The automatic gratuity is $240 \times 0.18 = $43.20. Total tip is $43.20 + $15.00 = $58.20. The percent tip is $\frac{58.20}{240} \times 100% = 24.25%$. Choice B results from calculating only 18% + additional tip percentage without considering the automatic gratuity amount. Choice C comes from calculation errors. Choice D results from rounding errors in intermediate steps.

Interest earned: $635 - $500 = $135. Using $I = PRT$: $135 = 500 \times 0.045 \times T$. Solving: $T = \frac{135}{500 \times 0.045} = \frac{135}{22.5} = 6$ years = 30 months. Choice A results from using 6% rate instead of 4.5%. Choice C comes from calculation errors. Choice D results from using the wrong formula or setup.

The actual increase is $3,680 - $3,200 = $480. The percent increase is $\frac{480}{3200} \times 100% = 15%$. Her boss is incorrect. Choice B would be correct if the increase were actually $384. Choice C results from calculating $\frac{480}{3680} \times 100%$ (using new salary as base). Choice D comes from estimation errors in the division.

This problem tests multi-step problems with percents, specifically percent error calculation. Using the given formula: percent error = |estimate - actual|/actual × 100%, we get |54 - 50|/50 × 100% = 4/50 × 100% = 0.08 × 100% = 8%. The percent error is 8%. A common error would be using the estimate in the denominator (4/54 ≈ 7.4%) or forgetting the absolute value (though not relevant here since estimate > actual). Strategy: (1) identify estimate (54g) and actual (50g), (2) find absolute difference |54 - 50| = 4, (3) divide by actual value (4/50 = 0.08), (4) convert to percent (0.08 × 100% = 8%), (5) verify reasonableness (4g error on 50g base is 8%✓). Percent error always uses actual value as the reference.

This question tests multi-step problems with percents, ratios, proportions: tax, tip, markup, markdown, commission, simple interest, percent change, percent error—calculating percent of amounts and combining operations. Percent operations: finding percent of amount ($8%$ of $40 = 40 \times 0.08 = 3.20$), increasing by percent (add: $40 + 3.20 = 43.20$, or multiply: $40 \times 1.08 = 43.20$ directly), decreasing (subtract or multiply by complement: $25%$ off $80 = 80 \times 0.75 = 60$). Multi-step: tax then tip (meal $45$, tax $7%$: $45 \times 1.07 = 48.15$, tip $20%$ on total: $48.15 \times 1.20 \approx 57.78$, or combined: $45 \times 1.07 \times 1.20$). Simple interest $I = P r t$ (principal $\times$ rate $\times$ time in years: $1000 \times 0.05 \times 2 = 100$). Percent change: $( \text{new} - \text{old} ) / \text{old} \times 100%$ ($200 \to 250$: $50 / 200 = 25%$ increase). For example, an item costs $18$, with $8%$ tax: $18 \times 0.08 = 1.44$ tax, total $18 + 1.44 = 19.44$; then add $10%$ donation on total: $19.44 \times 0.10 = 1.944$, altogether $19.44 + 1.944 = 21.384$ or directly $18 \times 1.08 \times 1.10 = 21.384$, rounded to $21.38$. The correct calculation is to first apply the $8%$ tax to $18$ getting $19.44$, then add $10%$ of that as donation, totaling $21.38$. A common error is calculating the donation on the pre-tax amount instead of the taxed total, leading to $18 \times 0.10 = 1.80$, then $18 + 1.44 + 1.80 = 21.24$, which is incorrect. Strategy: (1) identify operations needed (tax: multiply by $1 + \text{rate}$, tip: multiply by $1 + \text{rate}$ on appropriate base, interest: $I = P r t$), (2) sequence properly (tax before tip usually, markups before markdowns if both), (3) use decimal form of percents ($8% = 0.08$, $20% = 0.20$), (4) multiply for efficiency (increase by $8%$ then $20%$: $\times 1.08 \times 1.20$ in one calculation), (5) verify reasonable (total with tax and tip should be ~$30%$ more than meal: $45 \to \sim 58$ reasonable✓). Common formulas: simple interest $I = P r t$ (interest = principal $\times$ rate decimal $\times$ years), percent change = $( \text{new} - \text{old} ) / \text{old}$ (positive: increase, negative: decrease), percent error = $| \text{estimate} - \text{actual} | / \text{actual}$ (absolute difference over actual). Mistakes: percent as whole number (most common: $\times 8$ not $\times 0.08$), wrong base for sequential percents (compounding error), order wrong (operations applied in wrong sequence), formula errors ($I = P r$ without $t$, or wrong denominator in percent change).

When you see commission problems with changing rates and sales amounts, you need to work systematically through each scenario to find the unknown rate.

First, let's verify the January information. With sales of $180,000 and a 3.5% commission rate: $$180,000 \times 0.035 = 6,300$$. This matches the given commission, so we're on track.

Next, calculate February's sales. A 25% increase means: $$180,000 \times 1.25 = 225,000$$. Now we can find the new commission rate. Since commission equals sales times rate, we have: $$225,000 \times \text{rate} = 6,615$$. Solving for the rate: $$\text{rate} = \frac{6,615}{225,000} = 0.0294 = 2.94%$$.

Looking at the wrong answers: Choice A (3.15%) is too high—this would give a February commission of $7,087.50, much more than the actual $6,615. Choice C (2.85%) is too low—this would yield only $6,412.50 in commission. Choice D (3.25%) is also too high—this would result in $7,312.50 in commission.

The correct answer is B (2.94%).

Strategy tip: In multi-step percentage problems, always verify your intermediate calculations before moving to the next step. Here, confirming the January numbers helped ensure accuracy, and calculating February sales before finding the rate kept the solution organized and error-free.

When you encounter multi-step percent problems involving discounts and taxes, work through them systematically: first apply the discount to find the sale price, then calculate tax on that discounted amount.

Start with the 25% markdown. The laptop costs $$1,200 \times 0.25 = \300$$ less, so the sale price is $$1,200 - 300 = \900$$. Now apply the 8.5% sales tax to this discounted price: $$900 \times 0.085 = \76.50$$ in tax. The total amount paid is $$900 + 76.50 = \976.50$$.

Looking at the wrong answers: Choice A ($956.25) represents a common error where students subtract the tax from the sale price instead of adding it ($$900 - 76.50$$). Choice B ($900.00) is just the sale price before tax—students who choose this forget to include sales tax entirely. Choice C ($1,023.00) comes from incorrectly applying the 8.5% tax to the original $1,200 price instead of the discounted price, then subtracting the $300 discount afterward.

The correct sequence gives you D ($976.50).

Study tip: Always remember the order matters in discount and tax problems. Discounts come first, then taxes are calculated on the already-reduced price. Sales tax is never applied to the original price when there's a discount involved. Write out each step clearly to avoid mixing up the sequence.

When you see a problem about the same percentage decrease happening over multiple years, you're dealing with compound percent change. This is different from a simple percentage decrease because each year's decrease is calculated on the remaining amount from the previous year, not the original amount.

Let's work backwards from the final enrollment. If the school starts with 850 students and ends with 765 students after two identical percentage decreases, we need to find what percentage decrease each year would give us this result.

Let's call the annual decrease rate $$r$$. After the first year, enrollment would be $$850(1-r)$$. After the second year, it would be $$850(1-r)^2 = 765$$.

Solving for $$r$$: $$(1-r)^2 = \frac{765}{850} = 0.9$$

Taking the square root: $$1-r = \sqrt{0.9} = 0.95$$

Therefore: $$r = 1 - 0.95 = 0.05 = 5%$$

Let's check: $$850 × 0.95 × 0.95 = 765.625 ≈ 765$$ ✓

Now for the wrong answers: Choice A (4.76%) would result in about 773 students remaining, which is too high. Choice B (10%) is way too large—this would leave only about 688 students. Choice D (5.25%) would result in approximately 763 students, which is close but still incorrect.

Study tip: For compound percentage problems, remember that equal percentage changes over multiple periods require you to use exponents, not simple multiplication. Always check your work by calculating forward from your answer.

When you see commission problems, you're working with percentages and multi-step calculations. The key is to organize your work: find the total sales, calculate the commission, then apply any splits or deductions.

First, find the total value of homes sold: $$320,000 + 275,000 + 410,000 = 1,005,000$$. Next, calculate the 6% commission on this total: $$1,005,000 × 0.06 = 60,300$$. Since she splits this commission equally with her broker, divide by 2: $$60,300 ÷ 2 = 30,150$$. So she personally earned $30,150.

Looking at the wrong answers: Choice A ($31,200) likely comes from incorrectly calculating 6% of the total sales—perhaps rounding errors or miscalculating the percentage. Choice B ($60,300) is the total commission before splitting with the broker; this represents forgetting the final step of dividing by 2. Choice C ($28,750) might result from calculation errors in either the total sales amount or the percentage calculation, possibly confusing which operations to perform.

The correct answer is D ($30,150).

Remember this pattern for commission problems: total sales × commission rate = total commission, then apply any splits or deductions. Always read carefully to see if the person keeps the full commission or shares it. Many students forget that final step of splitting the commission, so double-check what the question asks for—total commission earned or the person's actual share.

Percent error questions test your ability to calculate how far off a measurement is from the true value. When you see "percent error," you're looking at the difference between what was measured and what the actual value is, expressed as a percentage of the actual value.

To find percent error, use this formula: $$\text{Percent Error} = \frac{|\text{Measured Value} - \text{Actual Value}|}{|\text{Actual Value}|} \times 100%$$

Here, the measured value is 24.8 grams and the actual value is 25.5 grams. First, find the absolute difference: $$|24.8 - 25.5| = |-0.7| = 0.7$$ grams. Then divide by the actual value: $$\frac{0.7}{25.5} = 0.02745...$$ Finally, convert to a percentage: $$0.02745 \times 100% = 2.745%$$, which rounds to 2.75%.

Choice A (2.90%) likely comes from dividing the error by the measured value instead of the actual value: $$\frac{0.7}{24.8} \times 100% = 2.82%$$ then rounding incorrectly. Choice B (2.82%) makes the same mistake but rounds correctly. Choice D (2.65%) might result from calculation errors or using an incorrect formula altogether.

Remember that percent error always uses the actual (true) value in the denominator, not the measured value. This is because you want to know how the error compares to what the measurement should have been. Also, always use absolute values since you only care about the size of the error, not its direction.