Fractions, Decimals, and Percents - ISEE Upper Level Quantitative Reasoning

1. Which of the following is closest to $\frac{7}{11}$ when expressed as a percent?
   Answer: 63.6% 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.
2. If $\frac{a}{b} = 0.75$ and $\frac{b}{c} = 1.6$, what is $\frac{a}{c}$ expressed as a percent?
   Answer: 120% 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.
3. A mixture contains water and alcohol in the ratio $5:3$. If $12.5\%$ of the mixture is removed, what percent of the remaining mixture is water?
   Answer: 62.5% 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?"
4. The decimal $0.024$ is what percent of $\frac{3}{5}$?
   Answer: 4% 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.
5. In a class, $40\%$ of students are boys. Of the boys, $\frac{3}{8}$ play soccer. What percent of the entire class consists of boys who play soccer?
   Answer: 15% 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."
6. A store marks up items by $33\frac{1}{3}\%$. If an item originally costs $24, what is the new price expressed as a decimal?
   Answer: $32.00 When you encounter percentage markup problems, you're dealing with finding a new value that's a certain percentage higher than the original. The key insight is that a markup increases the original price, so you'll always end up with more than you started with. To solve this, first convert the mixed number percentage to a decimal. $33\frac{1}{3}\% = 33.33...\% = \frac{1}{3}$ as a decimal (since $\frac{1}{3} = 0.3333...$). The markup amount is $\$24 \times \frac{1}{3} = \$8$. Adding this to the original price: $\$24 + \$8 = \$32.00$. Alternatively, you can think of this as finding 133⅓% of the original price: $\$24 \times 1.3333... = \$24 \times \frac{4}{3} = \$32.00$. Choice A ($31.20) represents a 30% markup instead of 33⅓%. This comes from incorrectly converting the mixed number or miscalculating $24 \times 0.30 = 7.20$, giving $24 + 7.20 = 31.20$. Choice C ($28.80) results from treating this as a 20$$24 \times 0.20 = 4.80$$, so $24 + 4.80 = 28.80$), likely from misreading the percentage entirely. Choice D ($30.40) comes from using approximately 26.67% instead of 33⅓%, possibly from calculation errors in converting the mixed number. The correct answer is B ($32.00). Remember: $33\frac{1}{3}\%$ appears frequently on standardized tests because it equals exactly $\frac{1}{3}$, making calculations cleaner. Always convert mixed number percentages to fractions when possible—it often simplifies your work significantly.
7. A number when expressed as a decimal is $0.8\overline{3}$. What is this number as a fraction in lowest terms?
   Answer: $\frac{5}{6}$ 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.
8. A baker uses $1\frac{1}{4}$ cups of flour for each batch of cookies. If she has $3.75$ cups of flour, what percent of the flour needed for 4 batches does she have?
   Answer: 75% 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.
9. In a survey, $0.375$ of the students prefer pizza, $\frac{2}{5}$ prefer burgers, and the rest prefer salad. What fraction of students prefer salad?
   Answer: $\frac{9}{40}$ 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: $\frac{15}{40} + \frac{16}{40} + \frac{9}{40} = \frac{40}{40} = 1$ ✓
10. A discount of $12.5\%$ reduces a price from $\$P$ to $42. What is $P$ expressed as a mixed number?
    Answer: $48 When you encounter discount problems, you're working with percentages and need to determine what remains after the discount is applied. A $12.5\%$ discount means the customer pays $100\% - 12.5\% = 87.5\%$ of the original price. Setting up the equation: if $87.5\%$ of the original price $P$ equals $42, then $0.875 \times P = 42$. To find $P$, divide both sides by $0.875$: $P = \frac{42}{0.875} = 48$. You can verify this: $12.5\%$ of $48 is $0.125 \times 48 = 6$, so the discounted price is $48 - 6 = 42$ ✓ Looking at the wrong answers: Choice A ($\$46\frac{1}{2}$) represents a common error where students might subtract the discount percentage directly from the original price without proper calculation. Choice B ($\$47\frac{1}{4}$) could result from incorrectly converting the percentage or making arithmetic errors in the division. Choice D ($\$50\frac{2}{3}$) might come from setting up the equation incorrectly, perhaps confusing which percentage to use in the calculation. The key strategy for discount problems is remembering that the final price represents what percentage remains after the discount. Convert $12.5\%$ to $87.5\%$ remaining, then set up your equation accordingly. Always check your answer by calculating the discount and ensuring it leads back to the given final price.
11. A tank is $\frac{5}{8}$ full of water. After using $37.5\%$ of the water in the tank, what fraction of the tank's total capacity remains filled with water?
    Answer: $\frac{25}{64}$ Find the water left, not the space used. The tank starts $\frac{5}{8}$ full, and 37.5% of that water is used. Since $37.5\% = \frac{3}{8}$, the fraction of the water that stays is $1 - \frac{3}{8} = \frac{5}{8}$. The water remaining is $\frac{5}{8} \times \frac{5}{8} = \frac{25}{64}$ of the tank's total capacity. This fraction is already in simplest form, because 25 and 64 share no common factor. The value $\frac{3}{8}$ is just 37.5% written as a fraction; it is the share of the water that was used, not what remains. Subtracting $\frac{3}{8}$ straight from $\frac{5}{8}$ would treat 37.5% as a percent of the whole tank and gives $\frac{1}{4}$, which is also wrong here. Values such as $\frac{7}{16}$ and $\frac{1}{2}$ are larger than the starting water times the kept share, so they cannot be right. The answer is $\frac{25}{64}$.
12. A car's value depreciates by $16\frac{2}{3}\%$ in the first year. If the car was originally worth $18,000, what is its value after one year?
    Answer: $15,000 When you encounter percentage depreciation problems, you're dealing with finding a remaining value after a percentage decrease. The key insight is that if something depreciates by a certain percentage, you keep the remaining percentage of the original value. First, convert the mixed number percentage to a decimal. $16\frac{2}{3}\% = 16.667\% = \frac{50}{3}\% = \frac{1}{6}$ of the original value. If the car depreciates by $\frac{1}{6}$, then it retains $1 - \frac{1}{6} = \frac{5}{6}$ of its original value. Calculate the remaining value: $\frac{5}{6} \times \$18,000 = \frac{5 \times 18,000}{6} = \frac{90,000}{6} = \$15,000$ Choice A ($14,800) likely comes from incorrectly calculating $16\frac{2}{3}\%$ as $17.78\%$ and subtracting $3,200 instead of $3,000. Choice C ($15,200) suggests someone calculated a $15.56\%$ depreciation ($2,800 loss) instead of $16\frac{2}{3}\%$. Choice D ($15,600) represents a $13.33\%$ depreciation ($2,400 loss), possibly from misreading the fraction. Remember that $16\frac{2}{3}\% = \frac{1}{6}$ is a common fraction that appears frequently on standardized tests. When you see this percentage, immediately think "one-sixth," which makes the mental math much easier. Always double-check that you're subtracting the depreciation from 100% to find what percentage remains, then multiply by the original value.
13. A recipe calls for ingredients in the ratio $3:4:5$. If the total amount is $2.4$ cups, how many cups of the second ingredient are needed, expressed as a decimal?
    Answer: 0.8 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.
14. A rectangle's length is increased by $25\%$ and its width is decreased by $\frac{1}{5}$. What is the percent change in the rectangle's area?
    Answer: 0% change 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%) The new area = $1.25L \times 0.8W = 1.00LW$ Since $1.00LW$ equals the original area $LW$, there's no change in area. Looking at the wrong answers: Choice B (5% increase) might tempt you if you incorrectly think $1.25 \times 0.8 = 1.05$. Choice C (10% decrease) could result from subtracting the percentages: $25\% - 20\% = 5\%$, then making a sign error. Choice D (15% increase) might come from incorrectly adding the percentage changes or miscalculating the product. The key insight is that when you multiply $1.25 \times 0.8$, you get exactly $1.00$, meaning the increases and decreases perfectly cancel out. Study tip: For dimension-change problems, always multiply the scaling factors rather than adding or subtracting percentages. Convert percentages to decimal multipliers first, then multiply them together to find the overall effect.
15. Express $\frac{3}{8} \div 0.375$ as a decimal.
    Answer: 1.0 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.
16. If $x$ percent of 60 is equal to $\frac{3}{4}$ of 32, find the value of $x$ as a decimal.
    Answer: 40.0 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.
17. A recipe calls for $\frac{3}{8}$ cup of sugar. If Maria wants to make $2\frac{2}{3}$ times the recipe, approximately what percent of a full cup of sugar will she need?
    Answer: 100% 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.
18. A fraction in lowest terms has a decimal expansion of $0.45$. If the denominator is increased by 5, what is the new decimal value?
    Answer: 0.36 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.
19. The fraction $\frac{7}{12}$ is between which two consecutive percentages?
    Answer: 58% and 59% 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.
20. Which expression is equivalent to $\frac{0.75 + \frac{1}{8}}{1.25}$?
    Answer: $\frac{7}{10}$ 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.

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