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7th Grade Math Quiz

7th Grade Math Quiz: Divide Rational Numbers

Practice Divide Rational Numbers in 7th Grade Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 20

0 of 20 answered

Which expression is equivalent to −15−3\frac{-15}{-3}−3−15​?

Select an answer to continue

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 7th Grade 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

Which expression is equivalent to −15−3\frac{-15}{-3}−3−15​?

  1. −5-5−5
  2. 315=15\frac{3}{15}=\frac{1}{5}153​=51​
  3. −15-\frac{1}{5}−51​
  4. 555 (correct answer)

Explanation: Dividing two negative numbers gives a positive result: -15 divided by -3 equals 5, matching choice D. Choice A treats a negative divided by a negative as still negative, missing the sign rule. Choice B inverts the fraction (dividing the wrong way) instead of computing the quotient. Choice C both inverts the fraction and keeps an incorrect negative sign.

Question 2

If x=−123x = \frac{-12}{3}x=3−12​ and y=−12−3y = \frac{-12}{-3}y=−3−12​, what is xy\frac{x}{y}yx​?

  1. 111
  2. −1-1−1 (correct answer)
  3. 14\frac{1}{4}41​
  4. −14-\frac{1}{4}−41​

Explanation: x=−123=−4x = \frac{-12}{3} = -4x=3−12​=−4 and y=−12−3=4y = \frac{-12}{-3} = 4y=−3−12​=4. Therefore xy=−44=−1\frac{x}{y} = \frac{-4}{4} = -1yx​=4−4​=−1. Choice A results from incorrectly thinking x=yx = yx=y. Choice C comes from computing 1∣x∣\frac{1}{|x|}∣x∣1​ instead of xy\frac{x}{y}yx​. Choice D comes from computing 1x\frac{1}{x}x1​ instead of xy\frac{x}{y}yx​.

Question 3

The temperature dropped 2142\frac{1}{4}241​ degrees over 34\frac{3}{4}43​ hour. At this rate, how much would the temperature change in 111 hour?

  1. Drop 333 degrees per hour (correct answer)
  2. Drop 2716\frac{27}{16}1627​ degrees per hour
  3. Drop 916\frac{9}{16}169​ degrees per hour
  4. Rise 333 degrees per hour

Explanation: Rate = −21434=−9434=−94×43=−3\frac{-2\frac{1}{4}}{\frac{3}{4}} = \frac{-\frac{9}{4}}{\frac{3}{4}} = -\frac{9}{4} \times \frac{4}{3} = -343​−241​​=43​−49​​=−49​×34​=−3 degrees per hour, meaning a 3-degree drop. Choice B incorrectly calculates 94×34\frac{9}{4} \times \frac{3}{4}49​×43​. Choice C uses 34÷94\frac{3}{4} \div \frac{9}{4}43​÷49​. Choice D gets the magnitude right but wrong sign interpretation.

Question 4

A submarine starts at sea level and descends 150150150 meters. It then ascends 25\frac{2}{5}52​ of the distance it descended. What is the submarine's final depth below sea level?

  1. 909090 meters below sea level (correct answer)
  2. 606060 meters below sea level
  3. 210210210 meters below sea level
  4. 303030 meters below sea level

Explanation: The submarine descends 150 meters, so it's at -150 meters. It ascends 25×150=60\frac{2}{5} \times 150 = 6052​×150=60 meters. Final position: −150+60=−90-150 + 60 = -90−150+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 15\frac{1}{5}51​ of the descent instead of the correct calculation.

Question 5

Which statement about dividing rational numbers is always true?

  1. When dividing two negative rational numbers, the result is always negative
  2. When dividing a positive by a negative rational number, the result has the same sign as the dividend
  3. The quotient ab\frac{a}{b}ba​ where aaa and bbb are integers with b≠0b \neq 0b=0 is always a rational number (correct answer)
  4. 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 6

The expression −ab\frac{-a}{b}b−a​ is equivalent to which of the following when aaa and bbb are integers and b≠0b \neq 0b=0?

  1. a−b\frac{a}{-b}−ba​ but not −ab-\frac{a}{b}−ba​
  2. −ab-\frac{a}{b}−ba​ but not a−b\frac{a}{-b}−ba​
  3. Both a−b\frac{a}{-b}−ba​ and −ab-\frac{a}{b}−ba​ (correct answer)
  4. Neither a−b\frac{a}{-b}−ba​ nor −ab-\frac{a}{b}−ba​

Explanation: From the property −pq=−pq=p−q-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}−qp​=q−p​=−qp​, we know that −ab=−ab=a−b\frac{-a}{b} = -\frac{a}{b} = \frac{a}{-b}b−a​=−ba​=−ba​. All three forms are equivalent. Choices A and B incorrectly suggest only one equivalence holds. Choice D incorrectly rejects both equivalences.

Question 7

Evaluate: −23−49+12−16\frac{-\frac{2}{3}}{-\frac{4}{9}} + \frac{\frac{1}{2}}{-\frac{1}{6}}−94​−32​​+−61​21​​

  1. 32\frac{3}{2}23​
  2. −32-\frac{3}{2}−23​ (correct answer)
  3. 92\frac{9}{2}29​
  4. −92-\frac{9}{2}−29​

Explanation: −23−49=23×94=1812=32\frac{-\frac{2}{3}}{-\frac{4}{9}} = \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}−94​−32​​=32​×49​=1218​=23​. 12−16=12×−61=−3\frac{\frac{1}{2}}{-\frac{1}{6}} = \frac{1}{2} \times \frac{-6}{1} = -3−61​21​​=21​×1−6​=−3. So 32+(−3)=32−62=−32\frac{3}{2} + (-3) = \frac{3}{2} - \frac{6}{2} = -\frac{3}{2}23​+(−3)=23​−26​=−23​. Choice A forgets the second term is negative. Choice C and D result from sign errors in the division steps.

Question 8

Calculate the quotient: 15÷(−3)15\div(-3)15÷(−3).

  1. −5-5−5 (correct answer)
  2. −8-8−8
  3. −15-\dfrac{1}{5}−51​
  4. 555

Explanation: This question tests dividing rational numbers applying sign rules (positive ÷ negative = negative) and understanding -(p/q) = (-p)/q = p/(-q) equivalence for negative quotients. Sign rules for division (same as multiplication): positive ÷ positive = positive (20 ÷ 4 = 5), negative ÷ negative = positive ((-20) ÷ (-4) = 5), positive ÷ negative = negative (20 ÷ (-4) = -5), negative ÷ positive = negative ((-20) ÷ 4 = -5). For example, 15 ÷ (-3) is positive divided by negative, so negative: -5. The correct division with proper sign is 15 ÷ (-3) = -5. A common error is treating positive ÷ negative as positive, getting 5, or miscalculating as -1/5. To divide: (1) determine sign (different signs → negative), (2) divide magnitudes (15 ÷ 3 = 5), (3) apply sign: -5, and express as fraction if needed (like 15/(-3) = -5). Negative quotients can be written three ways: -(15/3), (-15)/3, 15/(-3), all equal to -5.

Question 9

Any integer divided by a nonzero integer is a rational number. What is 15÷415\div 415÷4 written as a fraction in simplest form, and as a terminating decimal?​

  1. 154\dfrac{15}{4}415​ and 3.753.753.75 (correct answer)
  2. 35\dfrac{3}{5}53​ and 0.60.60.6
  3. 415\dfrac{4}{15}154​ and 0.266‾0.26\overline{6}0.266
  4. 154\dfrac{15}{4}415​ and 3.253.253.25

Explanation: This question tests that dividing integers (nonzero divisor) yields a rational number, expressible as fraction or decimal. Positive ÷ positive = positive, like 15 ÷ 4 = 15/4 = 3.75. The quotient is rational, fraction 15/4 in simplest form, decimal 3.75 terminating. For example, 15 ÷ 4 = 15/4 = 3.75. The correct forms are 15/4 and 3.75. An error is wrong fraction like 4/15 or incorrect decimal 3.25. To express, write as improper fraction, simplify, and convert to decimal if terminating or repeating.

Question 10

Which statement about division is correct?

  1. 5÷0=55\div 0 = 55÷0=5
  2. 5÷05\div 05÷0 is undefined (correct answer)
  3. 5÷0=05\div 0 = 05÷0=0
  4. 5÷0=∞5\div 0 = \infty5÷0=∞

Explanation: This question tests understanding that division by zero is undefined, a key rule in rational numbers. Division rules prohibit q = 0, so 5 ÷ 0 is undefined. No rational number satisfies it, unlike valid divisions like 5 ÷ 1 = 5. For example, you can't divide by zero in any context. The correct statement is 5 ÷ 0 is undefined. A mistake is claiming it's 0, 5, or infinity. Remember, for rational quotients, divisor must be nonzero; zero denominator is invalid.

Question 11

Which set of expressions are all equal to −4-4−4? (Use the idea that −(p/q)=(−p)/q=p/(−q)-(p/q)=(-p)/q=p/(-q)−(p/q)=(−p)/q=p/(−q).)

  1. −123, 123, 12−3-\dfrac{12}{3},\ \dfrac{12}{3},\ \dfrac{12}{-3}−312​, 312​, −312​
  2. −(123), 123, −12−3-\left(\dfrac{12}{3}\right),\ \dfrac{12}{3},\ \dfrac{-12}{-3}−(312​), 312​, −3−12​
  3. −123, −123, 12−3-\dfrac{12}{3},\ \dfrac{-12}{3},\ \dfrac{12}{-3}−312​, 3−12​, −312​ (correct answer)
  4. −123, 12−3, −12−3\dfrac{-12}{3},\ \dfrac{12}{-3},\ \dfrac{-12}{-3}3−12​, −312​, −3−12​

Explanation: This question tests understanding -(p/q) = (-p)/q = p/(-q) equivalence for negative quotients in rational numbers. Negative quotient three equivalent forms: -(12/3) = (-12)/3 = 12/(-3) = -4 (negative in front, numerator, or denominator all equal -4). Example: all forms simplify to -4, as 12/3=4, so negative versions are -4. Correct set: - (12/3), (-12)/3, 12/(-3), all = -4. Error: claiming forms different, like including positive 12/3 as equal when it's not. Equivalent forms: negative quotient written three ways, all equal -4. Mistakes: forms not recognized as equivalent, like thinking (-12)/(-3) = -4 when it's +4.

Question 12

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

  1. −123=123=12−3=−4-\dfrac{12}{3}=\dfrac{12}{3}=\dfrac{12}{-3}=-4−312​=312​=−312​=−4
  2. −123=−123=12−3=−4-\dfrac{12}{3}=\dfrac{-12}{3}=\dfrac{12}{-3}=-4−312​=3−12​=−312​=−4 (correct answer)
  3. −123=−123≠12−3-\dfrac{12}{3}=\dfrac{-12}{3}\ne\dfrac{12}{-3}−312​=3−12​=−312​, so the value is not the same
  4. −123=−123=12−3=4-\dfrac{12}{3}=\dfrac{-12}{3}=\dfrac{12}{-3}=4−312​=3−12​=−312​=4

Explanation: This question tests understanding equivalent forms of negative quotients, where −(p/q)=(−p)/q=p/(−q)-(p/q) = (-p)/q = p/(-q)−(p/q)=(−p)/q=p/(−q), all equal for q≠0q \neq 0q=0. Negative quotients have three forms: −(12/3)=(−12)/3=12/(−3)=−4-(12/3) = (-12)/3 = 12/(-3) = -4−(12/3)=(−12)/3=12/(−3)=−4. These are rational numbers expressing the same value. For example, all forms simplify to −4-4−4, showing equivalence. The correct set is −(12/3)=(−12)/3=12/(−3)=−4-(12/3) = (-12)/3 = 12/(-3) = -4−(12/3)=(−12)/3=12/(−3)=−4, with the right value. An error is claiming they are not equal or assigning positive 444. Remember, negative can be in front, numerator, or denominator, all equivalent for negative rational numbers.

Question 13

A science lab’s temperature changed by −45∘C-45^\circ\text{C}−45∘C over 999 hours (negative means it decreased). What was the temperature change per hour? Compute (−45)÷9(-45)\div 9(−45)÷9.

  1. −5-5−5 (correct answer)
  2. −54-54−54
  3. 945\dfrac{9}{45}459​
  4. 555

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 14

Which statement about division is true?

  1. 5÷05\div 05÷0 is undefined (correct answer)
  2. 5÷0=55\div 0=55÷0=5
  3. 5÷0=∞5\div 0=\infty5÷0=∞
  4. 5÷0=05\div 0=05÷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 15

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

  1. −12-12−12
  2. −5-5−5 (correct answer)
  3. 555
  4. 121212

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 16

A coach splits a team penalty of −60-60−60 points equally among 121212 players. What is each player’s share of the penalty? (Compute (−60)÷12(-60)\div 12(−60)÷12.)

  1. −5-5−5 (correct answer)
  2. 555
  3. 15\dfrac{1}{5}51​
  4. −72-72−72

Explanation: This question tests dividing rational numbers by applying sign rules, where a negative divided by a positive yields a negative quotient, and understanding the context of sharing a penalty. Sign rules for division are the same as for multiplication: negative divided by positive is negative, as in (-60) ÷ 12 = -5. The quotient of integers like -60 divided by 12 is a rational number, -5, which can also be expressed as a fraction -60/12. For example, sharing a debt of -$60 among 12 players means each player's share is (-60) ÷ 12 = -5, indicating each owes 5 points as a penalty. The correct calculation is (-60) ÷ 12 = -5, so each player’s share is -5 points. A common error is treating the penalty as positive, leading to 60 ÷ 12 = 5, but the negative sign must be preserved. To divide, determine the sign (different signs → negative), divide the magnitudes (60 ÷ 12 = 5), apply the sign to get -5, and recognize this as a rational number in the context of rates or shares.

Question 17

A coach splits a −60-60−60 point penalty evenly among 121212 players. What is each player’s share of the penalty? (Compute (−60)÷12(-60)\div 12(−60)÷12.)

  1. −5-5−5 (correct answer)
  2. −60-60−60
  3. −6012-\dfrac{60}{12}−1260​ is undefined
  4. 555

Explanation: This question tests dividing rational numbers by applying sign rules (negative divided by positive yields negative) and understanding real-world contexts like sharing a penalty. Sign rules for division are the same as for multiplication: positive divided by positive is positive, negative divided by negative is positive, positive divided by negative is negative, and negative divided by positive is negative, with quotients of integers being rational numbers like -60 ÷ 12 = -5. For example, sharing a -60 point penalty among 12 players means each gets -5 points, as (-60) ÷ 12 = -5, representing each player's share of the debt. The correct division is (-60) ÷ 12 = -5, applying the rule for negative divided by positive. A common error is ignoring the sign and getting 5, or confusing it with undefined like -60/12 is undefined, but it's defined and negative. To divide: (1) determine the sign (different signs → negative), (2) divide magnitudes (60 ÷ 12 = 5), (3) apply the sign to get -5. In contexts like penalties or debts, dividing a negative total by a positive number gives a negative per unit, meaning each owes.

Question 18

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

  1. −320-320−320 meters per hour
  2. −120-\dfrac{1}{20}−201​ meters per hour
  3. −20-20−20 meters per hour (correct answer)
  4. 202020 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 19

A student says, “Because 15÷415 \div 415÷4 is not an integer, it is not rational.” Which value shows the correct quotient and why the statement is false?

  1. 15÷4=415 \div 4 = 415÷4=4, which is rational
  2. 15÷4=15015 \div 4 = \dfrac{15}{0}15÷4=015​, which is rational
  3. 15÷4=41515 \div 4 = \dfrac{4}{15}15÷4=154​, which is rational
  4. 15÷4=15415 \div 4 = \dfrac{15}{4}15÷4=415​, which is rational (correct answer)

Explanation: This question tests that quotient of integers (p/q,q≠0)(p/q, q \neq 0)(p/q,q=0) is rational, even if not integer, like 15÷4=154=3.7515 \div 4 = \frac{15}{4} = 3.7515÷4=415​=3.75. Positive ÷ positive = positive, 15÷4=15415 \div 4 = \frac{15}{4}15÷4=415​. Rational includes fractions or decimals. Example: 15÷4=15415 \div 4 = \frac{15}{4}15÷4=415​, rational (terminating decimal 3.75). Correct: 15÷4=15415 \div 4 = \frac{15}{4}15÷4=415​, rational, falsifying 'not integer so not rational'. Error: wrong quotient like 415\frac{4}{15}154​, or division by zero. Express as fraction if needed, 15÷4=15415 \div 4 = \frac{15}{4}15÷4=415​ rational; mistakes: division by zero claimed valid, or reciprocal wrong.

Question 20

Divide and simplify: (−35)÷(610)\left(-\dfrac{3}{5}\right)\div\left(\dfrac{6}{10}\right)(−53​)÷(106​).

  1. −1-1−1 (correct answer)
  2. −925-\dfrac{9}{25}−259​
  3. −14-\dfrac{1}{4}−41​
  4. 111

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.