 2/3. The fractions are equal.

Question 62

For which real x is 1/(x − 2) undefined?

Accepted Answer

x = 2, the input that makes the denominator 0.

Question 63

If f(x) = x^2, what is f(−3)?

Accepted Answer

9, because (−3)^2 = 9. The sign disappears.

Question 64

A student graphs y = (x − 1)^2 + 4 and places the vertex at (0, 4), using only the constant. What correction fits the equation?

Accepted Answer

The vertex is (1, 4), the point where the squared factor is zero and the output is 4.