Percent Increase and Decrease - SSAT Middle Level Quantitative

The population of a town decreased by 15% in the first decade and increased by 25% in the second decade. If the population is now 25,500, what was the population 20 years ago?
Answer: 24,000 When you encounter percent change problems involving multiple time periods, you need to work backwards from the final value through each change in reverse order. Let's call the original population 20 years ago $x$ . After decreasing by 15% in the first decade, the population became $x \times 0.85$ . Then it increased by 25% in the second decade, giving us the current population: $x \times 0.85 \times 1.25 = 25,500$ . To solve for $x$ : $x \times 1.0625 = 25,500$ , so $x = 25,500 \div 1.0625 = 24,000$ . Let's verify: Starting with 24,000, a 15% decrease gives us $24,000 \times 0.85 = 20,400$ . Then a 25% increase: $20,400 \times 1.25 = 25,500$ ✓ Now for the wrong answers: Choice B (24,500) would result in a final population of 26,031 after both changes—too high. Choice C (25,000) gives us 26,562 as the final population, also too high. Choice D (26,000) results in 27,625, which is significantly too high. These incorrect answers likely come from common mistakes: adding and subtracting the percentages directly from the final answer, or calculating the percent changes incorrectly. Strategy tip: In multi-step percent problems, always multiply by the decimal form of each change (0.85 for a 15% decrease, 1.25 for a 25% increase) and remember that percent changes compound—they don't simply add or subtract.
After a 20% discount, a bicycle costs $160. The store manager decides to increase this discounted price by 25%. What is the final price of the bicycle?
Answer: $200 This question tests your understanding of sequential percentage changes - applying one percentage change, then applying another to the new result. When you see problems involving multiple discounts or markups in sequence, remember that each percentage applies to the current price, not the original. Let's work backwards from the discounted price. If the bicycle costs $160 after a 20% discount, this means $160 represents 80% of the original price (since 100% - 20% = 80%). To find the original price: $\text{Original price} = \frac{160}{0.80} = 200$ Now the manager increases the discounted price of $160 by 25%. This means the new price will be 125% of $160: $\text{Final price} = 160 \times 1.25 = 200$ Looking at the wrong answers: Choice (A) $190 likely comes from incorrectly calculating 25% of $160 as $40, then subtracting instead of adding ($160 + $40 - $10 = $190). Choice (C) $210 might result from adding 25% to the original price of $200 instead of the discounted price. Choice (D) $220 could come from taking 25% of the original price ($200) and adding it to the discounted price ($160 + $60 = $220). The key strategy here is to work step-by-step and always apply percentage changes to the current value, not mixing up which price serves as your base. Also, remember that a 25% increase means multiplying by 1.25, which often speeds up your calculations.
The enrollment at Lincoln Middle School decreased from 540 students to 486 students. What was the percent decrease in enrollment?
Answer: 10% Percent decrease problems test your ability to calculate how much a quantity has dropped relative to its original value. When you see enrollment numbers, prices, or populations changing, think about whether you're finding the change relative to the original or new amount. To find percent decrease, you need three steps: find the actual decrease, divide by the original amount, then convert to a percentage. The enrollment dropped from 540 to 486 students, so the decrease is $540 - 486 = 54$ students. Now divide this decrease by the original enrollment: $\frac{54}{540} = 0.10$ . Converting to a percentage: $0.10 \times 100\% = 10\%$ . Looking at the wrong answers, choice (A) 9% likely comes from calculation errors or rounding mistakes in the division. Choice (C) 11% might result from dividing by the new enrollment (486) instead of the original enrollment—a common trap that gives you $\frac{54}{486} \approx 0.11$ . Choice (D) 12% could come from arithmetic errors or incorrectly calculating the decrease amount. The key trap here is using the wrong denominator. Always remember that percent decrease uses the original (larger) value as the denominator, not the new (smaller) value. A helpful way to check your work is to verify that 10% of 540 is 54, and $540 - 54 = 486$ , which matches the final enrollment. This backward check can catch calculation errors quickly on test day.
The price of a movie ticket increased from $8 to $10, then later decreased to $9. What is the percent change from the original price to the final price?
Answer: 12.5% increase When you encounter percent change problems, remember that percent change is always calculated using the original value as your reference point, regardless of any intermediate steps. To find the percent change from the original price ($8) to the final price ($9), use the formula: $\text{Percent change} = \frac{\text{final value} - \text{original value}}{\text{original value}} \times 100\%$ Substituting the values: $\frac{9 - 8}{8} \times 100\% = \frac{1}{8} \times 100\% = 12.5\%$ Since the final price is higher than the original, this is a 12.5% increase, making C correct. Now let's examine why the other answers are wrong. Choice A (10% increase) likely comes from incorrectly calculating $\frac{1}{10} \times 100\% = 10\%$ , which suggests using $10 (the intermediate price) as the denominator instead of the original $8. Choice B (11.25% increase) might result from averaging the two price changes or making a calculation error. Choice D (15% increase) doesn't correspond to any logical calculation with these numbers and represents a significant miscalculation. The key trap here is getting distracted by the intermediate price change from $8 to $10. Many students make the mistake of calculating multiple percent changes or using the wrong reference point. Always remember: for overall percent change, you only need the starting point and ending point—ignore everything that happens in between.
The number of students in a school increased by 15% last year and is expected to increase by another 20% this year. If there are currently 1,380 students, how many students were there before last year's increase?
Answer: 1,200 The 15% increase already happened last year, and the 20% increase is only expected, so the current 1,380 students is the count after last year's increase and before this year's. To undo a 15% increase, divide by 1.15: $1{,}380 \div 1.15 = 1{,}200$ . Check: $1{,}200 \times 1.15 = 1{,}380$ . An answer of 1,000 divides by both 1.15 and 1.20, treating the expected increase as if it had already happened. An answer of 1,150 divides by 1.20, undoing the increase that has not happened yet. An answer of 1,173 subtracts 15% of 1,380, but a 15% increase is undone by dividing by 1.15, not by taking 85% of the new number.
The membership of a club increased by 25% to reach 75 members. Later, 15 members left the club. By what percent did the membership decrease from its peak?
Answer: 20% When you encounter percent change problems involving multiple steps, work through each change systematically and pay careful attention to what serves as your reference point for each calculation. First, find the original membership. If increasing by 25% resulted in 75 members, then $1.25 \times \text{original} = 75$ , so the original membership was $75 \div 1.25 = 60$ members. The peak membership was 75 members. When 15 members left, the club dropped to $75 - 15 = 60$ members. To find the percent decrease from the peak, use the peak (75) as your reference point: $\frac{15}{75} = 0.20 = 20\%$ . The membership decreased by 20% from its peak. Choice A (15%) represents a common error where students calculate $\frac{15}{100}$ instead of using the correct denominator. Choice B (18%) might result from incorrectly using the original membership of 60 as the denominator: $\frac{15}{60} = 25\%$ , though this doesn't match exactly. Choice D (22%) doesn't correspond to any logical calculation with the given numbers. Remember that in percent change problems, the denominator is always your reference point. When the question asks for "percent decrease from its peak," the peak value (75) must be your denominator, not the final value or any other number. Always identify what you're measuring the change from before setting up your fraction.
The number of subscribers to a newsletter increased by 40% in the first month, then decreased by 25% in the second month. If there were initially 800 subscribers, how many subscribers are there after two months?
Answer: 840 When you encounter percentage change problems involving multiple steps, work through each change sequentially rather than trying to combine them mentally. These problems test your understanding of how percentages compound. Start with the initial 800 subscribers. After a 40% increase in the first month: $800 + (0.40 × 800) = 800 + 320 = 1,120$ subscribers. Now apply the 25% decrease to this new total: $1,120 - (0.25 × 1,120) = 1,120 - 280 = 840$ subscribers after two months. Looking at the wrong answers: Choice B (920) represents a common error where students calculate 40% of 800 (getting 320) and 25% of 800 (getting 200), then compute 800 + 320 - 200 = 920. This fails because the decrease should be applied to the increased amount, not the original. Choice C (960) might result from incorrectly calculating the second step as a 15% net increase (40% - 25% = 15%), giving 800 × 1.15 = 920, then making an arithmetic error. Choice D (1,040) could come from applying only the first increase and forgetting the decrease entirely, or from other calculation mistakes. Remember: percentage changes always apply to the current amount, not the original. When working through multi-step percentage problems, calculate each step completely before moving to the next, and resist the urge to combine percentages directly since they compound rather than simply add or subtract.
A store's profit margin increased from 18% to 24%. This represents what percent increase in the profit margin?
Answer: 33⅓% When you see questions about percent increase, you're finding how much a value has grown relative to its original amount. The key is remembering that percent increase equals the change divided by the original value, then converted to a percentage. To find the percent increase in profit margin, start with the formula: $\text{Percent increase} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100\%$ The profit margin increased from 18% to 24%, so the change is $24\% - 18\% = 6\%$ . Now divide this change by the original value: $\frac{6\%}{18\%} = \frac{6}{18} = \frac{1}{3}$ Converting to a percentage: $\frac{1}{3} = 0.333... = 33\frac{1}{3}\%$ Choice A (6%) represents the absolute change in profit margin, not the percent increase. This is a common trap—don't confuse the raw difference with the percent change. Choice B (25%) might come from incorrectly using 24% as the denominator instead of 18%, or from calculation errors. Choice D (40%) doesn't match any logical calculation path for this problem and likely represents a computational mistake. Remember this pattern: when calculating percent increase, always divide the change by the original value, not the new value. Many students mistakenly use the final value as their denominator, leading to incorrect answers. Practice identifying which number represents your starting point—that's always your denominator.
The attendance at a concert was 20% higher than expected. If 1,440 people attended, how many people were originally expected?
Answer: 1,200 When you see a question about an actual value being a certain percentage higher than expected, you're working with percent increase problems. The key insight is that the actual attendance represents 120% of the expected attendance (100% + 20% increase). Let's call the expected attendance $x$ . Since the actual attendance is 20% higher than expected, we can write: $1.2x = 1440$ To find the original expected attendance, divide both sides by 1.2: $x = \frac{1440}{1.2} = 1200$ So 1,200 people were originally expected, making B correct. Let's examine why the other answers are wrong. Choice A (1,152) comes from incorrectly calculating 20% of 1,440 and subtracting it: $1440 - (0.20 \times 1440) = 1440 - 288 = 1152$ . This flawed approach assumes you subtract 20% of the actual attendance, but that's not how percent increases work backward. Choice C (1,300) might result from computational errors or misunderstanding the relationship between the percentages. Choice D (1,368) comes from subtracting only 5% instead of properly working backward from the 20% increase: $1440 - 72 = 1368$ . Strategy tip: When working backward from a percent increase, remember that if something increased by $n$ %, the new value represents $(100 + n)$ % of the original. Always divide the final value by this percentage (in decimal form) to find the original amount. Don't fall into the trap of simply subtracting the percentage from the given value.
A recipe calls for 3 cups of flour, but Maria wants to reduce this amount by 25%. Later, she decides the reduced amount is too little and increases it by 40%. How many cups of flour does she end up using?
Answer: 3.15 cups This question tests your ability to work with sequential percentage changes, which is a common pattern on the SSAT. The key insight is that when you apply multiple percentage changes, you must calculate them step by step rather than trying to combine them directly. Start with Maria's original amount: 3 cups of flour. First, she reduces this by 25%. To find 25% of 3 cups: $3 \times 0.25 = 0.75$ cups. So after the reduction: $3 - 0.75 = 2.25$ cups. Next, she increases this reduced amount by 40%. Now you're working with 2.25 cups as your new starting point. Calculate 40% of 2.25: $2.25 \times 0.40 = 0.90$ cups. Adding this increase: $2.25 + 0.90 = 3.15$ cups. Looking at the wrong answers: Choice A (2.85 cups) likely comes from incorrectly calculating one of the percentage changes or mixing up the order of operations. Choice C (3.30 cups) might result from trying to shortcut the problem by combining percentages incorrectly, such as thinking -25% + 40% = +15%, then applying that to the original 3 cups. Choice D (3.45 cups) could come from calculation errors or misunderstanding how the second percentage change applies to the already-modified amount, not the original. The correct answer is B: 3.15 cups. Strategy tip: With sequential percentage problems, always apply each change to the result of the previous step, never to the original amount. Work methodically through each step to avoid the tempting shortcuts that create wrong answer choices.
A car's value depreciates by 15% each year. If the car is worth $18,700 after one year of depreciation, what was its original value?
Answer: $22,000 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 15%, it retains 85% of its original value. Let's set up the equation. If the original value is $x$ , then after one year of 15% depreciation, the car is worth $0.85x = 18,700$ . To find the original value, divide both sides by 0.85: $x = \frac{18,700}{0.85} = 22,000$ . You can verify this: if the original value was $22,000, then 15% depreciation equals $3,300, leaving $18,700 after one year. Now let's examine why the other answers are wrong. Choice A ($21,505) represents a common error where students subtract the depreciation amount from the given value instead of working backwards properly. Choice C ($23,375) might result from incorrectly adding 15% to the current value ($18,700 × 1.15), which doesn't account for the fact that $18,700 is already the depreciated amount. Choice D ($24,500) could come from miscalculating the percentage or confusing the relationship between original and depreciated values. The correct answer is B ($22,000). Strategy tip: For depreciation problems, remember that the final value equals the original value times (100% minus the depreciation rate). Always work backwards by dividing the final value by this retention percentage. Watch out for the trap of simply adding or subtracting the depreciation percentage from the given amount.
A basketball player's free throw percentage improved from 60% to 75% over the season. By what percent did his free throw percentage increase?
Answer: 25% When you see a question asking "by what percent did something increase," you're looking for percent change, not the simple difference between the two percentages. To find percent change, you need the formula: $\frac{\text{new value} - \text{old value}}{\text{old value}} \times 100\%$ The player's free throw percentage went from 60% to 75%. The change is $75\% - 60\% = 15\%$ , but that's the absolute change in percentage points, not the percent increase. Using the percent change formula: $\frac{75 - 60}{60} \times 100\% = \frac{15}{60} \times 100\% = \frac{1}{4} \times 100\% = 25\%$ So the correct answer is C) 25%. Here's why the other choices are wrong: A) 15% represents the absolute difference between 75% and 60%, but this isn't the percent increase—it's just the change in percentage points. B) 20% would result from an incorrect calculation, possibly mixing up the original and new values. D) 30% is too high and likely comes from a computational error. The key insight is distinguishing between "percentage point change" and "percent change." If someone's approval rating goes from 20% to 40%, that's a 20 percentage point increase but a 100% increase (doubling). Strategy tip: Always identify your starting value (the denominator in percent change problems) and remember that percent change compares the amount of change to the original value, not to the new value.
A store increased the price of an item by 30%, then offered a 30% discount on the new price. Compared to the original price, the final price represents what percent change?
Answer: 9% decrease When you encounter percentage problems involving multiple changes, remember that each percentage change applies to the new value, not the original one. This creates a compounding effect that often surprises students. Let's trace through this step by step using a concrete example. Say the original price is $100. After a 30% increase, the new price becomes $100 + (0.30 × $100) = $130. Now comes the crucial part: the 30% discount applies to this new price of $130, not the original $100. So the discount is 0.30 × $130 = $39, making the final price $130 - $39 = $91. Compared to the original $100, the final price of $91 represents a $9 decrease, which is a 9% decrease. Looking at the wrong answers: Choice B (1% decrease) might tempt students who make small arithmetic errors in their calculations. Choice C (0% change) is the most common trap—many students incorrectly assume that a 30% increase followed by a 30% discount should cancel out perfectly. Choice D (1% increase) could result from confusion about whether the final result should be positive or negative. The key insight is that when you increase a value and then decrease it by the same percentage, you don't return to the original value because the decrease applies to the larger amount. Here's a quick formula: if you increase by $p\%$ then decrease by $p\%$ , the net change is $-p^2/100$ percent. For 30%, that's $-900/100 = -9\%$ .
A restaurant bill of $85 includes a 15% service charge that was added to the original food cost. What was the original cost of the food before the service charge?
Answer: $73.91 When you encounter a problem where a percentage has been added to an original amount, you're working with what's called a "reverse percentage" problem. The key insight is that the final amount represents more than 100% of the original. Here, the $85 total represents 115% of the original food cost (100% original + 15% service charge). To find the original amount, you need to work backwards by dividing the total by 1.15. Setting up the equation: Original cost × 1.15 = $85 Therefore: Original cost = $85 ÷ 1.15 = $73.91 You can verify this: $73.91 × 0.15 = $11.09 (service charge), and $73.91 + $11.09 = $85 ✓ Now let's examine why the other answers are incorrect: A) $72.25 is too low. If you add 15% to this ($10.84), you get $83.09, not $85. C) $74.50 is close but still incorrect. Adding 15% gives you $85.68, which exceeds the target. D) $76.30 is too high. This would result in a total of $87.75 after adding the 15% service charge. The most common mistake students make is subtracting 15% from $85 (getting $72.25), but this doesn't account for the fact that 15% of the original amount differs from 15% of the final amount. Remember: when a percentage is added to create a total, divide the total by (1 + percentage rate) to find the original amount.
The price of a jacket was reduced by 25% to $48. What was the original price of the jacket?
Answer: $64 When you see a percent decrease problem where you're given the final price, you're working backwards from a sale price to find the original price. The key insight is that if something was reduced by 25%, then the sale price represents 75% of the original price. Let's set up the equation. If the original price is $x$ , then after a 25% reduction, the jacket costs $0.75x = 48$ . To find the original price, divide both sides by 0.75: $x = \frac{48}{0.75} = \frac{48}{\frac{3}{4}} = 48 \times \frac{4}{3} = 64$ . So the original price was $64. Let's check why the other answers are wrong. Choice (A) $60 would mean a 25% reduction gives us $60 \times 0.75 = 45$ , not $48. Choice (C) $72 would give us $72 \times 0.75 = 54$ , which is too high. Choice (D) $75 would result in $75 \times 0.75 = 56.25$ , also too high. Only choice (B) $64 gives us the correct sale price when reduced by 25%. The most common trap in these problems is subtracting 25% of the sale price from the sale price itself. Remember: when something is reduced by a percentage, the final price represents what's left after the reduction. So if there's a 25% decrease, the sale price is 75% of the original, not the original minus 25% of the sale price.
A stock price fell 20% on Monday and rose 25% on Tuesday. If the stock was worth $60 per share on Sunday, what was its value after Tuesday?
Answer: $60 When you encounter percentage change problems, remember that each percentage change applies to the current value, not the original value. This is a key concept that creates compound effects. Let's track the stock price through each day. Starting with $60 on Sunday, the stock falls 20% on Monday. To find 20% of $60: $0.20 \times 60 = 12$ . So Monday's price is $60 - 12 = 48$ dollars. On Tuesday, the stock rises 25%, but this increase applies to Monday's price of $48, not the original $60. Calculate 25% of $48: $0.25 \times 48 = 12$ . Tuesday's final price is $48 + 12 = 60$ dollars. Answer choice A ($58) likely comes from incorrectly calculating the percentage changes or making arithmetic errors in the two-step process. Answer choice C ($63) probably results from applying the 25% Tuesday increase to the original Sunday price of $60 instead of Monday's reduced price. Answer choice D ($65) might come from adding the percentage points (25% - 20% = 5% gain) and applying that directly to $60, which ignores how percentage changes actually compound. The correct answer is B ($60) because the stock returns exactly to its original Sunday value. Remember this key strategy: when working with sequential percentage changes, always apply each percentage to the most recent value, not the original starting point. Write down the value after each step to avoid confusion and ensure accuracy.
The weight of a package increased by 40% when additional items were added. If the package now weighs 8.4 pounds, what was its weight before the items were added?
Answer: 6.00 pounds When you encounter percentage increase problems, you're working backwards from a final amount to find the original amount. The key insight is that if something increases by 40%, the new amount represents 140% (or 1.4 times) the original amount. Let's call the original weight $x$ pounds. After a 40% increase, the package weighs $x + 0.4x = 1.4x$ pounds. Since we know the final weight is 8.4 pounds, we can set up the equation: $1.4x = 8.4$ Solving for $x$ : $x = \frac{8.4}{1.4} = 6.0$ pounds So the original weight was 6.00 pounds, making B correct. Let's examine why the other choices are wrong. Choice A (5.04 pounds) comes from incorrectly calculating 40% of 8.4 and subtracting it: $8.4 - (0.4 \times 8.4) = 8.4 - 3.36 = 5.04$ . This is wrong because you're taking 40% of the final amount, not the original. Choice C (6.72 pounds) results from adding 40% to the correct answer: $6.0 + (0.4 \times 6.0) = 7.2$ , then making calculation errors. Choice D (7.00 pounds) might come from roughly estimating that the original should be close to the final amount. Remember this strategy: when dealing with percentage increases, if the final amount represents 100% plus the increase percentage, divide the final amount by this total percentage (as a decimal) to find the original.
A store's monthly revenue increased from $12,000 in January to $15,600 in February. What was the percent increase in revenue?
Answer: 30% Percent increase questions test your ability to calculate how much a quantity has grown relative to its original value. When you see "percent increase," you need to find the change and compare it to the starting amount. To find percent increase, use this formula: $\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$ Here, the revenue increased from $12,000 (January) to $15,600 (February). First, find the change: $15,600 - $12,000 = $3,600. Then divide by the original value: $\frac{3,600}{12,000} = 0.30$ . Convert to a percentage: $0.30 \times 100\% = 30\%$ . This confirms answer B is correct. Looking at the wrong answers: Choice A (23%) likely comes from incorrectly dividing the change by the new value instead of the original: $\frac{3,600}{15,600} ≈ 0.23$ . This is a common error—always use the original value as your denominator. Choice C (35%) might result from calculation mistakes or rounding errors. Choice D (43%) could come from more serious computational errors or misunderstanding the problem setup. Remember this key strategy: percent increase always compares the change to the original amount, not the final amount. A helpful way to check your work is to verify that 30% of $12,000 ($3,600) plus the original $12,000 equals the new value of $15,600. This backward check can catch calculation errors quickly.
A laptop computer was marked down 30% for a sale. During the sale, an additional 10% discount was applied to the already reduced price. If the final price was $504, what was the original price?
Answer: $800 When you encounter multiple discount problems, you need to apply each discount sequentially to the running price, not to the original price. Think of this as peeling back the discounts one layer at a time. Let's work backwards from the final price of $504. If the laptop received an additional 10% discount during the sale, then $504 represents 90% of the already-reduced price. So the price after the first discount was: $504 ÷ 0.90 = 560$ Now we know the price after the initial 30% markdown was $560. This $560 represents 70% of the original price (since 100% - 30% = 70%). Therefore, the original price was: $560 ÷ 0.70 = 800$ The original price was $800, making C correct. Let's check why the other answers are wrong. Choice A ($720) would give you a final price of $441 after both discounts ($720 × 0.70 × 0.90). Choice B ($780) would result in a final price of $490.20. Choice D ($840) would lead to a final price of $529.20. None of these match the given final price of $504. The key strategy here is recognizing that consecutive percentage discounts multiply together, not add. A 30% discount followed by a 10% discount gives you 70% × 90% = 63% of the original price, not a simple 40% discount. Always work backwards from the final amount when the original is unknown, and remember that each discount applies to the current price, not the starting price.
Maya's test score improved from 68 to 85. By approximately what percent did her score increase?
Answer: 25% When you encounter percent increase problems, you're working with the formula: $\text{Percent Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$ Maya's score increased from 68 to 85, so the change is $85 - 68 = 17$ points. Using our formula: $\frac{17}{68} \times 100\%$ To calculate this, divide: $17 ÷ 68 = 0.25$ , which equals 25%. This confirms answer choice B is correct. Let's examine why the other options are wrong. Choice A (20%) would result from incorrectly calculating $\frac{17}{85}$ instead of using the original value as the denominator—a common mistake where students use the new value instead of the starting point. Choice C (30%) might come from rounding errors or miscalculating the fraction, perhaps confusing $\frac{17}{68}$ with a simpler fraction like $\frac{1}{3}$ . Choice D (35%) is too high and suggests significant computational errors, possibly from incorrectly finding the change or denominator. Remember this key strategy: always use the original value as your denominator in percent change problems. A helpful check is that $\frac{1}{4} = 0.25 = 25\%$ , and since $\frac{17}{68}$ is exactly $\frac{1}{4}$ (because $17 \times 4 = 68$ ), you can quickly verify that 25% is reasonable. This fraction recognition can speed up your calculations on test day.

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