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.

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.

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.

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.

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.

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.

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.

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.

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.

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.