Middle School Math Quiz: Multiply Rational Numbers
20 questions · exam conditions
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Multiply Rational NumbersQuestion 1 of 20

A baker uses the expression (−34)×(−89)×(flour amount)(-\frac{3}{4}) \times (-\frac{8}{9}) \times (\text{flour amount}) to adjust a recipe. If this expression must equal 12\frac{1}{2} cup of flour, what was the original flour amount?

23\frac{2}{3} cup was the original flour amount needed
43\frac{4}{3} cup was the original flour amount needed
34\frac{3}{4} cup was the original flour amount needed
32\frac{3}{2} cup was the original flour amount needed
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Middle School Math Quiz

Middle School Math Quiz: Multiply Rational Numbers

Practice Multiply 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 Multiply 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

A baker uses the expression (−34)×(−89)×(flour amount)(-\frac{3}{4}) \times (-\frac{8}{9}) \times (\text{flour amount}) to adjust a recipe. If this expression must equal 12\frac{1}{2} cup of flour, what was the original flour amount?

  1. 23\frac{2}{3} cup was the original flour amount needed
  2. 43\frac{4}{3} cup was the original flour amount needed
  3. 34\frac{3}{4} cup was the original flour amount needed (correct answer)
  4. 32\frac{3}{2} cup was the original flour amount needed
Explanation: When you see an expression with unknown values that must equal a specific result, you're solving an equation. Here, you need to find what flour amount makes the entire expression equal 12\frac{1}{2} cup. First, simplify the known parts of the expression. When multiplying fractions with the same signs, the result is positive: (−34)×(−89)=3×84×9=2436=23(-\frac{3}{4}) \times (-\frac{8}{9}) = \frac{3 \times 8}{4 \times 9} = \frac{24}{36} = \frac{2}{3} Now your equation becomes: 23×(flour amount)=12\frac{2}{3} \times (\text{flour amount}) = \frac{1}{2} To solve for the flour amount, divide both sides by 23\frac{2}{3}, which is the same as multiplying by its reciprocal 32\frac{3}{2}: flour amount=12×32=34\text{flour amount} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} Answer C is correct: 34\frac{3}{4} cup was the original flour amount. Answer A (23\frac{2}{3}) would give you 23×23=49\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}, not 12\frac{1}{2}. Answer B (43\frac{4}{3}) would give you 23×43=89\frac{2}{3} \times \frac{4}{3} = \frac{8}{9}, which is too large. Answer D (32\frac{3}{2}) would give you 23×32=1\frac{2}{3} \times \frac{3}{2} = 1, which is double what you need. Remember: when solving equations with fractions, multiply by the reciprocal to "undo" division. Always check your answer by substituting it back into the original expression.

Question 2

Calculate: (−2.5)×4(-2.5) \times 4.

  1. −100-100
  2. −10-10 (correct answer)
  3. 1010
  4. −1-1
Explanation: This question tests multiplying rational numbers with decimals, applying sign rules (negative×positive=negativenegative \times positive = negative) and calculating accurately. Decimals: multiply magnitudes, apply sign; (−2.5)×4(-2.5) \times 4: magnitudes 2.5×4=102.5 \times 4 = 10, signs negative×positive=negativenegative \times positive = negative, result −10-10. For example, scaling a negative value by a positive factor keeps the negative direction. Correctly, it's −10-10, not 1010 (ignoring sign) or −1$/−-1$/-100(arithmeticerrors).Acommonmistakeisdecimalplacementerror,likethinking(arithmetic errors). A common mistake is decimal placement error, like thinking2.5 \times 4 = 1oror100.Sign:onenegative→odd,sonegative.Steps:sign(oddnegatives→−),magnitudes(. Sign: one negative → odd, so negative. Steps: sign (odd negatives → -), magnitudes (2.5 \times 4 = 10),apply(), apply (-10$).

Question 3

Verify the product. Is the statement 5×(−3)=−155\times(-3)=-15 true?

  1. No, because a positive times a negative is positive.
  2. No, because 5×3=85\times 3=8.
  3. Yes, because a positive times a negative is negative, and 5×3=155\times 3=15. (correct answer)
  4. Yes, because negative times negative is negative.
Explanation: This question tests verifying a multiplication of rational numbers using sign rules, where positive times negative equals negative. The sign rules state that positive × positive = positive (e.g., 3×5=15), negative × negative = positive (e.g., (-3)×(-5)=15), positive × negative = negative (e.g., 3×(-5)=-15), and negative × positive = negative (e.g., (-3)×5=-15). For example, 5×(-3)=-15, as positive × negative = negative, with magnitude 5×3=15. The statement is true because positive × negative is negative, and 5×3=15 leads to -15. A common error is claiming positive × negative = positive, or wrong magnitude like 5×3=8. Calculation: (1) sign (odd negatives → negative), (2) magnitudes (5×3=15), (3) apply sign (-15). Avoid sign rule errors or arithmetic mistakes in verification.

Question 4

Multiply: (−23)×(35)\left(-\frac{2}{3}\right)\times\left(\frac{3}{5}\right).

  1. 25\frac{2}{5}
  2. −615-\frac{6}{15}
  3. 68\frac{6}{8}
  4. −25-\frac{2}{5} (correct answer)
Explanation: This question tests multiplying rational numbers as negative and positive fractions, applying sign rules and simplifying. Multiply: (-2/3) × (3/5) = (-2×3)/(3×5) = -6/15, simplify by dividing by 3 to -2/5. Sign rules: negative × positive = negative, with fraction multiplication of numerators and denominators. Correctly, -2/5, not -6/15 (unsimplified) or positive versions (sign error) or 6/8 (wrong math). Error like multiplying denominators wrong or ignoring sign. Sign: one negative → negative. Steps: sign (-), magnitudes (2/3 × 3/5 = 6/15 = 2/5), apply (-2/5), simplify.

Question 5

Calculate the product: 6×(−4)6\times(-4).​

  1. 2424
  2. −10-10
  3. −24-24 (correct answer)
  4. −20-20
Explanation: This question tests multiplying rational numbers by applying sign rules, where a positive times a negative equals a negative, and interpreting products in contexts like scaling or debt. The sign rules state that positive × negative = negative, as in 6 × (-4) where the positive 6 and negative -4 result in a negative product. For example, if you have a rate of -4 units per step and take 6 steps, the total is -24 units, representing a loss or decrease. Here, multiplying the magnitudes 6 × 4 = 24 and applying the negative sign from the one negative factor gives -24. A common error is ignoring the sign and choosing 24, or miscalculating the magnitude like 6 × 4 = 10 or 20. To determine the sign, count the negatives: one negative means odd number, so negative result. Always multiply magnitudes first, apply the sign, and verify in context, such as a positive time period times a negative rate yielding a negative total change.

Question 6

The temperature drops 3∘C3^\circ\text{C} each hour for 4 hours. If a drop is negative, what is the total temperature change?

  1. −7∘C-7^\circ\text{C}
  2. 7∘C7^\circ\text{C}
  3. 12∘C12^\circ\text{C}
  4. −12∘C-12^\circ\text{C} (correct answer)
Explanation: This question tests multiplying rational numbers by applying sign rules (negative times negative is positive, positive times negative is negative) and interpreting products in contexts like debt or temperature changes. The sign rules are: positive times positive equals positive (like 3×5=15), negative times negative equals positive ((-3)×(-5)=15, since two negatives make a positive from properties like (-1)×(-1)=1), positive times negative equals negative (3×(-5)=-15), and negative times positive equals negative ((-3)×5=-15, due to commutativity). For fractions, multiply numerators and denominators, like (2/3)×(3/4)=6/12=1/2 simplified, and for decimals, multiply magnitudes and apply the sign, like (−2.5)×4 gives magnitudes 2.5×4=10, with negative×positive=negative, so -10. For temperature drop of 3°C per hour (negative rate -3) over 4 hours: 4×(-3)=-12°C total change. A common error is using positive for drop to get 12°C, or wrong math like 4+3=7. Count negatives: one (odd) means negative. In temperature contexts, negative rate times time gives total drop: 4×(-3)=-12°C, avoiding positive misinterpretation.

Question 7

Apply the sign rules and calculate: (−3)×(−5)(-3)\times(-5).

  1. −8-8
  2. 1515 (correct answer)
  3. 88
  4. −15-15
Explanation: This question tests multiplying rational numbers applying sign rules, specifically negative × negative = positive, and understanding products in contexts like double reversals. Sign rules: negative × negative = positive, as (-3) × (-5) = 15, since two negatives make a positive from properties like (-1) × (-1) = 1. For example, (-3) × (-5) both negative, so the rule gives positive 15; in context, like reversing a debt twice. Correctly, multiply magnitudes 3 × 5 = 15 and apply positive sign due to even number of negatives. A common mistake is treating negative × negative as negative, resulting in -15. Sign determination: count negatives (even number → positive). Calculation steps: determine sign (even negatives → +), multiply magnitudes (3 × 5 = 15), apply sign (+15).

Question 8

Which expression has a positive value?

  1. (−5)×3(-5)\times 3
  2. 7×(−1)7\times(-1)
  3. (−5)×(−3)(-5)\times(-3) (correct answer)
  4. 5×(−3)5\times(-3)
Explanation: This question tests multiplying rational numbers by applying sign rules (negative times negative is positive, positive times negative is negative) and interpreting products in contexts like debt or temperature changes. The sign rules are: positive times positive equals positive (like 3×5=15), negative times negative equals positive ((-3)×(-5)=15, since two negatives make a positive from properties like (-1)×(-1)=1), positive times negative equals negative (3×(-5)=-15), and negative times positive equals negative ((-3)×5=-15, due to commutativity). For fractions, multiply numerators and denominators, like (2/3)×(3/4)=6/12=1/2 simplified, and for decimals, multiply magnitudes and apply the sign, like (−2.5)×4 gives magnitudes 2.5×4=10, with negative×positive=negative, so -10. The expression with positive value is (-5)×(-3)=15, as two negatives make positive. A common error is picking one with odd negatives, like 5×(-3)=-15, thinking it's positive. To find positive, count negatives: even (two in C) → positive, odd → negative. Calculation steps help identify: determine sign by negative count, multiply magnitudes, apply sign.

Question 9

Which statement correctly verifies the product 5x(-3)?

  1. 5x(-3)=-8 because 5-3=2.
  2. 5x(-3)=15 because multiplication always gives a positive.
  3. 5x(-3)=-15 because one factor is negative. (correct answer)
  4. 5x(-3)=15 because the numbers are different.
Explanation: When multiplying a positive number by a negative number, the product is negative: 5 x (-3) = -15, matching choice C. Choice B incorrectly claims multiplication always gives a positive result, which is only true when both factors share the same sign. Choice A confuses multiplication with subtraction, treating the problem like 5-3=2. Choice D claims the product is positive simply because the numbers are different, which isn't a valid rule for determining sign.

Question 10

Calculate and simplify: 2/3 x 3/4.

  1. 6/12
  2. 1/2 (correct answer)
  3. 5/7
  4. 6/7
Explanation: Multiply the numerators and denominators: (2/3) x (3/4) = (2x3)/(3x4) = 6/12. Simplifying by dividing both by their common factor of 6 gives 1/2, matching choice B. Choice A (6/12) is the correct product before simplifying, so it isn't fully reduced. Choice C (5/7) comes from adding the numerators and denominators separately (2+3=5, 3+4=7) instead of multiplying, and choice D (6/7) correctly multiplies the numerators (2x3=6) but adds the denominators instead of multiplying (3+4=7 instead of 3x4=12).

Question 11

The area of a rectangle is found by multiplying its length by its width, and the original area is 5454 square feet. Suppose the length value is multiplied by −23-\frac{2}{3} and the width value is multiplied by −1.5-1.5 to create a new product. What is the new value, and how does it compare to the original area?

  1. 8181 square feet, larger than the original
  2. −54-54 square feet, the opposite of the original
  3. 5454 square feet, the same as the original (correct answer)
  4. 3636 square feet, smaller than the original
Explanation: To find the new value, multiply the original area by both scaling factors: 54 x (-2/3) x (-1.5). Multiplying the two scaling factors first: (-2/3) x (-1.5) = (-2/3) x (-3/2) = 1, since a negative times a negative gives a positive, and 2/3 times 3/2 equals 1. So the new value is 54 x 1 = 54, the same as the original. Choice A comes from mishandling the negative signs or fractions during multiplication. Choice B comes from forgetting that multiplying two negative numbers gives a positive result. Choice D comes from multiplying 2/3 x 1.5 without properly accounting for the negative signs.

Question 12

In a chemistry experiment, the temperature changes according to the expression (−2.5)×(−1.8)×(initial temperature)(-2.5) \times (-1.8) \times (\text{initial temperature}). If the initial temperature is −16°C-16°C, what is the final temperature?

  1. 20.3°C20.3°C as the final experimental temperature
  2. 72°C72°C as the final experimental temperature
  3. −36°C-36°C as the final experimental temperature
  4. −72°C-72°C as the final experimental temperature (correct answer)
Explanation: When you encounter a problem with multiple negative numbers being multiplied together, the key is carefully tracking the signs while working step by step through the calculation. Let's substitute the initial temperature into the expression: (−2.5)×(−1.8)×(−16)(-2.5) \times (-1.8) \times (-16). Start by multiplying the first two numbers: (−2.5)×(−1.8)=4.5(-2.5) \times (-1.8) = 4.5. Remember that when you multiply two negative numbers, the result is positive. Now multiply this result by the initial temperature: 4.5×(−16)=−724.5 \times (-16) = -72. Since you're multiplying a positive number by a negative number, the final result is negative. Looking at the wrong answers: Choice A (20.3°C20.3°C) appears to come from incorrectly calculating (−2.5)×(−1.8)(-2.5) \times (-1.8) as −4.5-4.5 instead of 4.54.5, then making another sign error. Choice B (72°C72°C) gets the correct numerical value but has the wrong sign—this happens when students forget that an odd number of negative factors (three negatives in this case) always produces a negative result. Choice C (−36°C-36°C) likely results from calculation errors in the multiplication steps, possibly confusing 2.5×16=402.5 \times 16 = 40 with 3636. The correct answer is D: −72°C-72°C. Study tip: When multiplying multiple numbers with mixed signs, count the negative signs first. An odd number of negatives gives a negative result, while an even number gives a positive result. This helps you catch sign errors before they happen.

Question 13

Two numbers have a product of −24-24. If one number is 35\frac{3}{5} and both numbers can be positive or negative, which of the following could NOT be the other number?

  1. −40-40
  2. 4040 (correct answer)
  3. −1203-\frac{120}{3}
  4. −2005-\frac{200}{5}
Explanation: Since the product must be -24 and one factor is 3/5 (positive), the other number must satisfy (3/5) times the other number equals -24, giving the other number as -24 divided by 3/5, which is -24 times 5/3, or -40. Choice A shows -40 directly, which does give the correct product of -24. Choice C simplifies to -120/3 = -40, which also gives the correct product. Choice D simplifies to -200/5 = -40, which again gives the correct product. Only choice B, positive 40, gives (3/5)(40) = 24, not -24, so it could not be the other number.

Question 14

In a science lab, the temperature changes by −3∘C-3^\circ\text{C} per hour (a drop). What is the total temperature change over 4 hours?

  1. 12∘C12^\circ\text{C}
  2. −12∘C-12^\circ\text{C} (correct answer)
  3. 7∘C7^\circ\text{C}
  4. −7∘C-7^\circ\text{C}
Explanation: This question tests multiplying rational numbers in a temperature context, where drop is negative, applying sign rules (positive hours × negative per hour = negative total). So, 4 × (-3) = -12°C, meaning a total drop of 12°C over 4 hours. Context: temperature drop as negative rate, positive time yields negative change. Correctly, -12°C, not 12°C (wrong sign) or -7°C (math error). Mistake like adding rates instead of multiplying. Sign: odd negatives → negative. Calculation: sign (-), magnitudes (4 × 3 = 12), apply (-12°C); contexts like this show real-world application.

Question 15

Find the product: (−12)×8\left(-\frac{1}{2}\right)\times 8.

  1. −4-4 (correct answer)
  2. 44
  3. −8-8
  4. 116\frac{1}{16}
Explanation: This question tests multiplying rational numbers mixing fractions and integers, with sign rules (negative ×\times positive = negative). For (−12)×8(- \frac{1}{2}) \times 8: multiply as −(12)×8=−4- (\frac{1}{2}) \times 8 = -4, or numerators −1×8=−8-1 \times 8 = -8 over denominator 2, then −8/2=−4-8/2 = -4. In context, like half in the opposite direction scaled by 8. Correctly, product is −4-4, not 44 (wrong sign) or −8-8 (not simplifying) or 116\frac{1}{16} (division error). Error like treating it as division or wrong fraction multiplication. Sign: one negative →\to negative. Calculation: sign (−)(-), magnitudes (12×8=4\frac{1}{2} \times 8 = 4), apply (−4)(-4); simplify if needed.

Question 16

Find the value of (−3)×(−5)(-3)\times(-5).

  1. 1515 (correct answer)
  2. −8-8
  3. −15-15
  4. 88
Explanation: This question tests multiplying rational numbers by applying sign rules, where negative times negative equals positive, and interpreting products in contexts like debt or temperature changes. The sign rules state that positive × positive = positive (e.g., 3×5=15), negative × negative = positive (e.g., (-3)×(-5)=15, as two negatives make positive from properties like (-1)×(-1)=1), positive × negative = negative (e.g., 3×(-5)=-15), and negative × positive = negative (e.g., (-3)×5=-15, due to commutativity). For example, (-3)×(-5) both negative, so the rule gives positive: 15. In this case, the correct calculation is (-3)×(-5)=15, as the magnitude 3×5=15 takes a positive sign from the two negatives. A common error is applying negative × negative = negative, like (-3)×(-5)=-15, or misadding like -3 + -5 = -8. To determine the sign, count the negatives: here, an even number (two) leads to positive, while odd would be negative, like (-3)×5=-15. For calculation: (1) determine sign (even negatives → positive), (2) multiply magnitudes (3×5=15), (3) apply sign (15), and avoid mistakes like wrong sign rules or arithmetic errors.

Question 17

Calculate the product: (−3)×(−5)(-3)\times(-5).

  1. 88
  2. 1515 (correct answer)
  3. −8-8
  4. −15-15
Explanation: This question tests multiplying rational numbers by applying sign rules (negative times negative is positive, positive times negative is negative) and interpreting products in contexts like debt or temperature changes. The sign rules are: positive times positive equals positive (like 3×5=15), negative times negative equals positive ((-3)×(-5)=15, since two negatives make a positive from properties like (-1)×(-1)=1), positive times negative equals negative (3×(-5)=-15), and negative times positive equals negative ((-3)×5=-15, due to commutativity). For fractions, multiply numerators and denominators, like (2/3)×(3/4)=6/12=1/2 simplified, and for decimals, multiply magnitudes and apply the sign, like (−2.5)×4 gives magnitudes 2.5×4=10, with negative×positive=negative, so -10. For (-3)×(-5), both negative, so the rule gives positive: 15. A common error is treating negative times negative as negative, like (-3)×(-5)=-15, or miscalculating the magnitude as 3+5=8. To determine the sign, count the negatives: an even number (two here) means positive, odd means negative. For calculation: (1) determine sign (even negatives → positive), (2) multiply magnitudes (3×5=15), (3) apply sign (15), and in contexts like two debts canceling out to a gain.

Question 18

A student owes $5 each month for 6 months. If owing money is represented by a negative number, which expression and value represent the total change in money?

  1. 6×5=306\times 5=30
  2. (−6)×5=−30(-6)\times 5=-30
  3. (−6)×(−5)=−30(-6)\times(-5)=-30
  4. 6×(−5)=−306\times(-5)=-30 (correct answer)
Explanation: This question tests multiplying rational numbers in a debt context, applying sign rules where positive times negative equals negative for total change. The sign rules state that positive × positive = positive (e.g., 3×5=15), negative × negative = positive (e.g., (-3)×(-5)=15), positive × negative = negative (e.g., 3×(-5)=-15), and negative × positive = negative (e.g., (-3)×5=-15). In a debt context, owing $5/month is -5, for 6 months: 6×(-5)=-30 (total debt of 30).Thecorrectexpressionis6×(−5)=−30,representingpositivemonthstimesnegativeamountpermonth.Acommonerrorisusingpositivefordebt,like6×5=30,orwrongsignslike(−6)×(−5)=−30whichimpliesnegativemonths.Contextslikedebtusenegativeamounts:6×(−5)=−30). The correct expression is 6×(-5)=-30, representing positive months times negative amount per month. A common error is using positive for debt, like 6×5=30, or wrong signs like (-6)×(-5)=-30 which implies negative months. Contexts like debt use negative amounts: 6×(-5)=-30 total. Mistakes include wrong context interpretation (debt as positive) or sign errors; calculate by determining sign (odd negatives → negative), multiplying magnitudes (6×5=30), applying sign (-30).

Question 19

A student owes $5 each month for 6 months. If owing money is represented by a negative number, what is the total change in the student's balance? (Use multiplication.)

  1. $11
  2. -\30$ (correct answer)
  3. $30
  4. -\11$
Explanation: This question tests multiplying rational numbers by applying sign rules (negative times negative is positive, positive times negative is negative) and interpreting products in contexts like debt or temperature changes. The sign rules are: positive times positive equals positive (like 3×5=15), negative times negative equals positive ((-3)×(-5)=15, since two negatives make a positive from properties like (-1)×(-1)=1), positive times negative equals negative (3×(-5)=-15), and negative times positive equals negative ((-3)×5=-15, due to commutativity). For fractions, multiply numerators and denominators, like (2/3)×(3/4)=6/12=1/2 simplified, and for decimals, multiply magnitudes and apply the sign, like (−2.5)×4 gives magnitudes 2.5×4=10, with negative×positive=negative, so -10. In this debt context, owing 5permonthis−5,times6months:6×(−5)=−30,totaldebtof−5 per month is -5, times 6 months: 6×(-5)=-30, total debt of -30. A common error is treating debt as positive to get 30,orarithmeticlike6+5=11.Countnegatives:one(odd)meansnegative.Contextslikedebtusenegativeamounts:6×(−5)=−30, or arithmetic like 6+5=11. Count negatives: one (odd) means negative. Contexts like debt use negative amounts: 6×(-5)=-30 total debt, avoiding mistakes like positive interpretation.

Question 20

Calculate the product: 6×(−4)6\times(-4).

  1. 1010
  2. 2424
  3. −24-24 (correct answer)
  4. −10-10
Explanation: This question tests multiplying rational numbers by applying sign rules (negative times negative is positive, positive times negative is negative) and interpreting products in contexts like debt or temperature changes. The sign rules are: positive times positive equals positive (like 3×5=15), negative times negative equals positive ((-3)×(-5)=15, since two negatives make a positive from properties like (-1)×(-1)=1), positive times negative equals negative (3×(-5)=-15), and negative times positive equals negative ((-3)×5=-15, due to commutativity). For fractions, multiply numerators and denominators, like (2/3)×(3/4)=6/12=1/2 simplified, and for decimals, multiply magnitudes and apply the sign, like (−2.5)×4 gives magnitudes 2.5×4=10, with negative×positive=negative, so -10. Here, 6×(-4) involves a positive times a negative, so the result is negative: -24. A common error is forgetting the sign rule and getting 24 (positive), or miscalculating the magnitude like 6+4=10 or 6-4=2. To determine the sign, count the negatives: an odd number (one here) means negative, even means positive. For calculation: (1) determine sign (odd negatives → negative), (2) multiply magnitudes (6×4=24), (3) apply sign (-24), and consider contexts like scaling in the opposite direction.