Middle School Math Quiz: Divide Rational Numbers

20 questions · exam conditions

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Divide Rational NumbersQuestion 1 of 20

The temperature dropped $2\frac{1}{4}$ degrees over $\frac{3}{4}$ hour. At this rate, how much would the temperature change in $1$ hour?

Drop

3

degrees per hour

Drop

\frac{27}{16}

degrees per hour

Drop

\frac{9}{16}

degrees per hour

Rise

3

degrees per hour

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Middle School Math Quiz

Middle School Math Quiz: Divide Rational Numbers

Practice Divide Rational Numbers in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Divide Rational Numbers, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The temperature dropped $2\frac{1}{4}$ degrees over $\frac{3}{4}$ hour. At this rate, how much would the temperature change in $1$ hour?

Drop $3$ degrees per hour (correct answer)
Drop $\frac{27}{16}$ degrees per hour
Drop $\frac{9}{16}$ degrees per hour
Rise $3$ degrees per hour

Explanation: Rate =

\frac{-2\frac{1}{4}}{\frac{3}{4}} = \frac{-\frac{9}{4}}{\frac{3}{4}} = -\frac{9}{4} \times \frac{4}{3} = -3

degrees per hour, meaning a 3-degree drop. Choice B incorrectly calculates

\frac{9}{4} \times \frac{3}{4}

. Choice C uses

\frac{3}{4} \div \frac{9}{4}

. Choice D gets the magnitude right but wrong sign interpretation.

Question 2

A submarine starts at sea level and descends $150$ meters. It then ascends $\frac{2}{5}$ of the distance it descended. What is the submarine's final depth below sea level?

$90$ meters below sea level (correct answer)
$60$ meters below sea level
$210$ meters below sea level
$30$ meters below sea level

Explanation: The submarine descends 150 meters, so it's at -150 meters. It ascends

\frac{2}{5} \times 150 = 60

meters. Final position:

-150 + 60 = -90

meters, which is 90 meters below sea level. Choice B gives the ascent distance, not final depth. Choice C adds instead of subtracts the ascent. Choice D gives

\frac{1}{5}

of the descent instead of the correct calculation.

Question 3

Which statement about dividing rational numbers is always true?

When dividing two negative rational numbers, the result is always negative
When dividing a positive by a negative rational number, the result has the same sign as the dividend
The quotient $\frac{a}{b}$ where $a$ and $b$ are integers with $b \neq 0$ is always a rational number (correct answer)
Division of rational numbers is only defined when both numbers are positive integers

Explanation: By definition, any quotient of integers (with non-zero divisor) is a rational number. Choice A is false: dividing two negatives gives a positive result. Choice B is false: positive ÷ negative = negative, which has the opposite sign of the dividend. Choice D is false: division works for all rational numbers with non-zero divisors.

Question 4

A science lab's temperature changed by $-45^\circ\text{C}$ over $9$ hours (negative means it decreased). What was the temperature change per hour? Compute $(-45)\div 9$ .

$-5$ (correct answer)
$-54$
$\dfrac{9}{45}$
$5$

Explanation: This question tests dividing rational numbers applying sign rules, neg ÷ pos = neg, in temperature change rate context. Negative ÷ positive = negative, (-45) ÷ 9 = -5. Quotient is rational: -45/9 = -5. Example: negative change ÷ time = negative rate, (-45) ÷ 9 = -5°C/h (decrease of 5°C/h). Correct: (-45) ÷ 9 = -5. Error: ignoring sign, 45 ÷ 9 = 5 (missing negative). Dividing: different signs → negative, magnitudes 45 ÷ 9 = 5, apply sign -5; context: negative rates indicate decrease.

Question 5

Which statement about division is true?

$5\div 0$ is undefined (correct answer)
$5\div 0=5$
$5\div 0=\infty$
$5\div 0=0$

Explanation: This question tests that division by zero is undefined in rational numbers, as q≠0 for p/q. Division rules require divisor ≠0; 5 ÷ 0 undefined. No sign issue, but fundamental property. Example: cannot divide by zero, no number times 0 gives 5. Correct: 5 ÷ 0 is undefined. Error: claiming 5 ÷ 0 = 0, 5, or ∞. Mistakes: division by zero claimed valid, like saying it's 0 or infinity.

Question 6

Calculate the quotient and apply sign rules: $15\div(-3)=$

$-12$
$-5$ (correct answer)
$5$
$12$

Explanation: This question tests dividing rational numbers by applying sign rules, specifically positive ÷ negative = negative, and recognizing equivalent forms of negative quotients. Sign rules for division are the same as multiplication: positive ÷ negative = negative, as in 15 ÷ (-3) = -5. The quotient of integers is rational, like 15 ÷ (-3) = -5, which can be written as - (15/3), (-15)/3, or 15/(-3), all equaling -5. For example, 15 ÷ (-3) involves positive divided by negative, giving -5. The correct division is 15 ÷ (-3) = -5, applying the proper sign. A mistake might be treating it as positive, like 5, forgetting the different signs rule. To divide, determine the sign (different signs → negative), divide magnitudes (15 ÷ 3 = 5), apply the sign to get -5, and note equivalent forms for negative quotients.

Question 7

A hiker's elevation changed by $-80$ meters over 4 hours (negative means going down). What was the average rate of change in elevation per hour, $(-80)\div 4$ ?

$-320$ meters per hour
$-\dfrac{1}{20}$ meters per hour
$-20$ meters per hour (correct answer)
$20$ meters per hour

Explanation: This question tests dividing rational numbers in a rate context, applying sign rules where negative ÷ positive = negative, representing downward change. Sign rules include negative ÷ positive = negative, as in (-80) ÷ 4 = -20. The quotient is rational, -20 meters per hour, meaning an average descent of 20 meters per hour. For example, a negative change of -80 meters over 4 hours gives (-80) ÷ 4 = -20, a negative rate. The correct rate is -20 meters per hour, as negative total divided by positive time yields negative. An error could be calculating as positive 20, ignoring the sign, or misdividing like -320. To find the rate, determine the sign (different signs → negative), divide magnitudes (80 ÷ 4 = 20), apply the sign for -20, and interpret as downward elevation change per hour.

Question 8

A student says, "Because $15 \div 4$ is not an integer, it is not rational." Which value shows the correct quotient and why the statement is false?

$15 \div 4 = 4$ , which is rational
$15 \div 4 = \dfrac{15}{0}$ , which is rational
$15 \div 4 = \dfrac{4}{15}$ , which is rational
$15 \div 4 = \dfrac{15}{4}$ , which is rational (correct answer)

Explanation: This question tests that quotient of integers

(p/q, q \neq 0)

is rational, even if not integer, like

15 \div 4 = \frac{15}{4} = 3.75

. Positive ÷ positive = positive,

15 \div 4 = \frac{15}{4}

. Rational includes fractions or decimals. Example:

15 \div 4 = \frac{15}{4}

, rational (terminating decimal 3.75). Correct:

15 \div 4 = \frac{15}{4}

, rational, falsifying 'not integer so not rational'. Error: wrong quotient like

\frac{4}{15}

, or division by zero. Express as fraction if needed,

15 \div 4 = \frac{15}{4}

rational; mistakes: division by zero claimed valid, or reciprocal wrong.

Question 9

Divide and simplify: $\left(-\dfrac{3}{5}\right)\div\left(\dfrac{6}{10}\right)$ .

$-1$ (correct answer)
$-\dfrac{9}{25}$
$-\dfrac{1}{4}$
$1$

Explanation: This question tests dividing rational numbers applying sign rules (negative ÷ positive = negative) and fraction division by reciprocal. Sign rules: negative ÷ positive = negative; ( -3/5 ) ÷ (6/10) = ( -3/5 ) × (10/6) = -30/30 = -1. For example, negative divided by positive yields negative, simplifying to -1. The correct division is -1, after simplifying -30/30. An error might be positives only to get 1, or wrong reciprocal like -3/5 × 6/10 = -18/50 = -9/25. Fraction division: multiply by reciprocal, apply sign rules to result. Simplify fractions: 10/6=5/3, but full calc: -3/5 * 5/3 = -3/3 = -1 after canceling.

Question 10

Which set shows three equivalent ways to write the same negative quotient and the correct value? (Remember $-(p/q)=(-p)/q=p/(-q)$ for $q \neq 0$ .)

$-\dfrac{12}{3}=\dfrac{12}{3}=\dfrac{12}{-3}=-4$
$-\dfrac{12}{3}=\dfrac{-12}{3}=\dfrac{12}{-3}=-4$ (correct answer)
$-\dfrac{12}{3}=\dfrac{-12}{3}\ne\dfrac{12}{-3}$ , so the value is not the same
$-\dfrac{12}{3}=\dfrac{-12}{3}=\dfrac{12}{-3}=4$

Explanation: This question tests understanding equivalent forms of negative quotients, where

-(p/q) = (-p)/q = p/(-q)

, all equal for

q \neq 0

. Negative quotients have three forms:

-(12/3) = (-12)/3 = 12/(-3) = -4

. These are rational numbers expressing the same value. For example, all forms simplify to

-4

, showing equivalence. The correct set is

-(12/3) = (-12)/3 = 12/(-3) = -4

, with the right value. An error is claiming they are not equal or assigning positive

4

. Remember, negative can be in front, numerator, or denominator, all equivalent for negative rational numbers.

Question 11

Divide and simplify: $\dfrac{2}{3}\div\dfrac{1}{4}$ .

$\dfrac{3}{8}$
$\dfrac{1}{6}$
$\dfrac{8}{3}$ (correct answer)
$\dfrac{2}{12}$

Explanation: This question tests dividing rational numbers applying sign rules (though both positive here) and fraction division by multiplying by reciprocal. Sign rules for division apply, but here positive÷positive=positive; quotient of fractions is rational:

(\frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3})

. For example,

(\frac{2}{3} \div \frac{1}{4})

means how many

1/4

2/3

, equaling

8/3

by multiplying by reciprocal. The correct division is

(\frac{2}{3}) \times (\frac{4}{1}) = \frac{8}{3}

, simplified. A common error is using wrong reciprocal, like

(\frac{2}{3}) \times (\frac{1}{4}) = \frac{2}{12} = \frac{1}{6}

, or multiplying numerators and denominators incorrectly to get

\frac{3}{8}

. Fraction division: multiply by reciprocal

(a/b \div c/d = (a/b) \times (d/c))

, apply sign rules (positive here). Mistakes include reciprocal errors or sign mishandling, though not applicable here.

Question 12

A science club tracked a total temperature change of $-45^\circ\text{C}$ over $9$ days. What was the average change per day? (Compute $(-45)\div 9$ .)

$5$
$-\dfrac{1}{5}$
$-54$
$-5$ (correct answer)

Explanation: This question tests dividing rational numbers applying sign rules (negative ÷ positive = negative) and contexts like average change. Sign rules: negative ÷ positive = negative; -45 ÷ 9 = -5. For example, total -45°C over 9 days averages -5°C per day. The correct division is -45 ÷ 9 = -5. An error might be positive 5, or -1/5, or miscalculating magnitude. To divide: (1) sign (different → negative), (2) magnitudes (45 ÷ 9 = 5), (3) apply -5. In averages, negative total ÷ positive units = negative average.

Question 13

A hiker's elevation change is $-80$ meters over $4$ hours (negative means going downhill). What is the average rate of change in meters per hour? Compute $(-80)\div 4$ .

$\dfrac{1}{20}$
$-20$ (correct answer)
$-320$
$20$

Explanation: This question tests dividing rational numbers applying sign rules, where negative ÷ positive = negative, and interpreting rates like elevation change. Sign rules: negative ÷ positive = negative, as in (-80) ÷ 4 = -20. Quotient is rational: -80/4 = -20. Example: negative change ÷ time = negative rate, (-80) ÷ 4 = -20 m/h (descending 20 m/h). Correct: (-80) ÷ 4 = -20. Error: ignoring sign, 80 ÷ 4 = 20 (missing negative). Dividing: different signs → negative, magnitudes 80 ÷ 4 = 20, apply sign -20; context: negative rates indicate decrease.

Question 14

A student has a total of $-60$ points after 12 rounds of a game (negative means they lost points). If the points were lost equally each round, what is the quotient $(-60)\div 12$ , and what does it mean?

$60$ points per round (they gained 60 each round)
$5$ points per round (they gained 5 each round)
$-72$ points per round (they lost 72 each round)
$-5$ points per round (they lost 5 each round) (correct answer)

Explanation: This question tests dividing rational numbers by applying sign rules, where a negative divided by a positive yields a negative quotient, and understanding real-world contexts like sharing losses equally. Sign rules for division mirror multiplication: negative ÷ positive = negative, as in (-60) ÷ 12 = -5. The quotient of integers like -60 ÷ 12 is a rational number, here -5, representing the average loss per round. For example, sharing a debt of -

60 among 12 people gives (-60) ÷ 12 = -

5 per person, meaning each owes $5. The correct quotient is -5, meaning they lost 5 points each round, as the total negative points divided by positive rounds results in a negative rate. A common error is interpreting the quotient as positive, like 5 points gained, ignoring the sign rule for different signs. To divide, determine the sign (different signs → negative), divide magnitudes (60 ÷ 12 = 5), apply the sign to get -5, and interpret in context as a loss per round.

Question 15

A class recorded a total change of $-45$ pages (negative means pages were removed from a display) over 9 days. If the change was the same each day, what is $(-45)\div 9$ ?

$-4$
$-\dfrac{1}{5}$
$5$
$-5$ (correct answer)

Explanation: This question tests dividing rational numbers in context, with negative ÷ positive = negative, for equal daily changes. Sign rules: negative ÷ positive = negative, as in (-45) ÷ 9 = -5. The quotient is rational, -5 pages per day, meaning removal rate. For example, total -45 pages over 9 days gives (-45) ÷ 9 = -5 per day. The correct value is -5, as different signs yield negative. A mistake is positive 5, ignoring signs, or wrong division like -4. To compute, determine sign (different → negative), divide magnitudes (45 ÷ 9 = 5), apply sign for -5, interpret as daily removal.

Question 16

A class recorded a total change of $-45$ pages (negative means pages were removed from a display) over 9 days. If the change was the same each day, what is $(-45)\div 9$ ?

$5$
$-4$
$-5$ (correct answer)
$-\dfrac{1}{5}$

Question 17

The expression $\frac{-a}{b}$ is equivalent to which of the following when $a$ and $b$ are integers and $b \neq 0$ ?

$\frac{a}{-b}$ but not $-\frac{a}{b}$
$-\frac{a}{b}$ but not $\frac{a}{-b}$
Both $\frac{a}{-b}$ and $-\frac{a}{b}$ (correct answer)
Neither $\frac{a}{-b}$ nor $-\frac{a}{b}$

Explanation: From the property

-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}

, we know that

\frac{-a}{b} = -\frac{a}{b} = \frac{a}{-b}

. All three forms are equivalent. Choices A and B incorrectly suggest only one equivalence holds. Choice D incorrectly rejects both equivalences.

Question 18

Calculate the quotient and apply sign rules: $15\div(-3)=$

$5$
$12$
$-12$
$-5$ (correct answer)

Question 19

A student says, " $5\div 0=0$ ." Which statement is correct?

$5\div 0=5$ because dividing keeps the number the same.
The student is correct because any number divided by $0$ equals $0$ .
The student is correct because $0$ goes into $5$ zero times.
$5\div 0$ is undefined (division by zero is not allowed). (correct answer)

Explanation: This question tests understanding that division by zero is undefined in rational numbers. Sign rules don't apply here since divisor is zero; quotient undefined as no number times 0 equals 5. For example, 5 ÷ 0 can't be defined, unlike 5 ÷ 1 = 5. The correct statement is 5 ÷ 0 is undefined. Student errors include claiming 5 ÷ 0 = 0 (wrong, as 0*0=0 not 5) or =5 (wrong). Dividing: can't divide by zero, as it leads to contradictions. Mistakes: thinking any number ÷0=0 or unchanged.

Question 20

Any integer divided by a nonzero integer is a rational number. What is $15\div 4$ , written as an improper fraction (already in simplest form, since no further simplification is possible) and as a terminating decimal?

$\dfrac{15}{4}$ and $3.25$
$\dfrac{3}{5}$ and $0.6$
$\dfrac{4}{15}$ and $0.26\overline{6}$
$\dfrac{15}{4}$ and $3.75$ (correct answer)

Explanation: Dividing 15 by 4 gives the improper fraction 15/4, which is already in simplest form since 15 and 4 share no common factors other than 1. Converting to a decimal, 15/4 = 3.75, a terminating decimal. Choice A shows the correct fraction but an incorrect decimal, 3.25, which doesn't match 15 divided by 4. Choice B shows an unrelated fraction, 3/5, not the result of 15 divided by 4. Choice C flips the fraction upside down, giving 4/15 instead of 15/4, along with its corresponding repeating decimal.