ISEE Upper Level Quantitative Reasoning Quiz: Fractions Decimals And Percents

What this quiz covers

This quiz focuses on Fractions Decimals And Percents, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level Quantitative Reasoning.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $\frac{1}{4}$
2. $\frac{9}{40}$ (correct answer)
3. $\frac{7}{40}$
4. $\frac{3}{20}$

Explanation: When you encounter problems involving fractions of a whole group, remember that all the parts must add up to 1 (or 100%). The key is converting everything to the same format before calculating. First, convert the decimal to a fraction: $0.375 = \frac{375}{1000} = \frac{3}{8}$ (dividing both numerator and denominator by 125). Now you have $\frac{3}{8}$ prefer pizza and $\frac{2}{5}$ prefer burgers. To subtract these from 1, you need a common denominator. The least common multiple of 8 and 5 is 40. Converting: $\frac{3}{8} = \frac{15}{40}$ and $\frac{2}{5} = \frac{16}{40}$ The fraction preferring salad is: $1 - \frac{15}{40} - \frac{16}{40} = \frac{40}{40} - \frac{31}{40} = \frac{9}{40}$ Choice B, $\frac{9}{40}$, is correct. Choice A ($\frac{1}{4} = \frac{10}{40}$) is too large and likely comes from incorrectly converting 0.375 or making an arithmetic error. Choice C ($\frac{7}{40}$) suggests subtracting incorrectly, possibly computing $\frac{16}{40} - \frac{15}{40}$ instead of subtracting both from 1. Choice D ($\frac{3}{20} = \frac{6}{40}$) likely results from computational errors with the common denominator or fraction conversions. Strategy tip: Always verify your fractions add to 1 as a check. Here: $\frac{15}{40} + \frac{16}{40} + \frac{9}{40} = \frac{40}{40} = 1$ ✓

Question 2

1. 0.36 (correct answer)
2. 0.40
3. 0.42
4. 0.50

Explanation: When you encounter decimal-to-fraction problems, start by converting the decimal to its simplest fractional form, then work with the given conditions. The decimal $0.45$ equals $\frac{45}{100}$. To reduce this to lowest terms, find the greatest common divisor of 45 and 100. Since $45 = 9 \times 5$ and $100 = 20 \times 5$, the GCD is 5. Therefore: $\frac{45}{100} = \frac{9}{20}$. The original fraction in lowest terms is $\frac{9}{20}$. When the denominator increases by 5, the new fraction becomes $\frac{9}{25}$. Converting to decimal: $\frac{9}{25} = \frac{9 \times 4}{25 \times 4} = \frac{36}{100} = 0.36$. Looking at the wrong answers: Choice B (0.40) might tempt you if you incorrectly calculated $\frac{9}{25}$ or confused it with $\frac{2}{5}$. Choice C (0.42) could result from arithmetic errors in the conversion process. Choice D (0.50) represents $\frac{1}{2}$, which you might get if you mistakenly thought the original fraction was $\frac{1}{2}$ instead of properly reducing $\frac{45}{100}$. The correct answer is A) 0.36. Strategy tip: Always reduce fractions to lowest terms first before applying any changes. When converting decimals to fractions, look for common factors immediately—this prevents errors and makes subsequent calculations cleaner. Practice converting common decimals like 0.25, 0.45, and 0.75 to fractions until it becomes automatic.

Question 3

1. $30$ (correct answer)
2. $24\frac{1}{2}$
3. $30\frac{1}{3}$
4. $32\frac{1}{4}$

Explanation: When you encounter percentage problems that set two expressions equal to each other, you're looking at an equation that you can solve algebraically. The key is to translate the words into mathematical expressions and then solve for the unknown. Let's translate this step by step. "$x\%$ of 80" means $\frac{x}{100} \times 80$, and "$\frac{3}{5}$ of 40" means $\frac{3}{5} \times 40$. Setting them equal: $\frac{x}{100} \times 80 = \frac{3}{5} \times 40$. First, simplify the right side: $\frac{3}{5} \times 40 = \frac{3 \times 40}{5} = \frac{120}{5} = 24$. Now solve: $\frac{x}{100} \times 80 = 24$. Simplify the left side: $\frac{80x}{100} = \frac{4x}{5} = 24$. Multiply both sides by $\frac{5}{4}$: $x = 24 \times \frac{5}{4} = \frac{120}{4} = 30$. Since 30 is a whole number, it's already in mixed number form. Looking at the wrong answers: Choice B ($24\frac{1}{2}$) is the trap of stopping at the value of the right side of the equation without solving for $x$. Choice C ($30\frac{1}{3}$) might result from calculation errors in the fraction arithmetic. Choice D ($32\frac{1}{4}$) could come from incorrectly setting up the initial equation or making arithmetic mistakes during solving. The answer is A. Strategy tip: Always translate percentage problems into equations first, then solve systematically. Double-check by substituting your answer back into the original problem to verify it works.

Question 4

1. $\frac{5}{6}$ (correct answer)
2. $\frac{25}{30}$
3. $\frac{83}{100}$
4. $\frac{75}{90}$

Explanation: When you encounter a repeating decimal like $0.8\overline{3}$, you need to convert it to a fraction using algebraic manipulation. The bar over the 3 means that digit repeats infinitely: $0.8333...$ Let $x = 0.8\overline{3} = 0.8333...$ To eliminate the repeating portion, multiply both sides by 10 (since one digit repeats): $10x = 8.333...$ Now subtract the original equation: $10x - x = 8.333... - 0.833...$ This gives you $9x = 7.5$, so $x = \frac{7.5}{9} = \frac{75}{90}$ To reduce to lowest terms, find the greatest common divisor of 75 and 90. Both are divisible by 15: $\frac{75}{90} = \frac{75 ÷ 15}{90 ÷ 15} = \frac{5}{6}$ Answer A, $\frac{5}{6}$, is correct. Answer B, $\frac{25}{30}$, reduces to $\frac{5}{6}$ but isn't in lowest terms as requested. Answer C, $\frac{83}{100} = 0.83$, represents the misconception of treating $0.8\overline{3}$ as simply $0.83$ without accounting for the infinite repetition. Answer D, $\frac{75}{90}$, is the unreduced form of the correct fraction—you correctly set up the equation but forgot to simplify. Strategy tip: For repeating decimals, always multiply by the appropriate power of 10 based on how many digits repeat (10 for one digit, 100 for two digits, etc.), then subtract to eliminate the repetition. Don't forget to reduce your final fraction to lowest terms.

Question 5

1. 12%
2. 15% (correct answer)
3. 18%
4. 20%

Explanation: When you encounter percentage problems involving subgroups, you need to find what portion of the entire population meets multiple criteria. Here, you're looking for students who are both boys AND play soccer. Start with what you know: 40% of the class are boys, and $\frac{3}{8}$ of those boys play soccer. To find the percentage of the entire class that consists of boys who play soccer, multiply these two fractions together. First, convert 40% to a fraction: $40\% = \frac{40}{100} = \frac{2}{5}$ Now multiply: $\frac{2}{5} \times \frac{3}{8} = \frac{6}{40} = \frac{3}{20}$ Convert back to a percentage: $\frac{3}{20} = 0.15 = 15\%$ This confirms answer choice B is correct. Looking at the wrong answers: Choice A (12%) likely comes from incorrectly calculating $\frac{2}{5} \times \frac{3}{8}$ or making an arithmetic error. Choice C (18%) might result from adding instead of multiplying the percentages, or miscalculating the fraction multiplication. Choice D (20%) could come from mistakenly using $\frac{1}{2}$ instead of $\frac{2}{5}$ for the percentage of boys, then multiplying $\frac{1}{2} \times \frac{3}{8} = \frac{3}{16} ≈ 18.75\%$, then rounding incorrectly. Remember: When finding the percentage of a total population that meets multiple criteria, multiply the individual percentages or fractions. Don't add them—that would give you something meaningless like "boys plus soccer players," not "boys who play soccer."

Question 6

1. $83\frac{1}{3}\%$ (correct answer)
2. $82\frac{1}{2}\%$
3. $85\frac{5}{6}\%$
4. $80\frac{2}{3}\%$

Explanation: When you encounter fraction-to-percentage conversions, remember that converting a fraction to a percentage means finding what the fraction equals out of 100. The given information that $\frac{2}{3} = 66\frac{2}{3}\%$ provides a helpful reference point. To find what percent $\frac{5}{6}$ equals, multiply the fraction by 100: $\frac{5}{6} \times 100 = \frac{500}{6}$. Now divide: $500 \div 6 = 83\frac{2}{6}$. Since $\frac{2}{6} = \frac{1}{3}$, we get $83\frac{1}{3}\%$. You can verify this makes sense by comparing it to the given information. Since $\frac{5}{6} = \frac{10}{12}$ and $\frac{2}{3} = \frac{8}{12}$, we know $\frac{5}{6}$ is larger than $\frac{2}{3}$, so its percentage should be greater than $66\frac{2}{3}\%$. Choice A gives us $83\frac{1}{3}\%$, which matches our calculation perfectly. Choice B ($82\frac{1}{2}\%$) results from incorrectly calculating $\frac{500}{6}$ or making an arithmetic error in the division. Choice C ($85\frac{5}{6}\%$) comes from the common mistake of not simplifying $\frac{500}{6}$ properly and leaving the fractional part as $\frac{5}{6}$ instead of converting to the mixed number correctly. Choice D ($80\frac{2}{3}\%$) likely results from calculation errors in the multiplication or division steps. Study tip: When converting fractions to percentages, always multiply by 100 first, then simplify your result completely. Double-check by ensuring your answer makes logical sense compared to benchmark fractions you know.

Question 7

1. $\frac{7}{10}$ (correct answer)
2. $\frac{3}{4}$
3. $\frac{4}{5}$
4. $\frac{5}{6}$

Explanation: When you encounter a complex fraction like this, your goal is to simplify it step by step by converting all numbers to the same form—either all decimals or all fractions—then performing the operations. Let's convert everything to fractions for easier calculation. First, convert the decimals: $0.75 = \frac{3}{4}$ and $1.25 = \frac{5}{4}$. Now our expression becomes $\frac{\frac{3}{4} + \frac{1}{8}}{\frac{5}{4}}$. To add the fractions in the numerator, find a common denominator. Since $\frac{3}{4} = \frac{6}{8}$, we have $\frac{6}{8} + \frac{1}{8} = \frac{7}{8}$. Our expression is now $\frac{\frac{7}{8}}{\frac{5}{4}}$. To divide by a fraction, multiply by its reciprocal: $\frac{7}{8} \times \frac{4}{5} = \frac{28}{40} = \frac{7}{10}$. This confirms answer choice A is correct. Looking at the wrong answers: B) $\frac{3}{4}$ might result from incorrectly using just the $0.75$ without adding $\frac{1}{8}$. C) $\frac{4}{5}$ could come from forgetting to divide by $1.25$ entirely. D) $\frac{5}{6}$ might result from calculation errors when finding common denominators or performing the division. For complex fraction problems on the ISEE, always convert mixed decimal-fraction expressions to a single form first. Work systematically through each step rather than trying mental shortcuts, as the test makers often include trap answers that correspond to common computational mistakes.

Question 8

1. 0.6
2. 0.8 (correct answer)
3. 1.0
4. 1.2

Explanation: When you encounter ratio problems with a given total, you're working with proportional relationships where each part represents a fraction of the whole. The key is to find what one "unit" of the ratio equals, then calculate each ingredient amount. The ratio $3:4:5$ means the ingredients are in proportional amounts of 3 parts, 4 parts, and 5 parts respectively. First, find the total number of ratio parts: $3 + 4 + 5 = 12$ parts total. Since the total amount is 2.4 cups, each part equals $\frac{2.4}{12} = 0.2$ cups per part. The second ingredient needs 4 parts, so: $4 \times 0.2 = 0.8$ cups. Looking at the wrong answers: Choice (A) 0.6 cups would result from incorrectly using 3 parts instead of 4 parts for the second ingredient ($3 \times 0.2 = 0.6$). Choice (C) 1.0 cups represents 5 parts ($5 \times 0.2 = 1.0$), which would be the third ingredient, not the second. Choice (D) 1.2 cups doesn't correspond to any single ingredient in this ratio—it would require 6 parts, which exceeds any individual component. The correct answer is (B) 0.8 cups. Strategy tip: Always set up ratio problems by first finding the total number of parts, then determining the value of one part by dividing the given total by this sum. This systematic approach prevents mix-ups between different ratio components and ensures accurate calculations.

Question 9

1. 17
2. 20 (correct answer)
3. 23
4. 25

Explanation: When you encounter a problem where a fraction equals a percentage, you're dealing with part-to-whole relationships. The key insight is that the fraction $\frac{17}{20}$ and 85% both represent the same ratio of correct answers to total questions. To solve this, convert the percentage to a fraction: $85\% = \frac{85}{100} = \frac{17}{20}$. Since we're told that $\frac{17}{20}$ of the questions were answered correctly and this equals 85%, we can set up the equation: $\frac{17}{20} = \frac{85}{100}$. Cross-multiplying: $17 \times 100 = 85 \times 20$, which gives us $1700 = 1700$. This confirms our fractions are equivalent. Looking at $\frac{17}{20}$, we can see that 17 represents the number of correct answers and 20 represents the total number of questions on the test. Choice A (17) represents only the number of correct answers, not the total questions. Choice C (23) might come from incorrectly adding 17 + 6, perhaps thinking you need to add something to get from the numerator to the total. Choice D (25) could result from the misconception that since $\frac{17}{20} = 0.85$, you might think 17 out of 25 gives 85% (but $\frac{17}{25} = 0.68 = 68\%$). Choice B (20) is correct because the denominator of $\frac{17}{20}$ directly tells us the total number of questions. Strategy tip: When a fraction equals a given percentage, convert the percentage to its simplest fraction form. The denominator of that simplified fraction often reveals the total you're looking for.

Question 10

1. 0% change (correct answer)
2. 5% increase
3. 10% decrease
4. 15% increase

Explanation: When you encounter problems about changing dimensions and their effect on area, remember that area changes are multiplicative, not additive. You need to calculate how each dimension change affects the overall area. Let's work with a rectangle that has original length $L$ and width $W$, so the original area is $L \times W$. After the changes:

• New length = $L + 0.25L = 1.25L$ (increased by 25%)
• New width = $W - \frac{1}{5}W = W - 0.2W = 0.8W$ (decreased by $\frac{1}{5}$ or 20%)

Question 11

1. 35.0
2. 40.0 (correct answer)
3. 42.5
4. 45.0

Explanation: When you encounter a problem that sets two expressions equal to each other, you're dealing with an equation that you can solve by translating the words into mathematical expressions and then isolating the variable. Start by translating each part of the equation. "$x$ percent of 60" means $\frac{x}{100} \times 60$, which simplifies to $\frac{60x}{100} = 0.6x$. The phrase "$\frac{3}{4}$ of 32" means $\frac{3}{4} \times 32 = 24$. So your equation becomes: $0.6x = 24$. To solve for $x$, divide both sides by 0.6: $x = \frac{24}{0.6} = 40$. You can verify this: 40% of 60 is $0.40 \times 60 = 24$, which equals $\frac{3}{4} \times 32 = 24$ ✓ Looking at the wrong answers: Choice A (35.0) would give you 35% of 60 = 21, which is less than 24. Choice C (42.5) would give you 42.5% of 60 = 25.5, which overshoots the target. Choice D (45.0) would give you 45% of 60 = 27, which is even further from 24. These incorrect answers likely result from computational errors in the division step or mistakes in setting up the initial equation. The key strategy here is to always translate word problems into mathematical expressions systematically, then solve step by step. Double-check your work by substituting your answer back into the original problem to verify both sides are equal.

Question 12

1. 4% (correct answer)
2. 6%
3. 8%
4. 10%

Explanation: When you see a question asking "what percent of," you're looking at a classic percent proportion problem. You need to find what percent one number represents when compared to another number. To solve this, set up the equation: $\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}$. Here, $0.024$ is the "part" and $\frac{3}{5}$ is the "whole." First, convert $\frac{3}{5}$ to a decimal: $\frac{3}{5} = 0.6$ Now substitute into your equation: $\frac{0.024}{0.6} = \frac{x}{100}$ Solve for $x$: $x = \frac{0.024 \times 100}{0.6} = \frac{2.4}{0.6} = 4$ Therefore, $0.024$ is $4\%$ of $\frac{3}{5}$, making A correct. The wrong answers likely come from calculation errors. Choice B (6%) might result from incorrectly converting $\frac{3}{5}$ or making an arithmetic mistake in the division. Choice C (8%) could come from confusing the setup—perhaps finding what percent $\frac{3}{5}$ is of $0.024$ instead of the reverse. Choice D (10%) might result from using $0.24$ instead of $0.024$ in your calculation, a common decimal placement error. Strategy tip: Always double-check which number is the "part" and which is the "whole" in percent problems. The number after "percent of" is always your whole (denominator). Also, verify your decimal conversions—fraction-to-decimal errors are frequent sources of wrong answers on these questions.

Question 13

1. 60.5%
2. 62.5% (correct answer)
3. 65.0%
4. 67.5%

Explanation: When you encounter ratio problems involving mixture removal, the key insight is that removing a portion of the mixture doesn't change the ratio of components within what remains—it affects only the total quantity. The mixture has water and alcohol in a $5:3$ ratio. This means for every 8 parts of mixture, 5 parts are water and 3 parts are alcohol. Therefore, water makes up $\frac{5}{8} = 0.625 = 62.5\%$ of the mixture. When $12.5\%$ of the mixture is removed, you're removing $12.5\%$ of both the water and the alcohol proportionally. The remaining $87.5\%$ still maintains the same $5:3$ ratio. Since the internal proportions haven't changed, water still comprises $62.5\%$ of the remaining mixture. Choice (A) 60.5% likely comes from incorrectly attempting to subtract some percentage related to the removal. Choice (C) 65.0% might result from adding rather than maintaining the original percentage. Choice (D) 67.5% could come from misapplying the $12.5\%$ removal figure to the water percentage calculation. The correct answer is (B) 62.5%. Study tip: Remember that when you remove a fraction of a mixture uniformly, the ratio of components stays constant. The percentage composition only changes if you selectively remove more of one component than another. Always ask yourself: "Does this operation change the internal ratio, or just the total amount?"

Question 14

1. 0.125
2. 0.5
3. 1.0 (correct answer)
4. 1.5

Explanation: When you encounter division problems mixing fractions and decimals, you have two main approaches: convert the fraction to a decimal, or convert the decimal to a fraction. Let's use the second approach here. First, convert 0.375 to a fraction. Since 0.375 has three decimal places, write it as $\frac{375}{1000}$. Now simplify by finding the GCD of 375 and 1000. Both are divisible by 125, so $0.375 = \frac{375÷125}{1000÷125} = \frac{3}{8}$. Now your problem becomes: $\frac{3}{8} ÷ \frac{3}{8}$ When dividing fractions, multiply by the reciprocal: $\frac{3}{8} × \frac{8}{3} = \frac{24}{24} = 1$ As a decimal, this is 1.0. Choice A (0.125) represents what you'd get if you incorrectly multiplied $\frac{3}{8} × 0.375$ instead of dividing. Choice B (0.5) might result from calculation errors or confusing the order of operations. Choice D (1.5) could come from adding instead of dividing, or from other arithmetic mistakes. The key insight here is recognizing that 0.375 equals $\frac{3}{8}$, making this division problem $\frac{3}{8} ÷ \frac{3}{8} = 1$. On standardized tests, when you see "nice" decimals like 0.375, 0.25, or 0.125, always check if they convert to simple fractions—this often reveals shortcuts that make the problem much easier to solve.

Question 15

1. 57% and 58%
2. 58% and 59% (correct answer)
3. 59% and 60%
4. 60% and 61%

Explanation: When you need to convert a fraction to a percentage and compare it to consecutive whole number percentages, the key is performing accurate decimal conversion and understanding where your result falls on the number line. To convert $\frac{7}{12}$ to a percentage, divide 7 by 12: $7 ÷ 12 = 0.58\overline{3}$ (where the 3 repeats infinitely). Converting to percentage form: $0.58\overline{3} × 100 = 58.\overline{3}\%$ or approximately 58.33%. Since 58.33% falls between 58% and 59%, the correct answer is B. Let's examine why the other choices are incorrect. Choice A (57% and 58%) places our fraction too low on the percentage scale—58.33% is greater than 58%, so it can't fall between 57% and 58%. Choice C (59% and 60%) positions the fraction too high—since 58.33% is less than 59%, it doesn't belong in this range. Similarly, choice D (60% and 61%) is far too high, as our fraction is nearly 2 percentage points below 60%. The trap here is rushing through the division or rounding incorrectly. Some students might stop at 0.58 and think the answer is A, forgetting that $\frac{7}{12}$ actually equals slightly more than 0.58. Study tip: When converting fractions to percentages for comparison questions, always carry your division to at least three decimal places to see exactly where the value falls between consecutive integers. This precision prevents you from selecting boundary answers incorrectly.

Question 16

1. 65%
2. 70%
3. 75% (correct answer)
4. 80%

Explanation: When you encounter percentage problems involving real-world scenarios like this, break them down into three clear steps: find what you have, find what you need, then calculate the percentage. First, let's convert everything to the same format. The baker has $3.75$ cups of flour. She uses $1\frac{1}{4} = 1.25$ cups per batch. For 4 batches, she needs: $4 \times 1.25 = 5$ cups of flour total. Now we can find what percentage she has: $\frac{\text{what she has}}{\text{what she needs}} \times 100\% = \frac{3.75}{5} \times 100\% = 0.75 \times 100\% = 75\%$ This confirms answer C is correct. Let's examine why the other options are wrong. Choice A (65%) would result from incorrectly calculating $\frac{3.25}{5}$, likely from converting $1\frac{1}{4}$ incorrectly. Choice B (70%) might come from computational errors in the division or rounding mistakes. Choice D (80%) could result from using 3 batches instead of 4 in the denominator ($\frac{3.75}{3 \times 1.25} = \frac{3.75}{3.75} = 100\%$ adjusted downward) or other calculation errors. Strategy tip: On percentage problems involving mixed numbers and decimals, always convert everything to the same format first—either all decimals or all fractions. This prevents conversion errors that lead to wrong answer traps. Also, double-check that you're using the correct quantities in your percentage formula.

Question 17

1. 63.6% (correct answer)
2. 64.7%
3. 65.2%
4. 66.1%

Explanation: When you need to convert a fraction to a percentage, you're essentially finding what the fraction equals out of 100. The key is to either multiply by 100 or use long division to find the decimal equivalent, then convert to a percent. To convert $\frac{7}{11}$ to a percentage, divide 7 by 11: $7 ÷ 11 = 0.636363...$ (the digits 63 repeat infinitely). Converting this decimal to a percentage means multiplying by 100, giving you 63.636...%, which rounds to 63.6%. Looking at the answer choices: A) 63.6% matches our calculated result exactly. B) 64.7% is too high—this would be closer to $\frac{7.1}{11}$, suggesting a calculation error where someone might have rounded up too aggressively or made an arithmetic mistake. C) 65.2% is significantly higher than the actual value, possibly resulting from confusing $\frac{7}{11}$ with a different fraction like $\frac{13}{20}$. D) 66.1% is even further off, potentially from mistakenly calculating $\frac{2}{3}$ instead of $\frac{7}{11}$. The answer is A) 63.6%. Study tip: When converting fractions to percentages on the ISEE, if the denominator doesn't divide evenly into 100, use long division to find the decimal (calculate to at least 3 decimal places for accuracy), then multiply by 100. Watch out for repeating decimals—they're common with denominators like 3, 6, 7, 9, and 11.

Question 18

1. $\frac{5}{16}$ (correct answer)
2. $\frac{3}{8}$
3. $\frac{7}{16}$
4. $\frac{1}{2}$

Explanation: This problem tests your ability to work with fractions and percentages together, which is a common pattern on quantitative reasoning tests. The key is to track what happens to the water step by step. Start with the tank being $\frac{5}{8}$ full. When you use 37.5% of the water currently in the tank, you need to find 37.5% of $\frac{5}{8}$. Convert 37.5% to a fraction: 37.5% = $\frac{37.5}{100} = \frac{3}{8}$. Now calculate: $\frac{3}{8} \times \frac{5}{8} = \frac{15}{64}$. This represents how much water was used. The remaining water is the original amount minus what was used: $\frac{5}{8} - \frac{15}{64}$. Convert $\frac{5}{8}$ to sixty-fourths: $\frac{5}{8} = \frac{40}{64}$. Therefore: $\frac{40}{64} - \frac{15}{64} = \frac{25}{64}$. Simplify by dividing both numerator and denominator by their greatest common factor of 5: $\frac{25}{64} = \frac{5}{16}$. The answer is A. Choice B ($\frac{3}{8}$) likely comes from incorrectly calculating 37.5% of the total tank capacity rather than 37.5% of the current water. Choice C ($\frac{7}{16}$) might result from subtracting 37.5% directly from $\frac{5}{8}$ without proper conversion. Choice D ($\frac{1}{2}$) could come from rough mental estimation errors. Remember: when dealing with percentages "of" something, always identify what that "something" refers to—here it's the current water amount, not the tank's total capacity.

Question 19

1. 120% (correct answer)
2. 125%
3. 135%
4. 150%

Explanation: When you see questions involving ratios in a chain like this, you need to connect them by finding a common variable. Think of these ratios as building blocks that you can multiply together to reach your target. Given that $\frac{a}{b} = 0.75$ and $\frac{b}{c} = 1.6$, you want to find $\frac{a}{c}$. Notice that both given ratios share the variable $b$. You can multiply these fractions to eliminate $b$ and connect $a$ directly to $c$: $\frac{a}{b} \times \frac{b}{c} = \frac{a}{c}$ The $b$ terms cancel out, leaving you with: $0.75 \times 1.6 = 1.2$ So $\frac{a}{c} = 1.2 = 120\%$ The answer is A) 120%. Let's see why the other choices represent common mistakes. B) 125% might result from incorrectly adding the decimals: $0.75 + 1.6 = 2.35$, then somehow getting 1.25. C) 135% could come from adding the percentages: $75\% + 60\% = 135\%$, which ignores the multiplicative relationship between ratios. D) 150% might result from converting to percentages first (75% and 160%) and then incorrectly manipulating them. Remember: when you have a chain of ratios sharing common variables, multiply them to create the ratio you need. The shared variables will cancel out naturally, giving you a direct relationship between the remaining variables. Always double-check by ensuring your multiplication eliminates the intermediate variable completely.

Question 20

1. 100% (correct answer)
2. 120%
3. 140%
4. 160%

Explanation: This question tests your ability to multiply fractions and mixed numbers, then convert the result to a percentage. When you see a recipe scaling problem, you need to multiply the original amount by the scaling factor. First, convert the mixed number $2\frac{2}{3}$ to an improper fraction: $2\frac{2}{3} = \frac{8}{3}$. Now multiply the original sugar amount by this scaling factor: $\frac{3}{8} \times \frac{8}{3} = \frac{24}{24} = 1$ cup of sugar. Since Maria needs exactly 1 cup of sugar and the question asks what percent of a full cup this represents, the answer is $\frac{1}{1} = 100\%$. Looking at the wrong answers: Choice B (120%) would result from incorrectly calculating $\frac{3}{8} \times 2\frac{2}{3}$ and getting $1.2$ cups, likely from computational errors with the fraction multiplication. Choice C (140%) might come from adding $\frac{3}{8}$ and $2\frac{2}{3}$ instead of multiplying them, or from other calculation mistakes. Choice D (160%) could result from mishandling the mixed number conversion or making errors in the fraction arithmetic. The key insight here is recognizing that $\frac{3}{8} \times \frac{8}{3}$ creates a situation where the numerators and denominators cancel perfectly, yielding exactly 1. Strategy tip: When multiplying fractions, look for opportunities to cancel common factors before multiplying. Also, double-check your mixed number to improper fraction conversions—this is where many errors occur in scaling problems.