7th Grade Math Quiz: Multiply Rational Numbers

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 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.

Question 1

1. $-40$
2. $40$ (correct answer)
3. $-\frac{120}{3}$
4. $-\frac{200}{5}$

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 2

1. $\frac{2}{3}$ cup was the original flour amount needed
2. $\frac{4}{3}$ cup was the original flour amount needed
3. $\frac{3}{4}$ cup was the original flour amount needed (correct answer)
4. $\frac{3}{2}$ 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 $\frac{1}{2}$ cup. First, simplify the known parts of the expression. When multiplying fractions with the same signs, the result is positive: $(-\frac{3}{4}) \times (-\frac{8}{9}) = \frac{3 \times 8}{4 \times 9} = \frac{24}{36} = \frac{2}{3}$ Now your equation becomes: $\frac{2}{3} \times (\text{flour amount}) = \frac{1}{2}$ To solve for the flour amount, divide both sides by $\frac{2}{3}$, which is the same as multiplying by its reciprocal $\frac{3}{2}$: $\text{flour amount} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}$ Answer C is correct: $\frac{3}{4}$ cup was the original flour amount. Answer A ($\frac{2}{3}$) would give you $\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$, not $\frac{1}{2}$. Answer B ($\frac{4}{3}$) would give you $\frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$, which is too large. Answer D ($\frac{3}{2}$) would give you $\frac{2}{3} \times \frac{3}{2} = 1$, 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 3

1. $20.3°C$ as the final experimental temperature
2. $72°C$ as the final experimental temperature
3. $-36°C$ as the final experimental temperature
4. $-72°C$ 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: $(-2.5) \times (-1.8) \times (-16)$. Start by multiplying the first two numbers: $(-2.5) \times (-1.8) = 4.5$. Remember that when you multiply two negative numbers, the result is positive. Now multiply this result by the initial temperature: $4.5 \times (-16) = -72$. Since you're multiplying a positive number by a negative number, the final result is negative. Looking at the wrong answers: Choice A ($20.3°C$) appears to come from incorrectly calculating $(-2.5) \times (-1.8)$ as $-4.5$ instead of $4.5$, then making another sign error. Choice B ($72°C$) 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 ($-36°C$) likely results from calculation errors in the multiplication steps, possibly confusing $2.5 \times 16 = 40$ with $36$. The correct answer is D: $-72°C$. 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 4

1. $-100$
2. $-10$ (correct answer)
3. $10$
4. $-1$

Explanation: This question tests multiplying rational numbers with decimals, applying sign rules ($negative \times positive = negative$) and calculating accurately. Decimals: multiply magnitudes, apply sign; $(-2.5) \times 4$: magnitudes $2.5 \times 4 = 10$, signs $negative \times positive = negative$, result $-10$. For example, scaling a negative value by a positive factor keeps the negative direction. Correctly, it's $-10$, not $10$ (ignoring sign) or $-1$/-$100$ (arithmetic errors). A common mistake is decimal placement error, like thinking $2.5 \times 4 = 1$ or $100$. Sign: one negative → odd, so negative. Steps: sign (odd negatives → -), magnitudes ($2.5 \times 4 = 10$), apply ($-10$).

Question 5

1. $12^\circ\text{C}$
2. $-12^\circ\text{C}$ (correct answer)
3. $7^\circ\text{C}$
4. $-7^\circ\text{C}$

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 6

1. $-4$ (correct answer)
2. $4$
3. $-8$
4. $\frac{1}{16}$

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

Question 7

1. No, because a positive times a negative is positive.
2. No, because $5\times 3=8$.
3. Yes, because a positive times a negative is negative, and $5\times 3=15$. (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 8

1. $\frac{2}{5}$
2. $-\frac{6}{15}$
3. $\frac{6}{8}$
4. $-\frac{2}{5}$ (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 9

1. $24$
2. $-10$
3. $-24$ (correct answer)
4. $-20$

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 10

1. $15$ (correct answer)
2. $-8$
3. $-15$
4. $8$

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 11

1. $8$
2. $15$ (correct answer)
3. $-8$
4. $-15$

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 12

1. $-7^\circ\text{C}$
2. $7^\circ\text{C}$
3. $12^\circ\text{C}$
4. $-12^\circ\text{C}$ (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 13

1. $-8$
2. $15$ (correct answer)
3. $8$
4. $-15$

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 14

1. $(-5)\times 3$
2. $7\times(-1)$
3. $(-5)\times(-3)$ (correct answer)
4. $5\times(-3)$

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 15

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 $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, 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 16

1. $4$
2. $\frac{1}{16}$
3. $-4$ (correct answer)
4. $-8$

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, (-1/2)×8 = - (1/2 × 8) = -4, applying the negative sign from the fraction. A common error is getting positive 4 by ignoring the sign, or wrong calculation like -8 or 1/16. Count negatives: one (odd) means negative. For calculation: (1) determine sign (odd → negative), (2) multiply magnitudes (1/2 × 8 = 4), (3) apply sign (-4), like reversing a quantity.

Question 17

1. $10$
2. $24$
3. $-24$ (correct answer)
4. $-10$

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.

Question 18

1. True, because multiplying by 6 makes the number larger.
2. False, because the product should be $-24$. (correct answer)
3. False, because the product should be $0$.
4. True, because the negative sign does not affect multiplication.

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 statement (-4)×6=24 is false, because negative times positive is negative: -24. A common error is agreeing it's true by ignoring the sign, or thinking it's 0. Count negatives: one (odd) means negative, so product is -24, not 24. Use steps: (1) sign (odd → negative), (2) magnitudes (4×6=24), (3) apply sign (-24) to verify.

Question 19

1. $100$
2. $-10$ (correct answer)
3. $10$
4. $-1$

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 (−2.5)×4, magnitudes 2.5×4=10, signs negative×positive=negative, result -10. A common error is ignoring the sign to get 10, or decimal mistakes like 2.5×4=1 or 100. To determine the sign, count the negatives: odd number (one) means negative. For calculation: (1) determine sign (odd → negative), (2) multiply magnitudes (2.5×4=10), (3) apply sign (-10), such as in contexts like negative scaling.

Question 20

1. -\30$ (correct answer)
2. -\11$
3. \11$
4. \30$

Explanation: This question tests interpreting multiplication of rational numbers in a debt context, where owing is negative, using sign rules (positive months × negative per month = negative total). Here, 6 × (-5) = -30, meaning a total debt of $30 over 6 months. The context shows debt as negative, so positive time × negative rate = negative total change. Correctly, it's -$30, not $30 (ignoring sign) or -$11 (wrong math). Common error: treating debt as positive, like 6 × 5 = 30. Sign: one negative → negative. Steps: sign (-), magnitudes (6 × 5 = 30), apply (-$30); context confirms debt accumulation.