Multi-Step Ratio and Percent Problems - 7th Grade Math

1. A video game originally costs $60. It is discounted by $25\%$, and then an additional $8\%$ sales tax is added to the discounted price. What is the final price (rounded to the nearest cent)?
   Answer: $48.60 This problem tests multi-step problems with percents, applying a discount followed by sales tax. First, we calculate the discounted price: $60 × 0.75 = $45 (25% off means paying 75%). Then we add 8% tax to the discounted price: $45 × 1.08 = $48.60. We can combine operations: $60 × 0.75 × 1.08 = $48.60. The final price is $48.60. A common error would be to calculate both percentages on the original price separately (25% of $60 = $15 discount, 8% of $60 = $4.80 tax) then compute $60 - $15 + $4.80 = $49.80, which incorrectly applies tax to the original rather than discounted price. Strategy: (1) apply discount first (multiply by 0.75 for 25% off), (2) then apply tax to the discounted amount (multiply by 1.08), (3) use complement for discounts (25% off = 75% of price), (4) sequence operations properly, (5) verify reasonableness (should be less than original $60 but not by full 25% due to tax✓).
2. A fundraiser collects $520$ in donations. The website charges a $3\%$ processing fee, then the organizer pays a flat $5$ transfer fee. How much money is left after both fees (rounded to the nearest cent)?
   Answer: $499.40$ 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: fees (donations $520$, 3% fee: $520 \times 0.97 = 504.40$ after, then subtract $5$: $504.40 - 5 = 499.40$). For this problem, $520$ minus 3% fee ($520 \times 0.03 = 15.60$, $520 - 15.60 = 504.40$), then minus $5$: $504.40 - 5 = 499.40$. A common error is subtracting fees in wrong order or using 3% as addition instead of deduction, or forgetting the flat fee. Strategy: (1) identify operations needed (percent fee: $\times(1-\text{rate})$, flat fee: subtract), (2) sequence properly (percent then flat), (3) use decimal form of percents (3% = 0.03), (4) calculate step-by-step, (5) verify reasonable (fees total about $20.60$, $520 - 20.60 = 499.40$ yes). Common formulas: percent error = $|\text{estimate} - \text{actual}| / \text{actual} \times 100\%$; mistakes include wrong sequence or percent as whole number.
3. A pair of sneakers is originally priced at $80. It is discounted by $15\%$, and then an additional $10\%$ coupon is applied to the discounted price. What is the final price (rounded to the nearest cent)?
   Answer: $61.20 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: successive discounts (item $80$, $15\%$ off: $80 \times 0.85 = 68$, then $10\%$ off: $68 \times 0.90 = 61.20$, or combined $80 \times 0.85 \times 0.90 = 61.20$). For this problem, original $80$, $15\%$ discount to $80 \times 0.85 = 68$, then $10\%$ coupon to $68 \times 0.90 = 61.20$. A common error is adding discounts directly ($15\% + 10\% = 25\%$, $80 \times 0.75 = 60$, wrong due to no compounding), or applying in wrong order. Strategy: (1) identify operations needed (discount: $\times 1$-rate), (2) sequence properly, (3) use decimal form of percents ($15\% = 0.15$, $10\% = 0.10$), (4) multiply for efficiency ($\times 0.85 \times 0.90 = 0.765$ overall, $80 \times 0.765 = 61.20$), (5) verify reasonable (total discount about $23.5\%$, $80 \rightarrow 61.20$ yes). Common formulas: percent change = $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$; mistakes include adding percents directly or wrong base for sequential percents.
4. A restaurant automatically adds an 18% gratuity to parties of 6 or more. If a party of 8 people has a bill of $240 before the gratuity, and they decide to leave an additional $15 cash tip, what is the total percent tip based on the original bill?
   Answer: 24.25% 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.
5. A student sells coupon books for a club. The student earns a $6\%$ commission on total sales. If the student sold $250 worth of coupon books, how much commission did the student earn?
   Answer: $15 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 = 40×0.08=$3.20), increasing by percent (add: $40+$3.20=$43.20, or multiply: 40×1.08=43.20 directly), decreasing (subtract or multiply by complement: 25% off $80 = 80×0.75=$60). Multi-step: tax then tip (meal $45, tax 7%: 45×1.07=48.15, tip 20% on total: 48.15×1.20≈57.78, or combined: 45×1.07×1.20). Simple interest I=Prt (principal×rate×time in years: $1000×0.05×2=$100). Percent change: (new-old)/old×100% (200→250: 50/200=25% increase). For example, 6% commission on $200 sales: 200×0.06=12. The correct calculation is 250×0.06=15. A common error is using 6 instead of 0.06, getting 250×6=1500, or adding to sales like 250+15=265. Strategy: (1) identify operations needed (tax: multiply by 1+rate, tip: multiply by 1+rate on appropriate base, interest: I=Prt), (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%: ×1.08×1.20 in one calculation), (5) verify reasonable (total with tax and tip should be ~30% more than meal: $45→~$58 reasonable✓). Common formulas: simple interest I=Prt (interest = principal × rate decimal × years), percent change = (new-old)/old (positive: increase, negative: decrease), percent error = |estimate-actual|/actual (absolute difference over actual). Mistakes: percent as whole number (most common: ×8 not ×0.08), wrong base for sequential percents (compounding error), order wrong (operations applied in wrong sequence), formula errors (I=Pr without t, or wrong denominator in percent change).
6. A video game costs $50 before tax. The tax rate is $8\%$. You have a coupon for $10\%$ off the pre-tax price, and the cashier applies the coupon first, then adds tax. What is the final price (rounded to the nearest cent)?
   Answer: $48.60 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 = 40×0.08=$3.20), increasing by percent (add: $40+$3.20=$43.20, or multiply: 40×1.08=43.20 directly), decreasing (subtract or multiply by complement: 25% off $80 = 80×0.75=$60). Multi-step: coupon then tax (game $50, 10% off: 50×0.90=45, then 8% tax: 45×1.08=48.60). For this problem, apply 10% coupon first: 50×0.90=45, then 8% tax: 45×1.08=48.60. A common error is applying tax before coupon (50×1.08=54, then 54×0.90=48.60 same here, but wrong if order specified), or discounting the tax amount. Strategy: (1) identify operations needed (discount: ×0.90, tax: ×1.08), (2) sequence properly (coupon before tax as stated), (3) use decimal form of percents, (4) multiply for efficiency (50×0.90×1.08=48.60), (5) verify reasonable (10% off saves $5, tax on $45 adds $3.60, total $48.60 yes). Common formulas: percent change = (new-old)/old×100%; mistakes include wrong order or adding percents directly.
7. A real estate agent charges a 6% commission on home sales. Last month, she sold 3 homes for $320,000, $275,000, and $410,000 respectively. If she splits her commission equally with her broker, how much money did she personally earn last month?
   Answer: $30,150 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.
8. Jordan sells raffle tickets for a school club and earns a $6\%$ commission on total sales. If Jordan sells $750 worth of tickets, how much commission does Jordan earn?
   Answer: $45 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 = 40×0.08=$3.20), increasing by percent (add: $40+$3.20=$43.20, or multiply: 40×1.08=43.20 directly), decreasing (subtract or multiply by complement: 25% off $80 = 80×0.75=$60). For commission, like 6% on $750: 750×0.06=$45. For this problem, sales $750 with 6% commission: 750×0.06=$45. A common error is using 6 instead of 0.06 (750×6=$4500, way too high), or dividing instead of multiplying. Strategy: (1) identify operations needed (commission: sales×rate decimal), (2) use decimal form of percents (6%=0.06), (3) multiply for efficiency, (4) verify reasonable (6% of $750 should be around $45, yes). Common formulas: simple interest I=Prt, percent change = (new-old)/old×100%; mistakes include percent as whole number or formula errors.
9. At a school fundraiser, a T-shirt costs $18.00$ and the sales tax is $8\%$. If you also donate an extra $10\%$ of the taxed total to support the club, how much do you pay altogether?
   Answer: $21.38$ 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).
10. A savings account uses simple interest. A student deposits $400 at $5\%$ simple interest per year for 3 years. How much interest will they earn?
    Answer: $60 This problem tests multi-step problems with percents, specifically simple interest calculation using the formula I = Prt. We have principal P = $400, rate r = 0.05 (5% as decimal), and time t = 3 years: I = $400 × 0.05 × 3 = $60. The interest earned is $60. A common error would be forgetting to multiply by time (calculating just $400 × 0.05 = $20 for one year) or using the percent as a whole number ($400 × 5 × 3 = $6,000). Strategy: (1) identify the three components: principal ($400), rate (5% = 0.05), time (3 years), (2) apply formula I = Prt, (3) ensure rate is in decimal form, (4) multiply all three values, (5) verify reasonableness (5% per year for 3 years = 15% total of $400 = $60✓). Simple interest accumulates linearly, not compounded.
11. At a restaurant, a meal costs $36.00. The tax is $7\%$, and you leave a $20\%$ tip on the total after tax. What is the total amount you pay (rounded to the nearest cent)?
    Answer: $46.22 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 = 40×0.08=$3.20), increasing by percent (add: $40+$3.20=$43.20, or multiply: 40×1.08=43.20 directly), decreasing (subtract or multiply by complement: 25% off $80 = 80×0.75=$60). Multi-step: tax then tip (meal $45, tax 7%: 45×1.07=48.15, tip 20% on total: 48.15×1.20≈57.78, or combined: 45×1.07×1.20). For this problem, meal is $36 with 7% tax (36×1.07=38.52), then 20% tip on total (38.52×1.20=46.224, rounded to $46.22). A common error is tipping on pre-tax amount instead of post-tax (tip on 36: 7.20, tax 2.52, total 36+2.52+7.20=45.72, wrong), or adding percents directly (7%+20%=27%, 36×1.27=45.72, ignores compounding). Strategy: (1) identify operations needed (tax: multiply by 1+rate, tip: multiply by 1+rate on appropriate base), (2) sequence properly (tax before tip usually), (3) use decimal form of percents (7%=0.07, 20%=0.20), (4) multiply for efficiency (×1.07×1.20 in one calculation), (5) verify reasonable (total with tax and tip should be ~27% more, but compounded to about 28.4%, $36→$46.22 reasonable). Common formulas: percent change = (new-old)/old×100%; mistakes include wrong base for sequential percents or order wrong.
12. At a restaurant, a meal costs $36.00. The tax is $7\%$, and you leave a $20\%$ tip on the total after tax. What is the total amount you pay (rounded to the nearest cent)?​
    Answer: $46.22 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 = 40×0.08=$3.20), increasing by percent (add: $40+$3.20=$43.20, or multiply: 40×1.08=43.20 directly), decreasing (subtract or multiply by complement: 25% off $80 = 80×0.75=$60). Multi-step: tax then tip (meal $45, tax 7%: 45×1.07=48.15, tip 20% on total: 48.15×1.20≈57.78, or combined: 45×1.07×1.20). For this problem, meal is $36 with 7% tax (36×1.07=38.52), then 20% tip on total (38.52×1.20=46.224, rounded to $46.22). A common error is tipping on pre-tax amount instead of post-tax (tip on 36: 7.20, tax 2.52, total 36+2.52+7.20=45.72, wrong), or adding percents directly (7%+20%=27%, 36×1.27=45.72, ignores compounding). Strategy: (1) identify operations needed (tax: multiply by 1+rate, tip: multiply by 1+rate on appropriate base), (2) sequence properly (tax before tip usually), (3) use decimal form of percents (7%=0.07, 20%=0.20), (4) multiply for efficiency (×1.07×1.20 in one calculation), (5) verify reasonable (total with tax and tip should be ~27% more, but compounded to about 28.4%, $36→$46.22 reasonable). Common formulas: percent change = (new-old)/old×100%; mistakes include wrong base for sequential percents or order wrong.
13. A savings account uses simple interest. You deposit $600$ at an annual interest rate of $5\%$ for $3$ years. How much interest will you earn? (Use $I=Prt$.)
    Answer: $90$ 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 = $80 \times 0.75 = 60$). For simple interest, like $600 at 5% for 3 years: $I = 600 \times 0.05 \times 3 = 90$. For this problem, $I = 600 \times 0.05 \times 3 = 90$. A common error is forgetting to multiply by time ($600 \times 0.05 = 30$, wrong), or using rate as 5 instead of 0.05 ($600 \times 5 \times 3 = 9000$, too high). Strategy: (1) identify formula $I=Prt$, (2) use decimal rate (5%=0.05), (3) multiply all terms, (4) verify reasonable (5% per year for 3 years = 15% total, 15% of 600 = 90 yes). Common formulas: simple interest $I=Prt$; mistakes include forgetting t or percent as whole number.
14. A smoothie shop offers a deal: take 15% off an $8.00 smoothie, then add 8% sales tax to the discounted price. What is the final price, rounded to the nearest cent?
    Answer: $7.34 First find the discounted price: $8.00 x 0.85 = $6.80 (paying 85% after a 15% discount). Then add 8% sales tax to that discounted price: $6.80 x 1.08 = $7.344, which rounds to $7.34. Choice B shows only the discounted price before tax is added. Choice C comes from adding tax to the original price instead of the discounted price. Choice D comes from a small rounding or calculation slip in the tax step.
15. A store offers a $10\%$ discount on a $50 backpack. After the discount, an additional $5\%$ fee is added to the discounted price for customization. What is the final price?
    Answer: $47.25 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: $40 \times 1.08 = 43.20$), 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=Prt (principal×rate×time in years: $1000 \times 0.05 \times 2 = 100$). Percent change: $(new-old)/old \times 100\%$ (200→250: 50/200=25% increase). For example, $50$ item, 10% discount: $50 \times 0.90 = 45$, then 5% fee: $45 \times 1.05 = 47.25$. The correct calculation is 50×0.90=45 discounted, then 45×1.05=47.25 final. A common error is adding the fee before discount, like 50×1.05=52.50, then 52.50×0.90=47.25 (same here, but wrong sequence could differ in other cases). Strategy: (1) identify operations needed (tax: multiply by 1+rate, tip: multiply by 1+rate on appropriate base, interest: I=Prt), (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%: ×1.08×1.20 in one calculation), (5) verify reasonable (total with tax and tip should be ~30% more than meal: $45$→~$58 reasonable✓). Common formulas: simple interest I=Prt (interest = principal × rate decimal × years), percent change = (new-old)/old (positive: increase, negative: decrease), percent error = |estimate-actual|/actual (absolute difference over actual). Mistakes: percent as whole number (most common: ×8 not ×0.08), wrong base for sequential percents (compounding error), order wrong (operations applied in wrong sequence), formula errors (I=Pr without t, or wrong denominator in percent change).
16. A science class estimated that a plant grew to $18$ cm, but the actual height was $16$ cm. What is the percent error? (Round to the nearest tenth of a percent.)
    Answer: $12.5\%$ 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 = 40×0.08=$3.20), increasing by percent (add: $40+$3.20=$43.20, or multiply: 40×1.08=43.20 directly), decreasing (subtract or multiply by complement: 25% off $80 = 80×0.75=$60). Multi-step: tax then tip (meal $45, tax 7%: 45×1.07=48.15, tip 20% on total: 48.15×1.20≈57.78, or combined: 45×1.07×1.20). Simple interest I=Prt (principal×rate×time in years: $1000×0.05×2=$100). Percent change: (new-old)/old×100% (200→250: 50/200=25% increase). For example, estimate 20 cm, actual 18 cm: |20-18|/18×100%≈11.1%, but note formula uses actual in denominator. The correct calculation is |18-16|/16×100%=2/16×100%=12.5%. A common error is using estimate in denominator: |18-16|/18×100%≈11.1%, or forgetting absolute value and getting negative. Strategy: (1) identify operations needed (tax: multiply by 1+rate, tip: multiply by 1+rate on appropriate base, interest: I=Prt), (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%: ×1.08×1.20 in one calculation), (5) verify reasonable (total with tax and tip should be ~30% more than meal: $45→~$58 reasonable✓). Common formulas: simple interest I=Prt (interest = principal × rate decimal × years), percent change = (new-old)/old (positive: increase, negative: decrease), percent error = |estimate-actual|/actual (absolute difference over actual). Mistakes: percent as whole number (most common: ×8 not ×0.08), wrong base for sequential percents (compounding error), order wrong (operations applied in wrong sequence), formula errors (I=Pr without t, or wrong denominator in percent change).
17. A car dealership offers a 3.5% commission on sales. In January, a salesperson sold cars worth $180,000 and earned $6,300 in commission. In February, total sales increased by 25%, but the commission rate was reduced. If the salesperson earned $6,615 in February, what was the new commission rate?
    Answer: 2.94% 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.
18. Maria's monthly salary increased from $3,200 to $3,680. Her boss told her this was a 12% raise, but Maria thinks it's more than 12%. What is the actual percent increase in Maria's salary, and is her boss correct?
    Answer: 15% increase; her boss is incorrect 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.
19. A school store buys notebooks for $2.50 each and marks them up by $40\%$. Later, the store has a sale and takes $10\%$ off the marked-up price. What is the sale price of one notebook?
    Answer: $3.15 This problem tests multi-step problems with percents, combining markup and markdown operations. First, we calculate the marked-up price: $2.50 \times 1.40 = 3.50$. Then we apply the 10% discount to this marked-up price: $3.50 \times 0.90 = 3.15$. We can also combine: $2.50 \times 1.40 \times 0.90 = 3.15$. The sale price is $3.15$. A common error would be to add the percentages (40% - 10% = 30% markup) and calculate $2.50 \times 1.30 = 3.25$, which incorrectly assumes the discount applies to the original price rather than the marked-up price. Strategy: (1) apply markup first (multiply by 1.40), (2) then apply discount to the new price (multiply by 0.90), (3) use decimal forms consistently, (4) remember that sequential percent changes compound rather than add, (5) verify the result makes sense (final price should be between original $2.50$ and full markup $3.50$✓).
20. Simple interest is calculated using the formula $I = PRT$, where $P$ is principal, $R$ is annual rate, and $T$ is time in years. If $500 is invested at 4.5% annual simple interest and grows to $556.25 total, how many months was the money invested?
    Answer: 30 months The interest earned is 556.25 - 500 = $56.25. Using I = PRT, this gives 56.25 = 500 x 0.045 x T, so T = 56.25 / (500 x 0.045) = 56.25/22.5 = 2.5 years, which is 30 months. Choice A, choice C, and choice D come from arithmetic slips or from using an incorrect rate or time conversion when solving for T.

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