Divide Rational Numbers - 7th Grade Math

Which statement about division is correct?
Answer: $5\div 0$ is undefined 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.
A student says, " $5\div 0=0$ ." Which statement is correct?
Answer: $5\div 0$ is undefined (division by zero is not allowed). 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.
Divide and simplify: $\dfrac{2}{3}\div\dfrac{1}{4}=$
Answer: $\dfrac{8}{3}$ This question tests dividing rational numbers as fractions, using the rule to multiply by the reciprocal and apply signs. Fraction division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ , like $\frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3}$ . The result is rational, $\frac{8}{3}$ , simplified. For example, dividing $\frac{2}{3}$ by $\frac{1}{4}$ means how many $\frac{1}{4}$ in $\frac{2}{3}$ , yielding $\frac{8}{3}$ . The correct quotient is $\frac{8}{3}$ , from proper reciprocal multiplication. A mistake is multiplying by the fraction instead, like $\frac{2}{3} \times \frac{1}{4} = \frac{1}{6}$ . To divide fractions, multiply by reciprocal, simplify, and apply sign rules if needed.
Evaluate: $\frac{-\frac{2}{3}}{-\frac{4}{9}} + \frac{\frac{1}{2}}{-\frac{1}{6}}$
Answer: $-\frac{3}{2}$ $\frac{-\frac{2}{3}}{-\frac{4}{9}} = \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}$ . $\frac{\frac{1}{2}}{-\frac{1}{6}} = \frac{1}{2} \times \frac{-6}{1} = -3$ . So $\frac{3}{2} + (-3) = \frac{3}{2} - \frac{6}{2} = -\frac{3}{2}$ . Choice A forgets the second term is negative. Choice C and D result from sign errors in the division steps.
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, $(-80)\div 4$ ?
Answer: $-20$ meters per hour 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.
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 $(-45)\div 9$ ?
Answer: $-5$ 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.
The expression $\frac{-a}{b}$ is equivalent to which of the following when $a$ and $b$ are integers and $b \neq 0$ ?
Answer: Both $\frac{a}{-b}$ and $-\frac{a}{b}$ 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.
A company's stock price changes by $-\$1.20$ over $4$ trading days. If the change was the same each day, what was the daily change? Then, what would be the total change over $7$ days at this rate?
Answer: Daily change: $-\$0.30$ , Total over $7$ days: $-\$2.10$ This problem tests your ability to work with negative numbers and apply division and multiplication in real-world contexts. When you see questions about consistent daily changes, think about using division to find the rate, then multiplication to project forward. To find the daily change, you need to divide the total change by the number of days: $-\$1.20 ÷ 4 = -\$0.30$ per day. The negative sign is crucial here—it means the stock lost value each day. To find the total change over 7 days, multiply the daily change by 7: $-\$0.30 × 7 = -\$2.10$ . Again, the result is negative because the stock continues losing value. Choice A incorrectly multiplies instead of dividing to find the daily change ( $-\$1.20 × 4 = -\$4.80$ ), which doesn't make sense—the daily change should be smaller than the total change over multiple days. Choice B makes two errors: it treats the change as positive ( $+\$0.30$ ) and then multiplies by 7 to get a positive total, completely ignoring that the stock was losing value. Choice C correctly calculates the daily change as $-\$0.30$ but then inexplicably makes the 7-day total positive ( $+\$2.10$ ), which contradicts the negative daily trend. Remember: when dividing negative numbers by positive numbers, your answer stays negative. Always check that your final answer makes logical sense—if a stock loses money daily, it should lose even more money over more days, not suddenly gain value.
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 $(-45)\div 9$ ?
Answer: $-5$ 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.
Which statement about division is true?
Answer: $5\div 0$ is undefined 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.
Which set shows three equivalent ways to write the same negative quotient and the correct value? (Remember $-(p/q)=(-p)/q=p/(-q)$ for $q\ne 0$ .)
Answer: $-\dfrac{12}{3}=\dfrac{-12}{3}=\dfrac{12}{-3}=-4$ This question tests understanding equivalent forms of negative quotients, where -(p/q) = (-p)/q = p/(-q), all equal for q ≠ 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.
Any integer divided by a nonzero integer is a rational number. What is $15\div 4$ written as a fraction in simplest form, and as a terminating decimal?
Answer: $\dfrac{15}{4}$ and $3.75$ 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.
A coach splits a team penalty of $-60$ points equally among $12$ players. What is each player's share of the penalty? (Compute $(-60)\div 12$ .)
Answer: $-5$ 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.
Calculate the quotient: $15\div(-3)$ .
Answer: $-5$ 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.
Which set shows three equivalent ways to write the same negative quotient and the correct value? (Remember $-(p/q)=(-p)/q=p/(-q)$ for $q \neq 0$ .)
Answer: $-\dfrac{12}{3}=\dfrac{-12}{3}=\dfrac{12}{-3}=-4$ 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.
Find the quotient: $(-18)\div(-6)=$
Answer: $3$ This question tests dividing rational numbers with sign rules, where negative ÷ negative = positive, as negatives cancel. Sign rules: negative ÷ negative = positive, like (-18) ÷ (-6) = 3. The quotient is rational, 3, from dividing integers with same signs. For example, (-18) ÷ (-6) both negative, so positive 3. The correct quotient is 3, applying the same-signs positive rule. A common mistake is neg ÷ neg = negative, like -3, wrong sign application. To divide, determine sign (same signs → positive), divide magnitudes (18 ÷ 6 = 3), apply positive sign, and recognize as rational number.
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?
Answer: $-5$ points per round (they lost 5 each round) 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.
Calculate the quotient: $(-18) \div (-6)$ .
Answer: $3$ This question tests dividing rational numbers applying sign rules ( $\text{neg} \div \text{neg} = \text{pos}$ ) and understanding $ -(p/q) = (-p)/q = p/(-q) $ equivalence for negative quotients. Sign rules for division (same as multiplication): positive ÷ positive = positive ($ 20 \div 4 = 5 $), negative ÷ negative = positive ($ (-20) \div (-4) = 5 $), negatives cancel, positive ÷ negative = negative ($ 20 \div (-4) = -5 $), negative ÷ positive = negative ($ (-20) \div 4 = -5 $). For example, ($ (-18) \div (-6) $) both negative, rule gives positive: 3. The correct division with proper sign is ($ (-18) \div (-6) = 3 $). A common error is neg ÷ neg = negative, like ($ (-18) \div (-6) = -3 $) with wrong sign rule. To divide: (1) determine sign (same signs → positive), (2) divide magnitudes ($ 18 \div 6 = 3 $), (3) apply sign: +3. Equivalent forms aren't directly needed here, but note that $ -18 / -6 = 18 / 6 = 3 $, showing negatives cancel.
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?
Answer: $\dfrac{15}{4}$ and $3.75$ 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.
Which set of expressions are all equal to $-4$ ? (Use the idea that $-(p/q)=(-p)/q=p/(-q)$ .)
Answer: $-\dfrac{12}{3},\ \dfrac{-12}{3},\ \dfrac{12}{-3}$ 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.

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