Percent of a Quantity - SHSAT Math

A company's quarterly profits were distributed as follows: 45% to shareholders, 30% reinvested in the company, and 25% set aside for taxes. If $180,000 was reinvested, how much was set aside for taxes?
Answer: $150,000 If $180,000 represents 30% of total profits, then total profits = $180,000 ÷ 0.30 = $600,000. Taxes are 25% of total: $600,000 × 0.25 = $150,000. Choice A results from calculating $180,000 × 25 ÷ 30 incorrectly. Choice B comes from estimation errors in the division. Choice D results from using the ratio 30:25 incorrectly as $180,000 × 0.90.
What is $6.25\%$ of $64$ ?
Answer: $4$ When you see a percentage problem like this, you're working with the fundamental relationship: part = percent × whole. Here, you need to find what $6.25\%$ of $64$ equals. First, convert the percentage to a decimal: $6.25\% = 6.25 ÷ 100 = 0.0625$ . Then multiply: $0.0625 × 64 = 4$ . There's also a helpful shortcut here. Notice that $6.25\% = \frac{6.25}{100} = \frac{1}{16}$ . So you're really finding $\frac{1}{16}$ of $64$ , which equals $64 ÷ 16 = 4$ . Looking at the wrong answers: Choice (A) gives $3$ , which would result from incorrectly calculating $6.25\% × 48$ instead of $64$ , or from other computational errors. Choice (C) gives $5$ , which might come from rounding $6.25\%$ to $8\%$ and then calculating $8\% × 64 = 5.12 ≈ 5$ . Choice (D) gives $8$ , which is exactly what you'd get if you calculated $12.5\%$ of $64$ instead—this represents doubling the given percentage. The correct answer is (B) $4$ . Study tip: Memorize that $6.25\% = \frac{1}{16}$ and $12.5\% = \frac{1}{8}$ . These percentages appear frequently on standardized tests, and recognizing them as simple fractions makes calculations much faster. Also, always double-check percentage problems by asking if your answer makes sense— $6.25\%$ is pretty small, so the result should be much smaller than the original number.
Out of $120$ eighth-grade students, $15\%$ were absent on Monday. How many students were absent?
Answer: $18$ This is a percentage calculation problem, which is fundamental to many math applications. When you see "15% of 120 students," you need to convert the percentage to a decimal and multiply. To find 15% of 120 students, convert 15% to decimal form by dividing by 100: $15\% = \frac{15}{100} = 0.15$ . Then multiply: $0.15 \times 120 = 18$ students were absent. You can also think of this as: $\frac{15}{100} \times 120 = \frac{15 \times 120}{100} = \frac{1800}{100} = 18$ . Looking at the wrong answers: Choice A (12) might result from incorrectly calculating 10% of 120 instead of 15%. Choice B (15) is a common trap—students sometimes give the percentage value itself rather than calculating what that percentage represents of the total. This would be correct if the question asked "what percentage were absent," but it asks "how many students." Choice D (20) could come from mistakenly calculating $\frac{20}{100} \times 120$ or confusing this with a different percentage. The correct answer is C (18). Study tip: For percentage problems, always identify what you're finding a percentage of (here, 120 students), convert the percentage to a decimal, then multiply. Watch out for the trap of giving the percentage number itself as your answer—always check that your final answer makes sense in context.
Sales tax in a city is $7\%$ . How much tax is added to a purchase of $65.00?
Answer: $4.55 When you encounter a sales tax problem, you're working with percentages and need to find what percent of the original price gets added as tax. To calculate $7\%$ tax on a $65.00 purchase, convert the percentage to a decimal and multiply: $7\% = 0.07$ , so $0.07 \times 65 = 4.55$ . The tax added is $4.55. Let's examine why each answer choice might appear: A) $4.20 represents a common calculation error. You might get this if you mistakenly calculated $6\%$ instead of $7\%$ ( $0.06 \times 65 = 3.90$ ), or made an arithmetic mistake in your multiplication. B) $4.55 is correct, as shown above. C) $5.11 could result from incorrectly calculating the tax as a higher percentage, perhaps confusing the setup or making errors in decimal placement during multiplication. **D) $7.00** is a trap that catches students who misunderstand what the problem is asking. This answer simply takes the tax percentage ($ $7$ $) and treats it as dollars, ignoring the actual purchase amount entirely. Study tip: Always double-check percentage problems by asking yourself: "Does this answer make sense?" A $7\%$ tax should be less than $10\%$ of the purchase price. Since $10\%$ of $65 is $6.50, your answer should be somewhat less than that, making $4.55 reasonable while $7.00 is clearly too high.
A clothing store has a "Buy 2, Get 30% off the third item" promotion, with the discount applied to the least expensive of the three items. Sarah buys three shirts priced at $45, $38, and $52. How much does she save compared to buying all three at full price?
Answer: $11.40 The promotion takes 30% off one item: the least expensive of the three. The three shirts cost $45, $38 and $52, so the discount applies to the $38 shirt. Savings = 0.30 × 38 = 11.40, so Sarah saves $11.40 compared with paying full price for all three shirts. A savings of $15.60 comes from taking 30% off the $52 shirt, the most expensive item, which the promotion does not allow. A savings of $13.50 comes from taking 10% off the whole $135 total, spreading the discount across all three shirts instead of applying 30% to one. Always identify which single item the discount applies to before computing.
A sauce recipe calls for $12$ teaspoons of liquid, $30\%$ of which should be lemon juice. How many teaspoons of lemon juice are needed?
Answer: $3.6$ When you encounter percentage problems, you're finding a part of a whole. Here, you need to determine what $30\%$ of $12$ teaspoons equals. To find $30\%$ of $12$ , convert the percentage to a decimal by dividing by $100$ : $30\% = 0.30$ . Then multiply: $0.30 \times 12 = 3.6$ teaspoons of lemon juice. Let's examine why the other answers are incorrect. Choice A ( $2.4$ ) results from calculating $20\%$ of $12$ instead of $30\%$ . This is a common error when you misread the percentage in the problem. Choice B ( $3.0$ ) comes from calculating $25\%$ of $12$ , which might happen if you round $30\%$ to $25\%$ for easier mental math. Choice D ( $4.8$ ) is what you'd get if you calculated $40\%$ of $12$ , another misreading error. You can verify the answer by checking if $3.6$ teaspoons represents $30\%$ of the total: $\frac{3.6}{12} = 0.30 = 30\%$ ✓ The correct answer is C. Strategy tip: Always double-check percentage problems by working backwards. Take your answer, divide it by the total, and see if you get the original percentage. This catches calculation errors and ensures you used the right percentage from the problem.
A laboratory solution contains $2{,}500$ milliliters of liquid. What volume is $0.4\%$ of the solution?
Answer: $10\text{ mL}$ When you see a percentage problem, you're converting a percent to a decimal and multiplying by the total amount. To find $0.4\%$ of $2{,}500$ milliliters, first convert the percentage: $0.4\% = \frac{0.4}{100} = 0.004$ . Now multiply: $0.004 \times 2{,}500 = 10$ milliliters. You can think of this as $\frac{4}{1000} \times 2{,}500 = \frac{4 \times 2{,}500}{1000} = \frac{10{,}000}{1000} = 10$ . Choice A ( $1\text{ mL}$ ) represents a common error where students might calculate $0.04\%$ instead of $0.4\%$ , or miscalculate the decimal conversion. Choice B ( $4\text{ mL}$ ) likely comes from forgetting to convert the percentage properly and just using $4 \times 2{,}500 \div 1000$ incorrectly. Choice D ( $100\text{ mL}$ ) suggests the student calculated $4\%$ instead of $0.4\%$ — this is what you'd get if you used $0.04$ instead of $0.004$ as your decimal. Choice C ( $10\text{ mL}$ ) is correct because $0.004 \times 2{,}500 = 10$ . Strategy tip: When working with small percentages like $0.4\%$ , double-check your decimal conversion. Remember that $0.4\% = 0.004$ , not $0.04$ . A quick reasonableness check helps too: $0.4\%$ is less than half a percent, so your answer should be much smaller than $1\%$ of the solution (which would be $25$ mL).
A store advertises a $20\%$ discount on a jacket that costs $45 before tax. What is the amount of the discount?
Answer: $9 When you see a discount problem, you're calculating what amount gets subtracted from the original price, not the final price you'll pay. To find the discount amount, multiply the original price by the discount percentage. The jacket costs $45 and has a $20\%$ discount, so: $20\% \text{ of } \$45 = 0.20 \times \$45 = \$9$ The discount amount is $9, making (B) correct. Let's examine why the other choices are wrong: (A) $5 would be the result if you incorrectly calculated $10\%$ of $45 instead of $20\%$ , or if you made an arithmetic error. (C) $36 is the sale price after the discount is applied ($ $\$45 - \$9 = \$36$ $), but the question asks for the discount amount, not the final price. (D) $90 appears to come from doubling the original price, perhaps from misunderstanding the percentage calculation entirely. The key distinction here is between "discount amount" and "sale price." The discount amount is how much money gets taken off, while the sale price is what you actually pay. Always read carefully to see which one the question is asking for. A helpful check: the discount amount should always be less than the original price, and when you subtract it from the original price, you should get a reasonable sale price.
A jar holds $2{,}000$ marbles, $55\%$ of which are blue. How many blue marbles are there?
Answer: $1{,}100$ When you encounter percentage problems, you're converting a percentage to a decimal and multiplying by the total amount. The key is remembering that "percent" means "per hundred," so $55\% = \frac{55}{100} = 0.55$ . To find the number of blue marbles, multiply the total number of marbles by the percentage that are blue: $2{,}000 \times 0.55 = 1{,}100$ . You can think of this as finding $55\%$ of $2{,}000$ , which gives you $1{,}100$ blue marbles. Let's examine why the other answers are incorrect. Choice A ( $900$ ) would represent $45\%$ of the marbles—this is the percentage that are NOT blue. It's a common error to calculate the complement instead of what's actually asked. Choice B ( $1{,}000$ ) represents exactly $50\%$ of the marbles, which might tempt you if you misremembered the percentage or rounded $55\%$ incorrectly. Choice D ( $1{,}150$ ) is what you'd get if you calculated $57.5\%$ instead of $55\%$ —this could result from a calculation error or misreading the problem. For percentage problems on the SHSAT, always double-check that you're finding the percentage of the right quantity and that you've converted the percentage correctly to a decimal. A quick sanity check: since $55\%$ is more than half, your answer should be greater than $1{,}000$ (which is half of $2{,}000$ ). Only choices C and D meet this criterion, making your final check easier.
A company allocates $\$7{,}500$ to advertising, with $4\%$ set aside for online ads. How many dollars are budgeted for online ads?
Answer: $300 When you encounter percentage problems, you're essentially finding a part of a whole. Here, you need to calculate what $4\%$ of $\$7{,}500$ equals. To find $4\%$ of $\$7{,}500$ , convert the percentage to a decimal by dividing by 100: $4\% = 0.04$ . Then multiply: $0.04 \times \$7{,}500 = \$300$ . You can also think of this as $\frac{4}{100} \times \$7{,}500 = \frac{4 \times 7{,}500}{100} = \frac{30{,}000}{100} = \$300$ . Let's examine why the other answers are incorrect. Choice A ($30) represents a common error where students might calculate $4 \times 7.5$ instead of properly handling the percentage conversion. Choice C ($375) could result from miscalculating $5\%$ of $\$7{,}500$ instead of $4\%$ , or from decimal placement errors. Choice D ( $\$3{,}000$ ) represents a major computational error, possibly from confusing $4\%$ with $40\%$ or making significant decimal mistakes. The correct answer is B ($300). Study tip: For percentage problems, always double-check your decimal conversion. Remember that $4\% = 0.04$ , not $0.4$ or $4.0$ . A quick reasonableness check helps too: $4\%$ should be a small portion of the total, so $300 out of $\$7{,}500$ makes sense, while $\$3{,}000$ would be nearly half the budget.
During a clearance sale, all items are marked down 40%. Additionally, customers with a membership card receive an extra 15% off the already discounted price. If a member pays $76.50 for an item, what was the original price before any discounts?
Answer: $150.00 Let P be the original price. After 40% off: P × 0.60. After additional 15% off: P × 0.60 × 0.85 = P × 0.51 = $76.50. Therefore P = $76.50 ÷ 0.51 = $150. Choice A results from incorrectly calculating the compound discount. Choice C comes from adding the percentages (55%) instead of multiplying. Choice D results from applying discounts in wrong order or miscalculating the final multiplier.
A water tank that holds $9{,}600$ liters is $87.5\%$ full. How many liters of water are in the tank?
Answer: $8{,}400$ When you see a percentage problem involving "part of a whole," you're calculating what fraction of the total amount you actually have. Here, you need to find 87.5% of the tank's 9,600-liter capacity. To find a percentage of a number, convert the percentage to a decimal and multiply. Convert 87.5% to a decimal by dividing by 100: $87.5 ÷ 100 = 0.875$ . Then multiply by the tank's capacity: $0.875 × 9{,}600 = 8{,}400$ liters. Let's examine why each answer choice is right or wrong. Choice A (7,000) would represent about 73% of the tank's capacity, which is significantly less than the stated 87.5%. Choice B (8,400) is correct—this equals exactly 87.5% of 9,600 liters. Choice C (8,750) represents about 91% of the tank, which overshoots the 87.5% mark. Choice D (9,875) is impossible since it exceeds the tank's total capacity of 9,600 liters. You can verify your answer by working backwards: $8{,}400 ÷ 9{,}600 = 0.875 = 87.5\%$ ✓ Strategy tip: When dealing with percentages, always check if your answer makes logical sense. If the percentage is less than 100%, your answer must be smaller than the original amount. Also, convert percentages to decimals carefully—87.5% becomes 0.875, not 0.0875.
The population of a town increases by 12% in the first year and then decreases by 8% in the second year. If the final population is 20,608, what was the original population?
Answer: 20,000 Let P be the original population. After the first year: P × 1.12. After the second year: P × 1.12 × 0.92 = P × 1.0304 = 20,608. Therefore P = 20,608 ÷ 1.0304 = 20,000. Choice A results from using simple addition/subtraction of percentages. Choice B comes from reversing the order of operations. Choice D results from adding 12% and 8% instead of applying them sequentially.
In a survey of $1{,}250$ people, $0.8\%$ chose option D. How many people chose option D?
Answer: $10$ When you see a percentage problem asking "how many people," you're converting a percentage to an actual count by multiplying the percentage (as a decimal) by the total number. To find how many people chose option D, multiply the total surveyed by the percentage: $1{,}250 \times 0.8\%$ . First, convert $0.8\%$ to a decimal by dividing by 100: $0.8\% = 0.008$ . Then multiply: $1{,}250 \times 0.008 = 10$ people. Let's examine why the other answers represent common mistakes. Choice A) $1$ likely comes from misunderstanding the decimal conversion—perhaps thinking $0.8\%$ means dividing 1,250 by 800 or making another calculation error. Choice B) $8$ results from incorrectly treating $0.8\%$ as $0.8$ (forgetting it's already a percentage) and then making an error in the decimal placement. Choice D) $100$ happens when you mistakenly use $8\%$ instead of $0.8\%$ —this gives you $1{,}250 \times 0.08 = 100$ . The correct answer is C) $10$ . Remember this two-step process for percentage problems: convert the percentage to a decimal by moving the decimal point two places left (or dividing by 100), then multiply by the total. Also, when you see small percentages like $0.8\%$ , expect relatively small results—this can help you catch calculation errors where your answer seems unreasonably large.
Refer to the table to answer the question. The museum plans to offer a special tour to exactly $25\%$ of Wednesday's visitors. How many people will receive the tour?
Answer: $50$ Wednesday shows $200$ visitors; $0.25\times200=50$ . Choices A and B result from using Monday or averaging two days, while D mistakenly applies $25\%$ to Friday's count.
What is $35\%$ of $120$ ?
Answer: $42$ When you see a percentage problem asking "What is X% of Y?", you're being asked to convert a percentage to a decimal and multiply it by the given number. To find $35\%$ of $120$ , first convert the percentage to a decimal by dividing by 100: $35\% = 35 ÷ 100 = 0.35$ . Then multiply: $0.35 × 120 = 42$ . You can also think of this as $\frac{35}{100} × 120 = \frac{35 × 120}{100} = \frac{4200}{100} = 42$ . Looking at the wrong answers: Choice A ( $36$ ) likely comes from miscalculating $30\%$ of $120$ instead of $35\%$ , since $0.30 × 120 = 36$ . This could happen if you misread the percentage or made an arithmetic error. Choice C ( $48$ ) represents $40\%$ of $120$ , since $0.40 × 120 = 48$ . Again, this suggests misreading $35\%$ as $40\%$ . Choice D ( $420$ ) is what you'd get if you forgot to convert the percentage to a decimal and simply multiplied $35 × 12 = 420$ . This is a classic trap—always remember that percentages must be converted before multiplying. The correct answer is B ( $42$ ). For percentage problems on the SHSAT, always double-check that you've converted the percentage correctly. A quick mental check: $35\%$ should give you slightly more than one-third of the number, and one-third of $120$ is $40$ , so $42$ makes perfect sense.
A farmer harvested $7.2$ tons of apples; $37.5\%$ are Granny Smith apples. How many tons of Granny Smith apples were harvested?
Answer: $2.7$ This is a percentage calculation problem where you need to find a part of a whole. When you see "X% of Y," you're looking for the product of the percentage (as a decimal) and the total amount. To find how many tons of Granny Smith apples were harvested, multiply the total harvest by the percentage that are Granny Smith apples. First, convert $37.5\%$ to a decimal by dividing by 100: $37.5 ÷ 100 = 0.375$ . Then multiply: $7.2 × 0.375 = 2.7$ tons. Looking at the wrong answers: Choice A ( $2.16$ ) likely comes from incorrectly converting the percentage—perhaps using $0.3$ instead of $0.375$ ( $7.2 × 0.3 = 2.16$ ). Choice C ( $3.2$ ) might result from rounding $37.5\%$ to $40\%$ and calculating $7.2 × 0.4 = 2.88$ , then rounding to $3.2$ . Choice D ( $4.5$ ) could come from calculating $37.5\%$ of $12$ instead of $7.2$ , or from another computational error involving the wrong base number. The correct answer is B ( $2.7$ ). Strategy tip: When converting percentages to decimals, remember that $37.5\% = 37.5 ÷ 100 = 0.375$ . Double-check your decimal conversion before multiplying, as this is where most errors occur in percentage problems.
On an $80$ -question test, a student must answer at least $65\%$ correctly to pass. What is the minimum number of questions the student must answer correctly?
Answer: $52$ When you encounter percentage problems involving minimum requirements, you need to calculate the exact percentage first, then determine whether rounding is appropriate based on the context. To find the minimum number of correct answers needed, calculate $65\%$ of $80$ questions: $0.65 \times 80 = 52$ questions exactly. Since this calculation yields a whole number, the student must answer exactly $52$ questions correctly to achieve the $65\%$ passing threshold. Let's verify why the other options don't work. Choice A ( $50$ ) represents $\frac{50}{80} = 0.625 = 62.5\%$ , which falls short of the required $65\%$ . Choice B ( $51$ ) gives $\frac{51}{80} = 0.6375 = 63.75\%$ , still below the passing threshold. Choice D ( $53$ ) would give $\frac{53}{80} = 0.6625 = 66.25\%$ , which exceeds the requirement but isn't the minimum needed. Choice C ( $52$ ) is correct because $\frac{52}{80} = 0.65 = 65\%$ exactly meets the requirement. The key insight here is recognizing that "minimum" means finding the smallest number that satisfies the condition. Since $52$ questions gives exactly $65\%$ , answering $51$ or fewer would result in failure, while $53$ or more would be more than necessary. Strategy tip: In minimum/maximum problems involving percentages, always calculate the exact percentage first. If you get a whole number, that's usually your answer. If you get a decimal, consider whether the context requires rounding up or down based on what the question is asking.
At a school fundraiser, 60% of the money raised came from ticket sales, and 35% of the ticket sales revenue came from student tickets. If student tickets generated $840, what was the total amount raised at the fundraiser?
Answer: $4,000 Student tickets ($840) represent 35% of ticket sales, so total ticket sales = $840 ÷ 0.35 = $2,400. Ticket sales represent 60% of total fundraising, so total raised = $2,400 ÷ 0.60 = $4,000. Choice A results from using $840 ÷ 0.35 × 0.60. Choice C comes from incorrectly adding percentages: $840 ÷ (0.35 + 0.60). Choice D results from $840 ÷ 0.35 × 2 instead of proper calculation.
A recipe calls for ingredients in the following ratios: 40% flour, 25% sugar, 20% butter, and 15% other ingredients. If you need 450 grams of flour for a large batch, how many grams of sugar do you need?
Answer: 281.25 grams If 450g represents 40% of the total, then the total batch is 450 ÷ 0.40 = 1,125g. Sugar is 25% of the total: 1,125 × 0.25 = 281.25g. Choice B results from incorrectly calculating 450 × 25 ÷ 40 + 15. Choice C comes from using 450 × 0.25 ÷ 40 × 100. Choice D results from simply calculating 450 × 0.80 without proper ratio conversion.

Start this quiz