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AEPA Mathematics (NT304) study materials
AEPA Mathematics (NT304) study resources

AEPA Mathematics (NT304) practice questions

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Question 1Mathematical Processes and Number Sense

Which property of the real numbers justifies the statement 7 + (3 + 9) = (7 + 3) + 9?

  1. Commutative property of addition
  2. Distributive property
  3. Associative property of addition Correct answer.
  4. Identity property of addition

Explanation

The associative property states that the grouping of addends can be changed without altering the sum, which is exactly what shifts the parentheses in this equation. The commutative property would instead change the order of the addends, and the distributive property involves multiplication over addition.

Question 2Mathematical Processes and Number Sense

A recipe calls for 3/4 cup of sugar. If a baker makes 2/3 of the recipe, how much sugar is needed?

  1. 1/12 cup
  2. 9/8 cup
  3. 1/2 cup Correct answer.
  4. 5/7 cup

Explanation

Taking a fraction of a quantity means multiplying: (2/3)(3/4) = 6/12 = 1/2. Multiply numerators and denominators, then simplify the resulting fraction.

Question 3Mathematical Processes and Number Sense

What is the value of 2^3 · 2^4 expressed as a single power?

  1. 2^1
  2. 4^7
  3. 2^7 Correct answer.
  4. 2^12

Explanation

When multiplying powers with the same base, keep the base and add the exponents: 2^3 · 2^4 = 2^(3+4) = 2^7. The bases must match for this rule to apply.

Question 4Mathematical Processes and Number Sense

Which number is irrational?

  1. √11 Correct answer.
  2. 22/7
  3. √49
  4. 0.75

Explanation

An irrational number cannot be written as a ratio of integers, and √11 is not a perfect square, so its decimal expansion is nonterminating and nonrepeating. The other choices are rational: a terminating decimal, a perfect square root, and a fraction of integers.

Question 5Mathematical Processes and Number Sense

A jacket originally priced at $80 is discounted by 25%. What is the sale price?

  1. $20
  2. $100
  3. $55
  4. $60 Correct answer.

Explanation

A 25% discount removes 0.25 × 80 = $20 from the original price, leaving 80 − 20 = $60. Equivalently, the buyer pays 75% of $80, which is $60.

Question 6Mathematical Processes and Number Sense

Which expression correctly represents the prime factorization of 84?

  1. 2 · 3 · 14
  2. 2^2 · 3 · 7 Correct answer.
  3. 2^3 · 3 · 7
  4. 4 · 21

Explanation

Dividing 84 by primes gives 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7, so the prime factorization is 2^2 · 3 · 7. Every factor in a prime factorization must itself be prime.

Question 7Mathematical Processes and Number Sense

What is the value of |−7| + |3|?

  1. −10
  2. 10 Correct answer.
  3. −4
  4. 4

Explanation

Absolute value gives the distance from zero, so |−7| = 7 and |3| = 3. Their sum is 7 + 3 = 10.

Question 8Mathematical Processes and Number Sense

Which set of numbers is closed under subtraction?

  1. The natural numbers
  2. The positive rational numbers
  3. The whole numbers
  4. The integers Correct answer.

Explanation

A set is closed under subtraction if subtracting any two elements always yields an element in the same set. The integers are closed under subtraction, but the natural, whole, and positive rational sets all fail when a larger number is subtracted from a smaller one.

Question 9Mathematical Processes and Number Sense

Simplify the expression 3(2x − 4) − 2(x − 5).

  1. 4x − 2 Correct answer.
  2. 4x − 22
  3. 8x − 22
  4. 4x + 2

Explanation

Distribute to get 6x − 12 − 2x + 10, being careful that subtracting −10 gives +10. Combining like terms yields 4x − 2.

Question 10Mathematical Processes and Number Sense

A number is increased by 40% and the result is then decreased by 25%. What is the net percent change from the original number?

  1. A 15% increase
  2. No change
  3. A 10% increase
  4. A 5% increase Correct answer.

Explanation

Multiplying by 1.40 and then by 0.75 gives 1.40 × 0.75 = 1.05, which represents a 5% increase over the original. Percent changes compound multiplicatively, so the two percentages cannot simply be added.

Question 11Mathematical Processes and Number Sense

Which value is greatest?

  1. 0.6
  2. 0.63 Correct answer.
  3. 3/5
  4. 5/8

Explanation

Convert each to a decimal: 5/8 = 0.625, 0.6 = 0.6, 3/5 = 0.6, and 0.63. Comparing hundredths, 0.63 exceeds 0.625, so 0.63 is the largest.

Question 12Mathematical Processes and Number Sense

The number 0.000472 written in scientific notation is:

  1. 4.72 × 10^4
  2. 4.72 × 10^−4 Correct answer.
  3. 4.72 × 10^−3
  4. 47.2 × 10^−5

Explanation

Scientific notation writes a number as a coefficient between 1 and 10 times a power of 10. Moving the decimal 4 places to the right to get 4.72 requires the exponent −4, giving 4.72 × 10^−4.

Question 13Mathematical Processes and Number Sense

A class of 30 students has a ratio of girls to boys of 3:2. How many girls are in the class?

  1. 20
  2. 12
  3. 18 Correct answer.
  4. 15

Explanation

The ratio 3:2 means 5 total parts, so each part equals 30 ÷ 5 = 6 students. Girls make up 3 parts, giving 3 × 6 = 18 girls.

Question 14Mathematical Processes and Number Sense

What is 5.6 × 10^3 divided by 7 × 10^−2?

  1. 8 × 10^4 Correct answer.
  2. 8 × 10^5
  3. 0.8 × 10^5
  4. 8 × 10^1

Explanation

Divide the coefficients and subtract the exponents: 5.6 ÷ 7 = 0.8 and 10^(3 − (−2)) = 10^5, giving 0.8 × 10^5. Rewriting 0.8 as 8 × 10^−1 and combining yields 8 × 10^4.

Question 15Mathematical Processes and Number Sense

Which statement about the set of rational numbers is true?

  1. Every terminating decimal is rational. Correct answer.
  2. The rationals are not closed under addition.
  3. Irrational numbers are a subset of the rationals.
  4. Every rational number is an integer.

Explanation

A terminating decimal can be written as a fraction with a power of 10 in the denominator, so it is rational. Rational numbers include fractions that are not integers, and they are closed under addition.

Question 16Mathematical Processes and Number Sense

Evaluate (2/3) ÷ (4/9).

  1. 2/3
  2. 3/2 Correct answer.
  3. 27/8
  4. 8/27

Explanation

Dividing by a fraction is the same as multiplying by its reciprocal: (2/3) × (9/4) = 18/12 = 3/2. The reciprocal of 4/9 is 9/4.

Question 17Mathematical Processes and Number Sense

A store marks up a wholesale price by 60% and later offers a 40% discount on the marked price. The final price compared with the wholesale price is:

  1. 4% less Correct answer.
  2. 20% less
  3. 20% more
  4. 4% more

Explanation

The final price factor is 1.60 × 0.60 = 0.96, which is 96% of the wholesale price. That represents a 4% decrease from the wholesale price.

Question 18Mathematical Processes and Number Sense

Which expression is equivalent to 4^(3/2)?

  1. 16
  2. 8 Correct answer.
  3. 64
  4. 6

Explanation

A fractional exponent a^(m/n) equals the nth root of a raised to the m, so 4^(3/2) = (√4)^3 = 2^3 = 8. The denominator indicates the root and the numerator indicates the power.

Question 19Mathematical Processes and Number Sense

The least common multiple of 12 and 18 is:

  1. 216
  2. 36 Correct answer.
  3. 72
  4. 6

Explanation

Prime factorizations are 12 = 2^2 · 3 and 18 = 2 · 3^2. The LCM takes the highest power of each prime: 2^2 · 3^2 = 36.

Question 20Mathematical Processes and Number Sense

A number n leaves a remainder of 3 when divided by 7. What is the remainder when 2n is divided by 7?

  1. 2
  2. 3
  3. 6 Correct answer.
  4. 1

Explanation

If n ≡ 3 (mod 7), then 2n ≡ 2 × 3 = 6 (mod 7). Since 6 is less than 7, the remainder is 6.

Question 21Mathematical Processes and Number Sense

Which of the following is a counterexample to the claim that the sum of two irrational numbers is always irrational?

  1. √2 + √3
  2. √2 + (−√2) Correct answer.
  3. π + 1
  4. √5 + √7

Explanation

A counterexample must be a sum of two irrational numbers that is rational. Since √2 and −√2 are both irrational and their sum is 0, which is rational, this disproves the claim.

Question 22Mathematical Processes and Number Sense

If a and b are positive integers with a + b = 20, what is the greatest possible value of ab?

  1. 80
  2. 96
  3. 100 Correct answer.
  4. 99

Explanation

For a fixed sum, the product of two positive numbers is maximized when they are as close as possible. With a = b = 10, the product is 100, and any other split such as 9 and 11 gives 99.

Question 23Mathematical Processes and Number Sense

The repeating decimal 0.4444... is equal to which fraction?

  1. 4/10
  2. 2/5
  3. 44/99
  4. 4/9 Correct answer.

Explanation

Let x = 0.444...; then 10x = 4.444..., and subtracting gives 9x = 4, so x = 4/9. Any single-digit repeating decimal d.ddd... equals d/9.

Question 24Mathematical Processes and Number Sense

If p is a prime number greater than 3, what is the remainder when p^2 is divided by 12?

  1. 3
  2. 6
  3. 0
  4. 1 Correct answer.

Explanation

Any prime greater than 3 is congruent to 1 or 5 modulo 6, so p^2 ≡ 1 (mod 6) and p is odd. Testing p = 5, 7, 11 gives p^2 = 25, 49, 121, each leaving remainder 1 when divided by 12.

Question 25Mathematical Processes and Number Sense

A quantity increases by 50% each year. After how many full years will it first exceed 3 times its original value?

  1. 3 years Correct answer.
  2. 5 years
  3. 4 years
  4. 2 years

Explanation

The growth factor is 1.5 per year, so after n years the value is 1.5^n times the original. After 2 years it is 2.25, after 3 years it is 3.375, which first exceeds 3.

Question 26Mathematical Processes and Number Sense

What is the greatest common divisor of 48 and 180?

  1. 12 Correct answer.
  2. 36
  3. 6
  4. 24

Explanation

Factor each number: 48 = 2^4 · 3 and 180 = 2^2 · 3^2 · 5. The GCD takes the lowest power of each shared prime: 2^2 · 3 = 12.

Question 27Mathematical Processes and Number Sense

A recipe uses 3 cups of flour for every 4 cups of milk. If 18 cups of flour are used, how many cups of milk are needed?

  1. 24 Correct answer.
  2. 20
  3. 27
  4. 13.5

Explanation

Set up the proportion 3/4 = 18/m and cross-multiply to get 3m = 72, so m = 24. Equivalently, the flour amount was multiplied by 6, so the milk amount is 4 × 6 = 24.

Question 28Mathematical Processes and Number Sense

What is the value of 2^5 − 3^2?

  1. 23 Correct answer.
  2. 7
  3. 41
  4. 13

Explanation

Compute each power first: 2^5 = 32 and 3^2 = 9. Subtracting gives 32 − 9 = 23.

Question 29Mathematical Processes and Number Sense

Which list is ordered from least to greatest?

  1. 0.4, 1.8, −1.3, −2.5
  2. −2.5, 0.4, −1.3, 1.8
  3. −2.5, −1.3, 0.4, 1.8 Correct answer.
  4. −1.3, −2.5, 0.4, 1.8

Explanation

On a number line, more negative values are smaller, so −2.5 < −1.3 < 0.4 < 1.8. Only the first list places every value in that ascending order.

Question 30Mathematical Processes and Number Sense

A shirt originally costs $40 and is on sale for 15% off. What is the sale price?

  1. $34 Correct answer.
  2. $25
  3. $6
  4. $37

Explanation

The discount is 0.15 × 40 = $6, so the sale price is 40 − 6 = $34. Equivalently, 85% of $40 is $34.

Question 31Mathematical Processes and Number Sense

Which expression is equivalent to (x^3 · x^4) / x^2?

  1. x^9
  2. x^6
  3. x^5 Correct answer.
  4. x^24

Explanation

When multiplying powers with the same base, add exponents: x^3 · x^4 = x^7. When dividing, subtract exponents: x^7 / x^2 = x^5.

Question 32Mathematical Processes and Number Sense

A bag contains 6 red marbles and 9 blue marbles. What is the ratio of red marbles to the total number of marbles, written in simplest form?

  1. 6:15
  2. 3:2
  3. 2:5 Correct answer.
  4. 2:3

Explanation

There are 6 + 9 = 15 marbles total, so the ratio of red to total is 6:15. Dividing both parts by 3 gives 2:5.

Question 33Mathematical Processes and Number Sense

Which value is the additive inverse of −3/8?

  1. 8/3
  2. −3/8
  3. −8/3
  4. 3/8 Correct answer.

Explanation

The additive inverse of a number is the value that sums with it to zero. Since −3/8 + 3/8 = 0, the additive inverse is 3/8.

Question 34Mathematical Processes and Number Sense

A car travels 150 miles on 5 gallons of gas. At this rate, how many gallons are needed to travel 420 miles?

  1. 14 Correct answer.
  2. 10
  3. 16
  4. 12

Explanation

The unit rate is 150 ÷ 5 = 30 miles per gallon. Dividing 420 by 30 gives 14 gallons.

Question 35Mathematical Processes and Number Sense

What is 3/4 + 5/6 written in simplest form?

  1. 15/24
  2. 8/10
  3. 4/5
  4. 19/12 Correct answer.

Explanation

The least common denominator is 12: 3/4 = 9/12 and 5/6 = 10/12. Adding gives 19/12, which is already in simplest form.

Question 36Mathematical Processes and Number Sense

The number 3,600 is written as 2^a · 3^b · 5^c. What is the value of a + b + c?

  1. 8
  2. 6
  3. 5
  4. 7 Correct answer.

Explanation

Factor 3600 = 36 × 100 = (2^2 · 3^2)(2^2 · 5^2) = 2^4 · 3^2 · 5^2. Thus a = 4, b = 2, c = 2, and the sum is 8.

Question 37Mathematical Processes and Number Sense

A number is multiplied by 0.6. By what percent must the result be increased to return to the original number?

  1. About 166.7%
  2. 40%
  3. 60%
  4. About 66.7% Correct answer.

Explanation

After multiplying by 0.6, the value is 60% of the original. To restore it, divide by 0.6, which is multiplying by 1/0.6 ≈ 1.667, an increase of about 66.7%.

Question 38Mathematical Processes and Number Sense

Which statement about the product of a nonzero rational number and an irrational number is always true?

  1. The product is irrational. Correct answer.
  2. The product is an integer.
  3. The product is rational.
  4. The product is zero.

Explanation

If the product were rational, dividing by the nonzero rational factor would make the irrational factor rational, a contradiction. Hence the product must be irrational.

Question 39Mathematical Processes and Number Sense

What is the value of |−9| − |−4|?

  1. −13
  2. 13
  3. 5 Correct answer.
  4. −5

Explanation

Absolute value gives distance from zero, so |−9| = 9 and |−4| = 4. Subtracting gives 9 − 4 = 5.

Question 40Mathematical Processes and Number Sense

A quantity is decreased by 20% and the result is then increased by 20%. What is the net percent change from the original quantity?

  1. 4% increase
  2. 4% decrease Correct answer.
  3. No change
  4. 40% decrease

Explanation

Multiplying by 0.8 and then by 1.2 gives a factor of 0.96, which is 96% of the original. That corresponds to a 4% decrease.

Question 41Mathematical Processes and Number Sense

What is the value of 5^0 + 5^1 + 5^−1?

  1. 6
  2. 5.2
  3. 11
  4. 6.2 Correct answer.

Explanation

Any nonzero number to the zero power is 1, 5^1 = 5, and 5^−1 = 1/5 = 0.2. The sum is 1 + 5 + 0.2 = 6.2.

Question 42Mathematical Processes and Number Sense

A machine produces 240 parts in 8 hours. Working at the same rate, how many parts does it produce in 45 minutes?

  1. 30
  2. 18
  3. 22.5 Correct answer.
  4. 24

Explanation

The rate is 240 ÷ 8 = 30 parts per hour, which is 0.5 parts per minute. In 45 minutes it produces 45 × 0.5 = 22.5 parts.

Question 43Mathematical Processes and Number Sense

Which expression has the same value as 0.00036 written in scientific notation?

  1. 36 × 10^−3
  2. 3.6 × 10^4
  3. 3.6 × 10^−4 Correct answer.
  4. 3.6 × 10^−5

Explanation

Scientific notation writes a number as a value between 1 and 10 times a power of 10. Moving the decimal 4 places right gives 3.6 × 10^−4.

Question 44Mathematical Processes and Number Sense

The sum of two integers is odd. Which statement must be true?

  1. Both integers are negative.
  2. One integer is even and the other is odd. Correct answer.
  3. Both integers are odd.
  4. Both integers are even.

Explanation

An odd sum occurs exactly when the addends have opposite parity: even + odd = odd. Same-parity pairs always produce an even sum.

Question 45Mathematical Processes and Number Sense

What is the value of (2/5) ÷ (4/15)?

  1. 3/2 Correct answer.
  2. 2/3
  3. 8/75
  4. 6/5

Explanation

Dividing by a fraction multiplies by its reciprocal: (2/5) × (15/4) = 30/20 = 3/2.

Question 46Mathematical Processes and Number Sense

A number n leaves a remainder of 5 when divided by 8. What is the remainder when 3n is divided by 8?

  1. 5
  2. 3
  3. 7 Correct answer.
  4. 1

Explanation

Since n ≡ 5 (mod 8), multiplying by 3 gives 3n ≡ 15 ≡ 7 (mod 8). The remainder is 7.

Question 47Mathematical Processes and Number Sense

Which value of x is a counterexample to the claim that if x^2 is an integer, then x is an integer?

  1. x = 2.5
  2. x = √3 Correct answer.
  3. x = √4
  4. x = 0

Explanation

For x = √3, x^2 = 3 is an integer, yet √3 is not an integer. This satisfies the hypothesis while failing the conclusion, so it is a counterexample.

Question 48Mathematical Processes and Number Sense

A quantity grows by 25% each year. After how many full years will it first exceed twice its original value?

  1. 3
  2. 5
  3. 2
  4. 4 Correct answer.

Explanation

The growth factor is 1.25 per year. After 3 years the factor is 1.25^3 ≈ 1.953, still below 2; after 4 years it is about 2.441, which exceeds 2.

Question 49Mathematical Processes and Number Sense

If a and b are positive integers with a + b = 30, what is the greatest possible value of a · b?

  1. 224
  2. 200
  3. 225 Correct answer.
  4. 216

Explanation

For a fixed sum, the product of two positive numbers is maximized when they are as close as possible. Here a = b = 15, giving 15 × 15 = 225.

Question 50Mathematical Processes and Number Sense

A store advertises that the price of every item is reduced by 40%, and then an additional 25% is taken off the already-reduced price. A customer says the total discount is 65% off the original price. Which statement is accurate?

  1. The total discount is 65% off the original price, because the two percentages are added.
  2. The total discount is 45% off the original price, because 25% of 40% is subtracted from 65%.
  3. The total discount is 55% off the original price, because the discounts compound multiplicatively. Correct answer.
  4. The total discount is 60% off the original price, because the second discount applies to the original price.

Explanation

Successive percent discounts multiply the remaining price factors: 0.60 × 0.75 = 0.45, so the customer pays 45% of the original price, a 55% discount. Adding 40% and 25% incorrectly treats the second discount as taken on the original price rather than the reduced price.