$$\frac{-15}{-3} = 5$$. Each expression equals 5: $$\frac{15}{3} = 5$$, $$\frac{-(-15)}{3} = \frac{15}{3} = 5$$, and $$\frac{15}{-(-3)} = \frac{15}{3} = 5$$. This tests understanding that $$-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$$. Choices A, B, and C each identify only one equivalent form, missing that all three are correct.

$$x = \frac{-12}{3} = -4$$ and $$y = \frac{-12}{-3} = 4$$. Therefore $$\frac{x}{y} = \frac{-4}{4} = -1$$. Choice A results from incorrectly thinking $$x = y$$. Choice C comes from computing $$\frac{1}{|x|}$$ instead of $$\frac{x}{y}$$. Choice D comes from computing $$\frac{1}{x}$$ instead of $$\frac{x}{y}$$.

Rate = $$\frac{-2\frac{1}{4}}{\frac{3}{4}} = \frac{-\frac{9}{4}}{\frac{3}{4}} = -\frac{9}{4} \times \frac{4}{3} = -3$$ degrees per hour, meaning a 3-degree drop. Choice B incorrectly calculates $$\frac{9}{4} \times \frac{3}{4}$$. Choice C uses $$\frac{3}{4} \div \frac{9}{4}$$. Choice D gets the magnitude right but wrong sign interpretation.

The submarine descends 150 meters, so it's at -150 meters. It ascends $$\frac{2}{5} \times 150 = 60$$ meters. Final position: $$-150 + 60 = -90$$ meters, which is 90 meters below sea level. Choice B gives the ascent distance, not final depth. Choice C adds instead of subtracts the ascent. Choice D gives $$\frac{1}{5}$$ of the descent instead of the correct calculation.

By definition, any quotient of integers (with non-zero divisor) is a rational number. Choice A is false: dividing two negatives gives a positive result. Choice B is false: positive ÷ negative = negative, which has the opposite sign of the dividend. Choice D is false: division works for all rational numbers with non-zero divisors.

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.

$$\frac{-\frac{2}{3}}{-\frac{4}{9}} = \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}$$. $$\frac{\frac{1}{2}}{-\frac{1}{6}} = \frac{1}{2} \times \frac{-6}{1} = -3$$. 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.

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.

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.