Percent Increase and Decrease - ISEE Upper Level Quantitative Reasoning

An online retailer increases shipping costs by 25%, then offers free shipping on orders over $50 (effectively a 100% discount on shipping for qualifying orders). For a qualifying order, what is the net change in shipping costs?
Answer: 100% decrease This question tests your understanding of sequential percentage changes and how discounts work in real-world scenarios. When analyzing percentage changes that happen in sequence, you need to track what happens to the original value step by step. Let's say the original shipping cost was $10. First, the retailer increases shipping costs by 25%, making the new cost $10 × 1.25 = $12.50. However, for qualifying orders (those over $50), they offer free shipping, which means a 100% discount on shipping costs. A 100% discount means you pay nothing for shipping - the entire shipping cost is eliminated. So regardless of whether the shipping cost was the original $10 or the increased $12.50, qualifying customers now pay $0. The net change from the original shipping cost is: $\frac{\text{Final Cost} - \text{Original Cost}}{\text{Original Cost}} = \frac{0 - 10}{10} = -100\%$ This confirms answer choice B: 100% decrease. Answer choice A (75% decrease) incorrectly suggests you subtract the 25% increase from the 100% discount (100% - 25% = 75%). Answer choice C (125% decrease) might come from adding the percentages (25% + 100% = 125%), but percentage decreases cannot exceed 100% since you cannot pay less than nothing. Answer choice D (no change possible to determine) ignores that free shipping means zero cost regardless of the previous increase. Remember: when you see "free" or "100% discount," the final cost is always zero, making the net change a 100% decrease from any positive original amount.
A car's value depreciates by 22% in the first year and by 18% in the second year. If the car is worth $28,142.40 after two years, what was its original value?
Answer: $44,000 Depreciation means the car keeps a percentage of its value each year. Losing 22% keeps 78%, and losing 18% keeps 82%. If the original value is $x$ , then after two years it is $x \times 0.78 \times 0.82 = 0.6396x$ . Set this equal to 28,142.40: $x = \frac{28{,}142.40}{0.6396} = 44{,}000$ Check: $44{,}000 \times 0.78 = 34{,}320$ , and $34{,}320 \times 0.82 = 28{,}142.40$ . A starting value of $42,000 would end at about $26,863, and $45,000 would end at $28,782, so neither matches. $48,000 would end at about $30,701, far too high. Tip: multiply the two keep-rates; do not add the loss percentages.
A real estate agent charges a 6% commission on home sales. Due to market conditions, the agent reduces the commission rate by 25%. If a house sells for $450,000, how much less commission does the agent earn compared to the original rate?
Answer: $6,750 This problem tests your ability to work with percentage changes and calculate the difference between two commission amounts. When you see questions involving percentage reductions, always calculate both the original and new values before finding the difference. First, calculate the original commission at 6%: $0.06 \times \$450,000 = \$27,000$ Next, find the reduced commission rate. A 25% reduction means the new rate is 75% of the original: $6\% \times 0.75 = 4.5\%$ Now calculate the new commission: $0.045 \times \$450,000 = \$20,250$ The difference between the original and reduced commission is: $\$27,000 - \$20,250 = \$6,750$ Looking at the wrong answers: Choice A ($4,500) represents 1% of the house price, which might result from confusing the percentage reduction with the actual commission difference. Choice C ($9,000) equals 2% of the house price, possibly from incorrectly thinking the reduction is 2 percentage points instead of 25% of the original rate. Choice D ($11,250) is 2.5% of the house price, which could come from miscalculating the percentage reduction as 2.5 percentage points. Remember that when dealing with percentage reductions, you must distinguish between a reduction "by" a certain percentage (multiply by the complement) versus a reduction "to" a certain percentage. Here, "reduces by 25%" means the new rate is 75% of the original, not 25% of the original.
A store marks up all items by 40%, then offers a "30% off sale" on the marked-up prices. If an item originally cost the store $60, what is the final selling price after both the markup and discount?
Answer: $58.80 When you encounter percentage problems involving multiple steps like markups and discounts, always work through each change sequentially, applying each percentage to the result of the previous step. Start with the original cost of $60. First, apply the 40% markup: $60 \times 1.40 = 84$ . The marked-up price is $84. Next, apply the 30% discount to this marked-up price. A "30% off" means the customer pays 70% of the marked price: $84 \times 0.70 = 58.80$ . The final selling price is $58.80. Let's examine why each answer choice appears: A) $58.80 is correct - this follows the proper sequential calculation of markup first, then discount. B) $60.00 represents a common misconception where students think the 40% markup and 30% discount "cancel out" because they're close in value. However, percentages don't work this way since they're applied to different base amounts. C) $62.40 occurs when students incorrectly apply both percentages to the original $60: they calculate $60 \times 1.40 \times 0.70 = 58.80$ , but then make an arithmetic error, or they add 40% and subtract 30% from the original price: $60 \times 1.10 = 66$ , then make calculation mistakes. D) $84.00 is the marked-up price before applying the discount - students who choose this forgot the second step entirely. Study tip: In multi-step percentage problems, never try to combine the percentages mentally. Always calculate each step completely before moving to the next, and remember that order matters when the base amount changes.
A store increases the price of all items by 24%, then offers a loyalty program that gives members a 15% discount on all purchases. What effective price change do loyalty program members experience?
Answer: 5.4% increase When you encounter questions about successive percentage changes, remember that you can't simply add or subtract the percentages—you must apply them sequentially to account for compounding effects. Let's trace through what happens to a $100 item. First, the store increases all prices by 24%: $100 \times 1.24 = 124$ . Then, loyalty members get a 15% discount on this new price: $124 \times 0.85 = 105.40$ . The final price is $105.40, representing a 5.4% increase from the original $100. You can also solve this algebraically. If the original price is P, the final price becomes: $P \times 1.24 \times 0.85 = P \times 1.054 = 1.054P$ . This confirms a 5.4% increase, making (A) correct. The wrong answers represent common misconceptions. Choice (B) 9.0% comes from incorrectly subtracting the percentages: 24% - 15% = 9%. This ignores the fact that the discount applies to the already-increased price. Choice (C) 12.6% might result from taking half of the original 24% increase, but this has no mathematical basis. Choice (D) 15.0% incorrectly assumes the discount exactly cancels out part of the increase on a one-to-one basis. Strategy tip: For successive percentage problems, always multiply the decimal forms of the changes together, or work through with a concrete number like $100. Never just add or subtract the percentages—the compounding effect means the second percentage applies to an already-changed amount.
A retailer buys items for $40 each and applies a 60% markup. During a clearance sale, the retailer offers 25% off the marked price. What is the profit margin, as a percent of the retailer's cost, on items sold during the clearance sale?
Answer: 20% profit margin Follow the price from cost to final sale. The markup is 60% of $40, which is $24, so the marked price is $64. The clearance discount is 25% of $64, which is $16, so the sale price is $48. The profit is $48 − $40 = $8. As a percent of the $40 cost, that is 8 ÷ 40 = 0.20, or 20%. A 35% margin comes from subtracting the percents, 60% − 25%, but the 25% discount is taken from the larger marked price, so the percents cannot simply be subtracted. A 25% margin confuses the discount rate with the profit. A 15% margin is close to the profit as a percent of the sale price (8 ÷ 48 ≈ 16.7%), but the question measures profit against the cost.
The number of employees at a company decreased by 18% due to layoffs, then increased by 25% due to new hires. If the final number of employees is 1,025, what was the original number of employees?
Answer: 1,000 employees When you encounter percent change problems involving multiple steps, you need to work backwards from the final result to find the original value. Think of this as reverse engineering the changes that occurred. Let's call the original number of employees $x$ . After an 18% decrease, the company had $0.82x$ employees (since they kept 82% of the original workforce). Then this reduced number increased by 25%, giving us $0.82x \times 1.25 = 1.025x$ . Since we know the final result is 1,025 employees, we can set up the equation: $1.025x = 1,025$ . Solving for $x$ : $x = 1,025 ÷ 1.025 = 1,000$ employees. Choice A (950 employees) represents a common error where students might subtract 18% and add 25%, thinking the net change is +7%, then working backwards incorrectly. Choice C (1,050 employees) might result from confusing the order of operations or miscalculating the compound percentage effects. Choice D (1,100 employees) could come from incorrectly assuming simple addition and subtraction of percentages without considering that the 25% increase applies to the already-reduced workforce. The key strategy here is recognizing that percentage changes are multiplicative, not additive, and that when working backwards from a final result, you need to set up an equation that accounts for all transformations in sequence. Always double-check by working forward: does 1,000 → 820 → 1,025 match the given conditions?
A subscription service increases its monthly fee by 30%, then offers existing customers a "grandfathered" rate that is 20% less than the new rate. What is the percent change in monthly fees for existing customers compared to the original rate?
Answer: 4% increase This is a multi-step percentage change problem that tests your ability to track cumulative effects. When dealing with sequential percentage changes, you need to apply each change to the result of the previous change, not the original amount. Let's work with an original monthly fee of $100 to make the math clear. First, the service increases the fee by 30%: $100 × 1.30 = $130. Next, existing customers get a "grandfathered" rate that's 20% less than this new rate: $130 × 0.80 = $104. Comparing the final amount ($104) to the original ($100), existing customers pay 4% more than they originally paid. Looking at the wrong answers: Choice B (8% increase) likely comes from incorrectly adding and subtracting the percentages directly (30% - 20% = 10%, then making an arithmetic error). Choice C (10% increase) is the trap of simply subtracting 30% - 20% = 10% without understanding that these percentages apply to different base amounts. Choice D (12% increase) might result from incorrectly multiplying the percentage changes or making calculation errors in the multi-step process. The key insight is that the 20% reduction applies to the already-increased amount of $130, not the original $100. This is why you can't simply subtract 20% from 30% to get the answer. Strategy tip: For sequential percentage problems, always apply each percentage to the result of the previous step, and use concrete numbers to avoid calculation errors. The final step should always compare back to the original baseline.
A mutual fund's value increased by 16% in the first quarter, decreased by 12% in the second quarter, and increased by 8% in the third quarter. What was the overall percent change for the three quarters?
Answer: 10.2% increase Percent changes in a row multiply rather than add, because each change applies to the new value. Start with 100. After a 16% increase: $100 \times 1.16 = 116$ . After a 12% decrease: $116 \times 0.88 = 102.08$ . After an 8% increase: $102.08 \times 1.08 = 110.2464$ . The fund ended about 10.2% higher than it started. A 12.0% increase comes from simply adding the changes, 16% − 12% + 8% = 12%, which ignores that the 12% decrease is taken from the larger value 116. The 11.4% and 13.2% increases are also too large; any answer of 11% or more overstates how much growth survives the decrease.
An investment account loses 20% of its value in January, gains 25% in February, and then loses 10% in March. What is the overall percent change in the account value over the three months?
Answer: A 10% decrease When you encounter percent change problems involving multiple periods, remember that these changes compound—they don't simply add or subtract. You must apply each percentage change to the result of the previous change. Let's start with an initial value of $100 to make the calculations clear. In January, the account loses 20%, so it becomes $100 × 0.80 = $80. In February, it gains 25% of its current value: $80 × 1.25 = $100. Finally, in March, it loses 10% of the February value: $100 × 0.90 = $90. The account went from $100 to $90, representing a $10 decrease on the original $100, which is a 10% decrease overall. Looking at the wrong answers: Choice B (5% decrease) likely comes from incorrectly adding the percentages: -20% + 25% - 10% = -5%. However, this ignores the compounding effect. Choice C (no change) might result from noticing that after February, the account returned to its original value, but this overlooks March's 10% loss. Choice D (5% increase) could come from miscalculating the percentage arithmetic or applying the changes in the wrong order. The key strategy here is to always work with the actual values step by step rather than trying to shortcut with percentage arithmetic. Set up a simple example with $100 as your starting point, apply each change sequentially, and then calculate the overall percent change from beginning to end.
The price of a concert ticket increased from $45 to $54 during the first quarter, then decreased by 25% during the second quarter. What was the overall percent change in the ticket price from the beginning of the first quarter to the end of the second quarter?
Answer: A 10% decrease When you encounter multi-step percent change problems, you need to track the actual dollar amounts through each change, then compare the final amount to the original amount. Let's trace the ticket price through both quarters. Starting price: $45. First quarter increase to $54 represents a $9 increase. For the second quarter, the price decreases by 25% from $54. Calculate 25% of $54: $0.25 \times 54 = 13.50$ . So the second quarter price becomes $54 - 13.50 = 40.50$ . Now compare the final price ($40.50) to the original price ($45). The change is $40.50 - 45 = -4.50$ , a $4.50 decrease. To find the percent change: $\frac{-4.50}{45} \times 100\% = -10\%$ . This confirms answer A: a 10% decrease. Looking at the wrong answers: B (10% increase) gets the magnitude right but the wrong direction—this comes from forgetting that the final price is lower than the original. C (5% decrease) and D (5% increase) both show 5%, which you'd get if you incorrectly added and subtracted the percentage changes directly (20% increase minus 25% decrease). This is a common trap because it seems logical but ignores that the 25% decrease applies to the higher $54 price, not the original $45. Remember: with multi-step percent changes, always work with actual amounts rather than trying to combine percentages directly. Percentages apply to different base amounts at each step, so you must calculate each change in dollars first.
A clothing store has a policy where all sale prices are first increased by 8% for tax, then decreased by 12% for a loyalty discount. If a jacket's final price after both adjustments is $95.04, what was the original sale price before any adjustments?
Answer: $100.00 When you encounter problems with sequential percentage changes, you need to work backwards from the final result. This question tests your ability to reverse multiple percentage adjustments to find an original value. Let's call the original sale price $x$ . The jacket undergoes two changes: first an 8% tax increase, then a 12% loyalty discount. After the tax increase, the price becomes $x \times 1.08$ . Then the 12% discount is applied to this new amount: $(x \times 1.08) \times 0.88 = x \times 1.08 \times 0.88 = x \times 0.9504$ . Since the final price is $95.04, we can set up the equation: $x \times 0.9504 = 95.04$ . Solving for $x$ : $x = \frac{95.04}{0.9504} = 100.00$ . Let's verify: $100.00 × 1.08 = $108.00 (after tax), then $108.00 × 0.88 = $95.04 ✓ Choice A ($98.00) would result in a final price of $92.14 after both adjustments—too low. Choice C ($102.50) would yield $97.39 as the final price—too high. Choice D ($108.00) represents a common error where students might think this is the price after just the tax increase, but applying the discount would give $95.04, not accounting for the tax properly. Study tip: For sequential percentage problems, multiply all the percentage factors together first (here: 1.08 × 0.88 = 0.9504), then divide the final result by this combined factor. Always verify by working forward through each step.
A stock price drops 30% on Monday, rises 40% on Tuesday, and drops 25% on Wednesday. If the stock price at the end of Wednesday is $77.175, what was the price at the beginning of Monday?
Answer: $105.00 When you encounter percentage change problems, remember that each change is applied to the result of the previous change, not the original value. You'll need to work backwards from the final amount to find the starting value. Let's call the original Monday price $x$ . After dropping 30% on Monday, the price becomes $0.70x$ . On Tuesday, it rises 40%, so we multiply by 1.40: $0.70x \times 1.40 = 0.98x$ . Wednesday brings a 25% drop, so we multiply by 0.75: $0.98x \times 0.75 = 0.735x$ . Since we know the final price is $77.175, we can set up the equation: $0.735x = 77.175$ . Solving for $x$ : $x = \frac{77.175}{0.735} = 105$ . Choice A ($98.00) would give you a final price of about $72.03 after all the changes. Choice C ($112.50) would result in approximately $82.69. Choice D ($126.00) would end at about $92.61. These are all incorrect because they don't account for the compounding effect of percentage changes. Choice B ($105.00) is correct because when you apply the sequence of changes: $105.00 → $73.50 (after Monday's 30% drop) → $102.90 (after Tuesday's 40% rise) → $77.175 (after Wednesday's 25% drop). Remember that percentage changes compound—each change builds on the previous result. Always work systematically through each step, and consider working backwards from the final value when that approach seems clearer.
The number of students in a school increased by 12% from 2020 to 2021, then decreased by 7.5% from 2021 to 2022. If there were 1,036 students in 2022, how many students were there in 2020?
Answer: 1,000 students Successive percent changes multiply, so work backward with the combined factor. A 12% increase multiplies by 1.12 and a 7.5% decrease multiplies by 0.925. If $x$ students were enrolled in 2020, then in 2022 there were $1.12 \times 0.925 \times x = 1.036x$ Setting $1.036x = 1{,}036$ gives $x = 1{,}000$ . Check: $1{,}000 \times 1.12 = 1{,}120$ , and $1{,}120 \times 0.925 = 1{,}036$ . Starting from 950 would give only $950 \times 1.036 = 984.2$ , too few. Starting from 1,050 gives $1{,}087.8$ and from 1,100 gives $1{,}139.6$ , both too many. Subtracting the percents to get a net 4.5% change and dividing by 1.045 also misses, since the decrease applies to the larger 2021 count. Tip: convert each change to a multiplier and multiply them; never add or subtract the percents.
A company's quarterly revenue increased by 28% in Q1, decreased by 15% in Q2, and increased by 20% in Q3. What single percent increase would have achieved the same result over the three quarters?
Answer: 30.6% increase Percent changes in a row multiply; they do not add. Write each change as a multiplier: a 28% increase is 1.28, a 15% decrease is 0.85, and a 20% increase is 1.20. The overall multiplier is 1.28 × 0.85 × 1.20. First, 1.28 × 0.85 = 1.088. Then 1.088 × 1.20 = 1.3056. A multiplier of 1.3056 means the revenue ended 30.56% higher than it started, which rounds to a 30.6% increase. Adding the percents gives 28 − 15 + 20 = 33, which is the 33.0% trap; it ignores that each change acts on a different starting amount. The 31.9% and 35.2% values do not come from the product 1.28 × 0.85 × 1.20, so they are not equivalent to the three changes. The single equivalent change is a 30.6% increase.
The membership fee for a gym increased by 25% in 2022 and then increased by another 20% in 2023. If the fee in 2023 was $180, what was the fee in 2021?
Answer: $120 When you encounter percentage increase problems that work backwards from a final value, you need to reverse multiple percentage changes step by step. Let's call the 2021 fee $x$ . After a 25% increase in 2022, the fee became $x \times 1.25$ . Then after a 20% increase in 2023, it became $x \times 1.25 \times 1.20 = x \times 1.5$ . Since we know the 2023 fee was $180, we can write: $x \times 1.5 = 180$ , so $x = 180 ÷ 1.5 = 120$ . Choice A ($110) represents a common error where students might subtract the percentage increases rather than working with the multiplication factors correctly. Choice C ($144) likely comes from miscalculating one of the percentage increases or applying them in the wrong order. Choice D ($150) appears when students incorrectly assume they can simply divide $180 by 1.2 (the 2023 increase only), forgetting about the 2022 increase entirely. The correct answer is B ($120). You can verify this: $120 × 1.25 = $150 after 2022, then $150 × 1.20 = $180 after 2023. Strategy tip: For multi-step percentage problems working backwards, always multiply all the growth factors together first (here: 1.25 × 1.20 = 1.5), then divide the final amount by this total factor. This prevents the confusion that leads to most wrong answers on these questions.
A laptop's price is reduced by 20% for a sale, then an additional 10% is taken off the sale price for a student discount. If the final price is $576, what was the original price?
Answer: $800 When you encounter percent decrease problems with multiple discounts, remember that each discount applies to the price after the previous discount, not to the original price. This creates a compound effect that's crucial to track correctly. Let's work backwards from the final price of $576. If we call the original price $x$ , then after a 20% reduction, the laptop costs $0.8x$ . The additional 10% student discount applies to this sale price, so we multiply by $0.9$ again: $0.8x \times 0.9 = 0.72x$ . Setting up the equation: $0.72x = 576$ Solving: $x = \frac{576}{0.72} = 800$ So the original price was $800. Let's check why the other answers are wrong. Choice A ($720) represents what you'd get if you incorrectly assumed the discounts were simple rather than compound - subtracting 30% total instead of applying them sequentially. Choice B ($750) might result from calculation errors in the division. Choice D ($850) could come from misunderstanding which direction to apply the percentage calculations. You can verify answer C by working forward: $800 \times 0.8 = 640$ (after 20% off), then $640 \times 0.9 = 576$ (after additional 10% off). Strategy tip: For compound percent problems, always multiply the decimal equivalents together first (here: $0.8 \times 0.9 = 0.72$ ), then work backwards from the final amount. This prevents the common trap of simply adding or subtracting the percentages.
The value of a cryptocurrency dropped by 35% on Monday, rose by 50% on Tuesday, and dropped by 20% on Wednesday. What is the overall percent change from Monday's opening to Wednesday's closing?
Answer: 22% decrease When you encounter percent change problems with multiple steps, you need to apply each percentage change sequentially to the running total, not to the original value. Let's say the cryptocurrency starts at $100 on Monday morning. After dropping 35% on Monday, its value becomes $100 × (1 - 0.35) = $100 × 0.65 = $65. On Tuesday, it rises 50% from the Monday closing value of $65: $65 × (1 + 0.50) = $65 × 1.50 = $97.50. On Wednesday, it drops 20% from Tuesday's closing value of $97.50: $97.50 × (1 - 0.20) = $97.50 × 0.80 = $78. The overall change is from $100 to $78, which represents a decrease of $22, or 22%. Choice A (22% decrease) is correct. Choice B (5% decrease) likely comes from incorrectly adding and subtracting the percentages: -35% + 50% - 20% = -5%. This is wrong because percentages compound. Choice C (18% increase) might result from calculation errors or misunderstanding which direction the changes go. Choice D (25% increase) could come from adding the absolute values of changes and assuming a net positive result. Remember that consecutive percentage changes multiply, they don't add. A quick way to solve this is: $1.00 × 0.65 × 1.50 × 0.80 = 0.78$ , confirming a 22% decrease from the original value.
The population of a town decreased by 15% in the first year and then increased by 20% in the second year. If the population at the end of two years was 10,200, what was the original population?
Answer: 10,000 people When you see percentage changes applied sequentially, remember that each percentage change applies to the result of the previous change, not the original amount. This creates a compound effect that requires working backwards from the final result. Let's call the original population $P$ . After a 15% decrease in year one, the population becomes $0.85P$ (since decreasing by 15% means keeping 85%). In year two, this reduced population increases by 20%, so it becomes $0.85P \times 1.20 = 1.02P$ . We know this final amount equals 10,200, so: $1.02P = 10,200$ $P = \frac{10,200}{1.02} = 10,000$ Let's verify: Starting with 10,000, after a 15% decrease we get $10,000 \times 0.85 = 8,500$ . Then after a 20% increase: $8,500 \times 1.20 = 10,200$ ✓ Now for the wrong answers: Choice A (9,500) would give us a final population of $9,500 \times 0.85 \times 1.20 = 9,690$ , which is too low. Choice C (10,500) would yield $10,500 \times 0.85 \times 1.20 = 10,710$ , which exceeds our target. Choice D (12,000) would produce $12,000 \times 0.85 \times 1.20 = 12,240$ , far too high. The key strategy for sequential percentage problems is to set up one equation with all changes applied in order, then solve backwards. Don't try to reverse each percentage change separately—this often leads to calculation errors and is more time-consuming than the direct algebraic approach.

Start this quiz