Since the product must be $$-24$$ and one factor is $$\frac{3}{5}$$ (positive), the other number must be $$\frac{-24}{3/5} = -24 \times \frac{5}{3} = -40$$. Choice A gives $$\frac{3}{5} \times(-40) = -24$$ ✓. Choice B gives $$\frac{3}{5} \times 40 = 24 \neq -24$$ ✗. Choice C: $$-\frac{120}{3} = -40$$, so $$\frac{3}{5} \times(-40) = -24$$ ✓. Choice D gives $$\frac{3}{5} \times 40 = 24 \neq -24$$ ✗. Both B and D are impossible, but B is listed first.

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.

When you see a problem about changing dimensions of a rectangle, you need to understand how multiplying dimensions affects area. Remember that area equals length times width, so when both dimensions change, you multiply the original area by both scaling factors.

Let's work through this step by step. The original area is 54 square feet. The length gets multiplied by $$-\frac{2}{3}$$ and the width gets multiplied by $$-1.5$$. To find the new area, multiply the original area by both factors:

New area = $$54 \times \left(-\frac{2}{3}\right) \times(-1.5)$$

First, multiply the scaling factors: $$\left(-\frac{2}{3}\right) \times(-1.5) = \left(-\frac{2}{3}\right) \times \left(-\frac{3}{2}\right) = +1$$

Since negative times negative equals positive, and $$\frac{2}{3} \times \frac{3}{2} = 1$$, the combined scaling factor is 1.

Therefore: New area = $$54 \times 1 = 54$$ square feet.

Answer A incorrectly calculates the area as 81, likely from mishandling the negative signs or fractions. Answer B shows -54, which would happen if you forgot that multiplying two negative numbers gives a positive result. Answer D gives 36, probably from multiplying $$\frac{2}{3} \times 1.5$$ without considering the negative signs properly.

The key insight here is that when you multiply by two negative scaling factors, the negatives cancel out, leaving you with the same area. Always pay careful attention to signs when multiplying—two negatives make a positive!

Each week, the debt becomes $$1 - \frac{3}{8} = \frac{5}{8}$$ of its previous amount. After week 1: $$45.60 \times \frac{5}{8} = \frac{228}{8} = 28.50$$. After week 2: $$28.50 \times \frac{5}{8} = \frac{142.50}{8} = 11.40$$. Choice B shows only one week's reduction. Choice C reflects incorrectly subtracting $$\frac{3}{8} \times 45.60$$ twice. Choice D reflects finding the average of the starting amount and one-week amount.

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.

When you see a problem asking you to apply the distributive property, remember that you're multiplying a number outside parentheses by each term inside the parentheses. The distributive property states that $$a(b + c) = ab + ac$$.

For $$(-0.6) \times[4.2 + (-2.7)]$$, you need to multiply $$(-0.6)$$ by both $$4.2$$ and $$(-2.7)$$. This gives you $$(-0.6)(4.2) + (-0.6)(-2.7)$$.

Now let's calculate: $$(-0.6)(4.2) = -2.52$$ and $$(-0.6)(-2.7) = +1.62$$ (remember that negative times negative equals positive). So the expression becomes $$-2.52 + 1.62 = -0.9$$.

Choice A incorrectly changes $$(-0.6)$$ to $$(0.6)$$ in the second term, which completely alters the problem. Choice B makes a sign error when multiplying $$(-0.6)(-2.7)$$, writing the result as $$-1.62$$ instead of $$+1.62$$. This leads to the wrong final answer of $$-4.14$$. Choice C incorrectly writes the second term as $$(-0.6)(2.7)$$ instead of $$(-0.6)(-2.7)$$, missing the negative sign on $$2.7$$, though it compensates with subtraction instead of addition.

Choice D correctly applies the distributive property and performs all calculations accurately.

Study tip: When applying the distributive property with negative numbers, pay extra attention to signs. Write out each multiplication step separately, and remember that multiplying two negatives gives a positive result.

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 ($-10$).

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.

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.

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.