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GACE Middle Grades Mathematics (4-8) (704) study materials
GACE Middle Grades Mathematics (4-8) (704) study resources

GACE Middle Grades Mathematics (4-8) (704) practice questions

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Question 1Number and Quantity

A student writes the prime factorization of 504. Which expression is correct?

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

Explanation

Dividing 504 repeatedly by the smallest primes gives 504 = 8 × 63 = 2^3 × 3^2 × 7. The prime factorization must use only primes, and the exponents must multiply back to 504 exactly.

Question 2Number and Quantity

Which number is irrational?

  1. −7/3
  2. √81
  3. 0.121212… (repeating)
  4. the square root of 50 Correct answer.

Explanation

A number is irrational when it cannot be written as a ratio of two integers; √50 = 5√2, and √2 is not rational. Repeating decimals and fractions of integers are rational, and √81 equals the integer 9.

Question 3Number and Quantity

A recipe uses 3/4 cup of oil for every 2/3 cup of vinegar. What is the ratio of oil to vinegar in simplest form?

  1. 1/2
  2. 9/8 Correct answer.
  3. 8/9
  4. 6/8

Explanation

Divide the oil amount by the vinegar amount: (3/4) ÷ (2/3) = (3/4) × (3/2) = 9/8. The ratio 9/8 is already in simplest form because 9 and 8 share no common factor.

Question 4Number and Quantity

A store marks a jacket down 25%, then takes an additional 20% off the sale price. What single percent discount is equivalent to the two reductions?

  1. 45%
  2. 40% Correct answer.
  3. 35%
  4. 55%

Explanation

After a 25% markdown the price is 0.75 of the original; the second reduction leaves 0.80 of that, so 0.75 × 0.80 = 0.60. Paying 60% means a single discount of 40%, not the sum of 25% and 20%.

Question 5Number and Quantity

Which value is greatest?

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

Explanation

Evaluating each power gives 2^5 = 32, 4^2 = 16, 3^3 = 27, and 5^2 = 25. The largest of these values is 32, produced by 2^5.

Question 6Number and Quantity

Simplify (2 × 10^5)(3 × 10^−8) and write the result in scientific notation.

  1. 6 × 10^13
  2. 5 × 10^−3
  3. 6 × 10^−4
  4. 6 × 10^−3 Correct answer.

Explanation

Multiply the coefficients: 2 × 3 = 6, and add the exponents: 5 + (−8) = −3. The product is 6 × 10^−3, which is already in scientific notation because 6 lies between 1 and 10.

Question 7Number and Quantity

A bag holds 5 red, 3 blue, and 2 green marbles. One marble is drawn at random. What is the probability that it is blue or green?

  1. 3/5
  2. 2/5
  3. 3/10
  4. 1/2 Correct answer.

Explanation

There are 3 + 2 = 5 favorable marbles out of 5 + 3 + 2 = 10 total, so the probability is 5/10 = 1/2. Because blue and green are mutually exclusive, their probabilities 3/10 and 2/10 simply add.

Question 8Number and Quantity

Which list is ordered from least to greatest?

  1. −2, −2.5, −7/3, 0.5
  2. −7/3, −2.5, −2, 0.5 Correct answer.
  3. −2.5, −7/3, −2, 0.5
  4. 0.5, −2, −2.5, −7/3

Explanation

Since −7/3 ≈ −2.33, the negatives order as −7/3 < −2.5 < −2, and 0.5 is the only positive. Placing them in that order gives the only correct arrangement.

Question 9Number and Quantity

A car travels 150 miles in 2.5 hours. At this constant rate, how many miles does it travel in 4 hours?

  1. 240 Correct answer.
  2. 260
  3. 225
  4. 250

Explanation

The unit rate is 150 ÷ 2.5 = 60 miles per hour. Multiplying by 4 hours gives 240 miles, so the proportional relationship is used consistently.

Question 10Number and Quantity

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

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

Explanation

If the product were rational, dividing by the nonzero rational factor would make the irrational factor rational, a contradiction. Therefore the product of a nonzero rational and an irrational number is always irrational.

Question 11Number and Quantity

Evaluate the expression 3 + 4 × 2^2 − 6 ÷ 3 using the standard order of operations.

  1. 19
  2. 17 Correct answer.
  3. 15
  4. 25

Explanation

First evaluate the exponent: 2^2 = 4. Then multiply and divide left to right: 4 × 4 = 16 and 6 ÷ 3 = 2, giving 3 + 16 − 2 = 17.

Question 12Number and Quantity

A number is multiplied by 6, then 9 is subtracted, and the result is 27. What is the number?

  1. 5
  2. 6 Correct answer.
  3. 3
  4. 4.5

Explanation

Solving 6x − 9 = 27 gives 6x = 36, so x = 6. Checking: 6 × 6 − 9 = 27, which confirms the solution.

Question 13Number and Quantity

Which expression is equivalent to 4(2x − 3) − 2(x − 5)?

  1. 6x − 2 Correct answer.
  2. 6x + 2
  3. 10x − 22
  4. 6x − 22

Explanation

Distribute: 8x − 12 − 2x + 10. Combining like terms gives 6x − 2, and the sign of +10 comes from distributing −2 over −5.

Question 14Number and Quantity

The greatest common factor of 84 and 126 is used to rewrite their sum. Which expression shows 84 + 126 written as the GCF times a sum of two whole numbers with no common factor?

  1. 14(6 + 9)
  2. 6(14 + 21)
  3. 21(4 + 6)
  4. 42(2 + 3) Correct answer.

Explanation

The GCF of 84 and 126 is 42, since 84 = 42 × 2 and 126 = 42 × 3. The remaining addends 2 and 3 share no common factor, so 42(2 + 3) is the fully factored form.

Question 15Number and Quantity

Which value of x satisfies the inequality −3x + 7 < 1?

  1. x > 2 Correct answer.
  2. x < 2
  3. x < −2
  4. x > −2

Explanation

Subtract 7 to get −3x < −6, then divide by −3 and reverse the inequality symbol, giving x > 2. Reversing the symbol is required whenever both sides are divided by a negative number.

Question 16Number and Quantity

A chemist mixes a 20% acid solution with a 50% acid solution to obtain 60 liters of a 30% solution. How many liters of the 50% solution are used?

  1. 24
  2. 20 Correct answer.
  3. 30
  4. 36

Explanation

Let y be the liters of 50% solution; then 0.20(60 − y) + 0.50y = 0.30(60). Solving gives 12 + 0.30y = 18, so y = 20 liters.

Question 17Number and Quantity

What is the value of 5^0 + 5^−2?

  1. 1/25
  2. 25/26
  3. 26/25 Correct answer.
  4. 1/10

Explanation

Any nonzero number to the zero power equals 1, and 5^−2 = 1/5^2 = 1/25. Adding gives 1 + 1/25 = 26/25.

Question 18Number and Quantity

A square has a side length of √18 units. What is its area in simplest radical form?

  1. 9√2
  2. 18 Correct answer.
  3. 3√2
  4. 36

Explanation

The area of a square is the side length squared, so (√18)^2 = 18. Squaring a square root of a nonnegative number returns the radicand.

Question 19Number and Quantity

Which number has a decimal expansion that terminates?

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

Explanation

A fraction in lowest terms has a terminating decimal exactly when its denominator has only 2 and 5 as prime factors. Since 8 = 2^3, 5/8 = 0.625 terminates.

Question 20Number and Quantity

On a map, 1.5 inches represents 40 miles. Two towns are 4.5 inches apart on the map. What is the actual distance between them?

  1. 135 miles
  2. 90 miles
  3. 120 miles Correct answer.
  4. 180 miles

Explanation

Set up the proportion 1.5/40 = 4.5/d. Cross-multiplying gives 1.5d = 180, so d = 120 miles, which is three times the 40-mile unit.

Question 21Number and Quantity

A population grows by 8% each year. Which expression gives the population after 3 years if it starts at P?

  1. P(1.08)^3 Correct answer.
  2. P + 0.08P
  3. P(1.24)
  4. 3P(1.08)

Explanation

Each year the population is multiplied by 1.08, so after three years the factor is applied three times: P(1.08)^3. This is exponential growth rather than repeated addition.

Question 22Number and Quantity

Which statement correctly compares the rational numbers −5/8 and −0.6?

  1. −5/8 = −0.6
  2. −5/8 > −0.6 because −0.625 < −0.6 Correct answer.
  3. −5/8 < −0.6 because −0.625 < −0.6
  4. −5/8 > −0.6 because 5/8 > 0.6

Explanation

Converting gives −5/8 = −0.625, and on the number line −0.625 lies to the left of −0.6. Therefore −5/8 is less than −0.6, which means −5/8 > −0.6 is false; the correct comparison is −5/8 < −0.6.

Question 23Number and Quantity

A student claims that multiplying any number by 10^−3 always makes it smaller. Which counterexample shows the claim is false?

  1. Multiplying 5 by 10^−3 gives 0.005.
  2. Multiplying −5 by 10^−3 gives −0.005. Correct answer.
  3. Multiplying 1000 by 10^−3 gives 1.
  4. Multiplying 0 by 10^−3 gives 0.

Explanation

Multiplying by 10^−3 divides by 1000. For a negative number such as −5, the result −0.005 is greater than −5, so the number became larger, not smaller.

Question 24Number and Quantity

A number n is divisible by both 6 and 10. Which statement must be true about n?

  1. n is divisible by 15 but not necessarily by 30.
  2. n is divisible by 60.
  3. n is divisible by 30. Correct answer.
  4. n is divisible by 12.

Explanation

The least common multiple of 6 and 10 is 30, so any common multiple of 6 and 10 is a multiple of 30. Divisibility by 60 or 12 does not follow, since n could be 30 itself.

Question 25Number and Quantity

Evaluate |−7| − |3 − 9|.

  1. −1
  2. 13
  3. 7
  4. 1 Correct answer.

Explanation

The absolute value of −7 is 7, and 3 − 9 = −6, whose absolute value is 6. Subtracting gives 7 − 6 = 1.

Question 26Number and Quantity

Which expression is equivalent to (2 × 10^5)(3 × 10^−8) written in scientific notation?

  1. 6 × 10^3
  2. 6 × 10^−3 Correct answer.
  3. 6 × 10^−4
  4. 5 × 10^−3

Explanation

Multiply the coefficients: 2 × 3 = 6. Add the exponents: 5 + (−8) = −3. The result is 6 × 10^−3, which is already in scientific notation because 6 is between 1 and 10.

Question 27Number and Quantity

A bag contains 5 red, 3 blue, and 2 green marbles. One marble is drawn at random. What is the probability that it is blue or green?

  1. 2/5
  2. 1/2 Correct answer.
  3. 3/10
  4. 1/5

Explanation

There are 3 + 2 = 5 favorable marbles (blue or green) out of 10 total marbles. The probability is 5/10 = 1/2.

Question 28Number and Quantity

The greatest common factor of 84 and 126 is used to rewrite their sum. Which expression shows 84 + 126 written as the GCF times a sum?

  1. 21(4 + 6)
  2. 14(6 + 9)
  3. 42(2 + 3) Correct answer.
  4. 6(14 + 21)

Explanation

The greatest common factor of 84 and 126 is 42. Then 84 = 42 × 2 and 126 = 42 × 3, so 84 + 126 = 42(2 + 3).

Question 29Number and Quantity

A chemist mixes a 20% acid solution with a 50% acid solution to obtain 60 liters of a 30% solution. How many liters of the 20% solution are used?

  1. 20
  2. 30
  3. 40 Correct answer.
  4. 10

Explanation

Let x be the liters of 20% solution. Then 0.20x + 0.50(60 − x) = 0.30(60). Simplify: 0.20x + 30 − 0.50x = 18, so −0.30x = −12, giving x = 40 liters.

Question 30Number and Quantity

On a map, 1.5 inches represents 40 miles. Two towns are 4.5 inches apart on the map. What is the actual distance between the towns?

  1. 135 miles
  2. 120 miles Correct answer.
  3. 180 miles
  4. 90 miles

Explanation

Set up the proportion 1.5 inches / 40 miles = 4.5 inches / d miles. Cross-multiply: 1.5d = 180, so d = 120 miles.

Question 31Number and Quantity

A store sells notebooks for $2.50 each and pens for $1.25 each. A teacher buys 6 notebooks and 8 pens and pays with a $50 bill. How much change should the teacher receive?

  1. $25.00 Correct answer.
  2. $20.00
  3. $35.00
  4. $30.00

Explanation

The notebooks cost 6 × $2.50 = $15.00 and the pens cost 8 × $1.25 = $10.00, so the total is $25.00. Subtracting from $50.00 gives $25.00 in change.

Question 32Number and Quantity

Which list orders the values 3/4, 0.7, 2/3, and 0.75 from least to greatest?

  1. 2/3, 0.7, 3/4, 0.75
  2. 2/3, 0.7, 0.75, 3/4
  3. 0.7, 2/3, 3/4, 0.75 Correct answer.
  4. 0.7, 3/4, 2/3, 0.75

Explanation

Writing each value as a decimal gives 0.7, 0.666..., 0.75, and 0.75. Since 0.666... < 0.7 < 0.75, and 3/4 equals 0.75 exactly, the order is 0.7, 2/3, 3/4, 0.75.

Question 33Number and Quantity

A recipe calls for 2 1/4 cups of flour. A baker wants to make 1 1/3 batches of the recipe. How many cups of flour are needed?

  1. 2 7/12 cups
  2. 3 1/2 cups
  3. 2 1/3 cups
  4. 3 cups Correct answer.

Explanation

Convert to improper fractions: 2 1/4 = 9/4 and 1 1/3 = 4/3. Multiplying gives 36/12 = 3 cups.

Question 34Number and Quantity

A number written in scientific notation is 4.2 × 10^5. Which standard-form value is equivalent?

  1. 4,200,000
  2. 42,000
  3. 420,000 Correct answer.
  4. 0.000042

Explanation

Multiplying by 10^5 moves the decimal point five places to the right. Starting from 4.2 gives 420,000.

Question 35Number and Quantity

A square has an area of 72 square centimeters. What is the length of one side in simplest radical form?

  1. 2√6 cm
  2. 12√6 cm
  3. 3√8 cm
  4. 6√2 cm Correct answer.

Explanation

The side length is √72. Since 72 = 36 × 2, this simplifies to √36 × √2 = 6√2.

Question 36Number and Quantity

A shirt is priced at $80. The store first applies a 25% discount, then adds 8% sales tax to the discounted price. What is the final cost?

  1. $60.00
  2. $64.80 Correct answer.
  3. $66.40
  4. $86.40

Explanation

A 25% discount leaves 75% of $80, or $60. Adding 8% tax gives $60 × 1.08 = $64.80.

Question 37Number and Quantity

A number n has a prime factorization of 2^3 × 3^2 × 5. How many positive divisors does n have?

  1. 24 Correct answer.
  2. 12
  3. 18
  4. 36

Explanation

For a prime factorization with exponents a, b, c, the number of divisors is (a+1)(b+1)(c+1). Here that is (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24.

Question 38Number and Quantity

A car travels at a constant speed of 45 miles per hour. A second car leaves the same point 2 hours later traveling at 60 miles per hour in the same direction. How many hours after the second car starts will it catch the first car?

  1. 6 hours Correct answer.
  2. 10 hours
  3. 8 hours
  4. 4 hours

Explanation

When the second car starts, the first has a 90-mile lead (45 × 2). The second car closes the gap at 60 − 45 = 15 mph, so it takes 90 ÷ 15 = 6 hours.

Question 39Number and Quantity

A student evaluates 3/4 ÷ 2/5 by writing (3/4) × (2/5) = 6/20 = 3/10. Which statement identifies the error in the student's reasoning and gives the correct quotient?

  1. The student should have found a common denominator first; the correct quotient is 7/20.
  2. The student multiplied by the divisor instead of its reciprocal; the correct quotient is 15/8. Correct answer.
  3. The student divided numerators and denominators separately; the correct quotient is 6/5.
  4. The student simplified 6/20 incorrectly; the correct quotient is 3/5.

Explanation

Dividing by a fraction is equivalent to multiplying by its reciprocal, so 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8. The student multiplied by 2/5, the divisor itself, which is the operation for multiplication rather than division. Since 15/8 is greater than 1 while the divisor 2/5 is less than 1, the quotient must exceed 3/4, confirming the result.

Question 40Number and Quantity

A machine component's length is specified as 4.6 × 10^−3 meters. A second component is 2.3 × 10^−4 meters long. How many times as long is the first component as the second, expressed in scientific notation?

  1. 2.0 × 10^0
  2. 2.0 × 10^1 Correct answer.
  3. 2.0 × 10^−7
  4. 2.0 × 10^−1

Explanation

Divide the lengths: (4.6 × 10^−3) ÷ (2.3 × 10^−4) = (4.6 ÷ 2.3) × 10^(−3 − (−4)) = 2.0 × 10^1. The exponent rule for division subtracts exponents, and −3 − (−4) = 1. Since 20 is the ratio, writing it as 2.0 × 10^1 keeps proper scientific-notation form.

Question 41Algebra and Functions

A cell-phone plan charges a monthly fee of $18 plus $0.10 per minute of use. Which function gives the total monthly cost C(m) for m minutes of use?

  1. C(m) = 0.10(m + 18)
  2. C(m) = 18m + 0.10
  3. C(m) = 0.10m + 18 Correct answer.
  4. C(m) = 18 − 0.10m

Explanation

A linear model has the form C(m) = (rate per minute)·m + (fixed fee). The fixed monthly fee of $18 is the constant term and $0.10 per minute is the coefficient of m, giving C(m) = 0.10m + 18.

Question 42Algebra and Functions

What is the slope of the line passing through the points (−2, 5) and (4, −7)?

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

Explanation

Slope equals the change in y divided by the change in x: (−7 − 5)/(4 − (−2)) = −12/6 = −2. The line falls 12 units while moving 6 units right, so the slope is negative.

Question 43Algebra and Functions

Which expression is equivalent to (3x²y)(−2xy³)?

  1. 6x³y⁴
  2. −6x³y⁴ Correct answer.
  3. −6x²y³
  4. −5x³y⁴

Explanation

Multiply coefficients: 3·(−2) = −6. Add exponents of like bases: x²·x = x³ and y·y³ = y⁴. The product is therefore −6x³y⁴.

Question 44Algebra and Functions

A sequence begins 4, 7, 10, 13, … and continues with the same pattern. Which expression gives the nth term?

  1. 3n + 4
  2. n + 3
  3. 4n + 3
  4. 3n + 1 Correct answer.

Explanation

The sequence is arithmetic with common difference 3, so the nth term has the form 3n + b. Using n = 1 gives 3(1) + b = 4, so b = 1 and the rule is 3n + 1.

Question 45Algebra and Functions

Which statement correctly describes the effect of the transformation y = f(x) + 5 on the graph of y = f(x)?

  1. The graph shifts 5 units to the left.
  2. The graph shifts 5 units up. Correct answer.
  3. The graph stretches vertically by a factor of 5.
  4. The graph shifts 5 units to the right.

Explanation

Adding a constant outside the function changes each output value by that constant while leaving inputs unchanged. Every point (x, y) moves to (x, y + 5), a vertical shift upward by 5 units.

Question 46Algebra and Functions

A line is parallel to y = 4x − 3 and passes through the point (1, 2). What is the equation of this line?

  1. y = −4x + 6
  2. y = 4x + 2
  3. y = 4x − 2 Correct answer.
  4. y = −(1/4)x + 2

Explanation

Parallel lines share the same slope, so the new line has slope 4. Using point-slope form with (1, 2): y − 2 = 4(x − 1), which simplifies to y = 4x − 2.

Question 47Algebra and Functions

If f(x) = 2x² − 3x + 1, what is f(−2)?

  1. 3
  2. −1
  3. 11
  4. 15 Correct answer.

Explanation

Substitute −2 for every x: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 15. Squaring first and handling the negative sign carefully gives the correct value.

Question 48Algebra and Functions

What is the solution set of the inequality −3x + 4 > 19?

  1. x < 5
  2. x > 5
  3. x < −5 Correct answer.
  4. x > −5

Explanation

Subtract 4 from both sides to get −3x > 15, then divide by −3. Dividing by a negative number reverses the inequality, giving x < −5.

Question 49Algebra and Functions

A quadratic function has zeros at x = −3 and x = 5. Which expression could define the function?

  1. f(x) = (x − 3)(x + 5)
  2. f(x) = (x − 3)(x − 5)
  3. f(x) = (x + 3)(x + 5)
  4. f(x) = (x + 3)(x − 5) Correct answer.

Explanation

By the zero product property, a factor (x − r) produces a zero at x = r. To get zeros at −3 and 5, the factors must be (x + 3) and (x − 5).

Question 50Algebra and Functions

The system 2x + 3y = 12 and 4x + 6y = 20 is graphed in the coordinate plane. Which statement about the system is true?

  1. The system has exactly two solutions.
  2. The system has infinitely many solutions.
  3. The system has exactly one solution.
  4. The system has no solution because the lines are parallel. Correct answer.

Explanation

The second equation is 2 times the left side of the first but not 2 times the right side, so the lines have the same slope and different intercepts. Parallel distinct lines never intersect, so the system is inconsistent.