Multiply Rational Numbers - 7th Grade Math

1. Calculate the product: $\left(-\frac{1}{2}\right)\times 8$.
   Answer: $-4$ 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.
2. 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?
   Answer: $6\times(-5)=-30$ 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). 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).
3. Multiply and simplify: $\frac{2}{3}\times\frac{3}{4}$.
   Answer: $\frac{1}{2}$ Multiplying the numerators and denominators gives (2x3)/(3x4) = 6/12, which simplifies to 1/2. Choice D comes from adding the numerators and denominators separately, (2+3)/(3+4) = 5/7, instead of multiplying. Choice B comes from a similar mix-up between adding and multiplying the numerators and denominators. Choice C comes from cross-multiplying incorrectly, pairing each numerator with the wrong denominator to get 8/9 instead of multiplying straight across.
4. Which statement correctly verifies the product 5x(-3)?
   Answer: 5x(-3)=-15 because one factor is negative. 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.
5. Which expression has a positive value?
   Answer: $(-5)\times(-3)$ 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.
6. Find the product: $\left(-\frac{1}{2}\right)\times 8$.
   Answer: $-4$ 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.
7. 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.)
   Answer: $-\$30$ 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.
8. Multiply: $\left(-\frac{2}{3}\right)\times\left(\frac{3}{5}\right)$. Give your answer in simplest form.
   Answer: $-\frac{2}{5}$ 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.
9. Apply the sign rules and calculate: $(-3)\times(-5)$.
   Answer: $15$ 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).
10. Calculate the product: $(-3)\times(-5)$.
    Answer: $15$ 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.
11. The area of a rectangle is found by multiplying its length by its width, and the original area is $54$ square feet. Suppose the length value is multiplied by $-\frac{2}{3}$ and the width value is multiplied by $-1.5$ to create a new product. What is the new value, and how does it compare to the original area?
    Answer: $54$ square feet, the same as the original 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.
12. Verify the product. Is the statement $5\times(-3)=-15$ true?
    Answer: Yes, because a positive times a negative is negative, and $5\times 3=15$. 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.
13. Find the value of $(-3)\times(-5)$.
    Answer: $15$ 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.
14. Is the statement true or false? $(-4)\times 6 = 24$.
    Answer: False, because the product should be $-24$. 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.
15. A student tracks a monthly change to their lunch account as negative when money is owed. If they owe $5 each month for 6 months, what is the total change? (Use multiplication.)
    Answer: $-\$30$ 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.
16. Calculate and simplify: 2/3 x 3/4.
    Answer: 1/2 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).
17. Which statement is true based on the sign rules for multiplication?
    Answer: Negative $\times$ positive is negative. This question tests understanding sign rules for multiplying rational numbers, identifying the true statement among options. The sign rules state that positive × positive = positive (e.g., 3×5=15), negative × negative = positive (e.g., (-3)×(-5)=15, from (-1)×(-1)=1), positive × negative = negative (e.g., 3×(-5)=-15), and negative × positive = negative (e.g., (-3)×5=-15). For example, negative × positive = negative, like (-3)×5=-15. The correct statement is 'Negative × positive is negative,' as it matches the rule. A common error is thinking negative × negative = negative, or positive × negative = positive. Sign determination: count negatives (even → positive, odd → negative, like one negative → negative). In contexts, this applies to debt or scaling; avoid mistakes like inverting rules.
18. In a science lab, the temperature changes by $-3^\circ\text{C}$ per hour (a drop). What is the total temperature change over 4 hours?
    Answer: $-12^\circ\text{C}$ 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.
19. Calculate the product: $6\times(-4)$.
    Answer: $-24$ 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.
20. Calculate: $(-2.5) \times 4$.
    Answer: $-10$ 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 ($-10$).

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