Finding Percent of a Quantity - Accuplacer Arithmetic

1. A car's value depreciates by 18% each year. After 2 years, the car is worth $13,448. What was the car's original purchase price?
   Answer: $20,000 was the original price When you encounter depreciation problems, you're working backwards from a final value to find an original value. The key insight is that if something depreciates by 18% each year, it retains 82% of its value (100% - 18% = 82%). Let's set up the equation. If the original price is P, then after one year the car is worth 0.82P. After two years, it's worth 0.82 × (0.82P) = (0.82)²P = 0.6724P. We know this final value equals $13,448. So: 0.6724P = $13,448 Solving for P: P = $13,448 ÷ 0.6724 = $20,000 Let's verify: $20,000 × 0.82 = $16,400 (after year 1), then $16,400 × 0.82 = $13,448 (after year 2) ✓ Answer C ($20,000) is correct. Answer A ($18,800) would give you $10,668 after two years, which is too low. Answer B ($19,500) would result in $13,111 after two years, still short of our target. Answer D ($19,200) would yield $12,912 after two years, also too low. These incorrect answers likely come from calculation errors or misunderstanding the compounding nature of depreciation. Strategy tip: For depreciation problems, always convert the percentage to what remains (100% - depreciation rate), then work backwards by dividing the final value by the retention rate raised to the number of years. Double-check by calculating forward from your answer.
2. A savings account containing $2,500 earns 1.4% simple interest annually. How much interest is earned in one year?
   Answer: $35.00 To find the interest earned, multiply the principal amount by the interest rate. First, convert the percentage to a decimal: $1.4\% = 0.014$. Then, multiply: $\$2,500 \times 0.014 = \$35.00$.
3. A clothing store offers a "buy 2, get 1 at 50% off" promotion on items that are already marked 30% off the original price. If someone buys 3 identical shirts originally priced at $45 each, what percent of the total original price do they actually pay?
   Answer: Approximately 58% of the original total price When you encounter multi-step discount problems, work through each promotion systematically to avoid compounding errors. First, calculate the 30% markdown on each $45 shirt: $45 \times 0.30 = 13.50$, so each shirt costs $45 - 13.50 = 31.50$ after the initial discount. Next, apply the "buy 2, get 1 at 50% off" promotion to the already-discounted prices. You pay full price for two shirts ($31.50 \times 2 = 63.00$) and half price for the third shirt ($31.50 \times 0.50 = 15.75$). Your total payment is $63.00 + 15.75 = 78.75$. The original total for three shirts was $45 \times 3 = 135$. To find what percentage you actually pay: $\frac{78.75}{135} = 0.583$, or approximately 58%. Answer B is correct at approximately 58% of the original total price. Answer A (70%) likely comes from only applying the 30% initial discount and ignoring the second promotion. Answer C (65%) might result from incorrectly averaging the two discount percentages or applying them in the wrong order. Answer D (75%) probably comes from applying the 50% discount to the original price of just one shirt rather than to the already-discounted price. Remember: when multiple promotions stack, apply them sequentially in the order given, and always check that your final percentage makes sense—it should be less than what either discount would achieve alone.
4. A restaurant bill for a group of diners is $45.50 for food and $12.50 for drinks. What is the total cost, including an 18% tip calculated on the entire bill?
   Answer: $68.44 First, calculate the total bill before the tip: $\$45.50 + \$12.50 = \$58.00$. Next, calculate the 18% tip on this amount: $0.18 \times 58.00 = \$10.44$. Finally, add the tip to the bill to find the total cost: $\$58.00 + \$10.44 = \$68.44$.
5. A manufactured part is designed to be 80 centimeters long. The quality control standards allow for a margin of error of 0.5%. What is the maximum acceptable length for the part?
   Answer: 80.4 cm First, calculate the allowed error amount: 0.5% of 80 cm. Convert 0.5% to a decimal: $0.5\% = 0.005$. Then multiply: $0.005 \times 80 = 0.4$ cm. The maximum acceptable length is the designed length plus the error margin: $80 + 0.4 = 80.4$ cm.
6. A granola bar contains 250 calories, and 20% of the calories come from fat. If a person eats 60% of the bar, how many calories from fat did they consume?
   Answer: 30 First, find the total number of fat calories in the entire bar: 20% of 250 is $0.20 \times 250 = 50$ fat calories. The person eats 60% of the bar, so they consume 60% of the fat calories: 60% of 50 is $0.60 \times 50 = 30$ fat calories.
7. A retail store marks up all items by 35% above wholesale cost. During a clearance sale, the store offers 20% off the marked retail price. If a customer buys an item that originally cost the store $60 wholesale, what percent of the wholesale cost does the customer actually pay?
   Answer: 108% of the wholesale cost When you encounter percentage markup and discount problems, you need to track how each percentage change affects the price step by step, working from the original wholesale cost through each transformation. Let's trace through this problem systematically. The wholesale cost is $60. First, the store marks up by 35%, so the retail price becomes: $60 × 1.35 = 81$ dollars. Next, during the clearance sale, customers get 20% off this marked retail price. The sale price becomes: $81 × 0.80 = 64.80$ dollars. To find what percent of the wholesale cost the customer pays, divide the final price by the wholesale cost: $\frac{64.80}{60} = 1.08 = 108\%$ Looking at the wrong answers: Choice A (85%) likely comes from mistakenly thinking the customer pays 100% - 15% = 85% of wholesale, incorrectly combining the markup and discount. Choice B (115%) might result from adding the markup percentage (35%) and subtracting the discount percentage (20%) to get 15% above wholesale. Choice C (92%) could come from applying the 20% discount directly to the 115% figure, getting 115% - 23% ≈ 92%. The key insight is that percentages compound—you can't simply add and subtract them. Each percentage operates on the result of the previous calculation, not the original amount. Strategy tip: In multi-step percentage problems, always calculate each step sequentially and convert percentages to decimals (35% = 1.35 for increases, 20% off means multiply by 0.80). This prevents the common error of trying to combine percentages arithmetically.
8. A salesperson earns a 5% commission on the first $10,000 of sales and an 8% commission on all sales above $10,000. If the salesperson makes $22,000 in sales, what is their total commission?
   Answer: $1,460 The commission is calculated in two parts. First part: 5% of the first $10,000 is $0.05 \times 10,000 = \$500$. Second part: The amount of sales over $10,000 is $\$22,000 - \$10,000 = \$12,000$. The commission on this amount is 8% of $12,000, which is $0.08 \times 12,000 = \$960$. The total commission is the sum of the two parts: $\$500 + \$960 = \$1,460$.
9. What is the sum of 20% of 180 and 30% of 120?
   Answer: 72 This is a two-part calculation. First, find 20% of 180: $0.20 \times 180 = 36$. Second, find 30% of 120: $0.30 \times 120 = 36$. Finally, find the sum of these two results: $36 + 36 = 72$.
10. The price of a television is increased by 10%. The new price is then decreased by 10%. If the original price was $150, what is the final price?
    Answer: $148.50 First, calculate the 10% increase: 10% of $150 is $0.10 \times 150 = \$15$. The new price is $\$150 + \$15 = \$165$. Next, calculate the 10% decrease on the new price: 10% of $165 is $0.10 \times 165 = \$16.50$. The final price is $\$165 - \$16.50 = \$148.50$.
11. A library has an annual budget of $400,000. Sixty percent of the budget is used for salaries. Of the remaining funds, 25% is spent on new books. How much money is spent on new books?
    Answer: $40,000 First, calculate the amount spent on salaries: 60% of $400,000 is $0.60 \times 400,000 = \$240,000$. Next, find the remaining funds: $\$400,000 - \$240,000 = \$160,000$. Finally, calculate the amount for new books, which is 25% of the remaining funds: $0.25 \times 160,000 = \$40,000$.
12. A project is estimated to take 80 hours to complete. After some work is done, the project manager reports that 35% of the project is complete. How many hours of work are left to finish the project?
    Answer: 52 First, calculate the number of hours already completed: 35% of 80 hours is $0.35 \times 80 = 28$ hours. To find the hours left, subtract the completed hours from the total: $80 - 28 = 52$ hours.
13. A jacket originally priced at $80 is on sale for 15% off. If the sales tax is 5% of the sale price, what is the final cost of the jacket?
    Answer: $71.40 First, calculate the discount amount: 15% of $80 is $0.15 \times 80 = \$12$. The sale price is $\$80 - \$12 = \$68$. Next, calculate the sales tax on the sale price: 5% of $68 is $0.05 \times 68 = \$3.40$. The final cost is the sale price plus the tax: $\$68 + \$3.40 = \$71.40$.
14. Maria's monthly salary increased by 8% in January, then decreased by 12% in March due to reduced hours. If her final salary is $2,376 per month, what was her original salary before the January increase?
    Answer: $2,500 per month originally When you encounter percentage change problems with multiple steps, you need to work backwards from the final result to find the original value. This requires setting up an equation that accounts for all the changes in sequence. Let's call Maria's original salary $x$. After an 8% increase in January, her salary became $x \times 1.08$. Then after a 12% decrease in March, her salary became $(x \times 1.08) \times 0.88 = x \times 1.08 \times 0.88 = x \times 0.9504$. Since we know her final salary is $2,376, we can write: $x \times 0.9504 = 2376$. Solving for $x$: $x = \frac{2376}{0.9504} = 2500$. Let's verify: $2,500 × 1.08 = $2,700 (after January increase), then $2,700 × 0.88 = $2,376 (after March decrease). This confirms our answer. Choice A ($2,400) gives us $2,400 × 0.9504 = $2,281, which is too low. Choice B ($2,475) gives us $2,475 × 0.9504 = $2,352, still too low. Choice D ($2,525) gives us $2,525 × 0.9504 = $2,399, which is too high. These wrong answers likely represent common calculation errors or incorrect interpretations of the percentage changes. Remember that when working with sequential percentage changes, multiply the decimal forms together first (1.08 × 0.88 = 0.9504), then divide the final result by this combined factor to find the original value. Always verify your answer by working forward through the problem.
15. A container holds $\frac{3}{4}$ of a gallon of liquid. If 40% of the liquid is poured out, how many gallons of liquid were poured out?
    Answer: $\frac{3}{10}$ The amount poured out is 40% of $\frac{3}{4}$ gallons. First, convert the percent and fraction to decimals for easier multiplication: 40% is 0.4 and $\frac{3}{4}$ is 0.75. Then, multiply: $0.4 \times 0.75 = 0.3$. Finally, convert the decimal answer back to a fraction: $0.3 = \frac{3}{10}$.
16. A school has 850 students. A survey reveals that 28% of the students own a pet. How many students at the school do NOT own a pet?
    Answer: 612 First, determine the percentage of students who do not own a pet: $100\% - 28\% = 72\%$. Now, find 72% of the total number of students: $0.72 \times 850 = 612$. So, 612 students do not own a pet.
17. What is $\frac{1}{2}\%$ of 800?
    Answer: 4 First, convert $\frac{1}{2}\%$ to a decimal. $\frac{1}{2}\%$ is equal to 0.5%. To convert a percent to a decimal, divide by 100. So, $0.5 \div 100 = 0.005$. Now, multiply this decimal by 800: $0.005 \times 800 = 4$.
18. A town with a population of 12,000 experiences a 5.5% increase in population in one year. What is the town's population after the increase?
    Answer: 12,660 First, calculate the number of new people: 5.5% of 12,000 is $0.055 \times 12,000 = 660$. Then, add this increase to the original population: $12,000 + 660 = 12,660$.
19. A recipe that makes 24 muffins requires 300 grams of sugar. A baker wants to make a batch that is 175% of the standard recipe size. How many grams of sugar are needed?
    Answer: 525 To find the amount of sugar needed for the larger batch, calculate 175% of the original amount. Convert 175% to a decimal, which is 1.75. Then multiply by the original amount of sugar: $1.75 \times 300 = 525$ grams.
20. A school's enrollment increased by 15% from last year to this year, reaching 1,380 students. Due to budget constraints, the school must reduce enrollment by 8% next year. How many students will be enrolled next year?
    Answer: 1,270 students will be enrolled next year When you encounter percentage problems involving sequential changes, you need to work step-by-step through each change rather than trying to combine percentages directly. First, find last year's enrollment. Since this year's 1,380 students represents a 15% increase from last year, you can set up: last year's enrollment × 1.15 = 1,380. Solving for last year: $1,380 ÷ 1.15 = 1,200$ students. Next, calculate next year's enrollment by reducing this year's count by 8%. An 8% reduction means keeping 92% of current enrollment: $1,380 × 0.92 = 1,269.6$, which rounds to 1,270 students. Looking at the wrong answers: Choice A (1,284) likely comes from incorrectly reducing last year's enrollment (1,200) by 8% instead of this year's enrollment, giving $1,200 × 0.92 = 1,104$, or from some other calculation error. Choice C (1,296) might result from reducing this year's enrollment by only 6% instead of 8% ($1,380 × 0.94 = 1,297.2$). Choice D (1,310) could come from mistakenly applying a 5% reduction instead of 8%. The key strategy here is to resist the urge to work backwards from the final answer or combine percentage changes. Instead, identify what each percentage change applies to: the 15% increase applied to an unknown previous amount, while the 8% decrease applies to the known current amount of 1,380 students. Always read carefully to determine the base amount for each percentage calculation.

Start this quiz