Converting Percents - Accuplacer Arithmetic

1. A parking lot has 250 spaces. 30% of the spaces are reserved. Of the non-reserved spaces, $\frac{1}{5}$ are for compact cars. What simplified fraction represents the portion of the total parking lot that is for non-reserved compact cars?
   Answer: $\frac{7}{50}$ First, find the percentage of non-reserved spaces: 100% - 30% = 70%. Next, find the portion of these that are for compact cars: $\frac{1}{5}$ of 70%. To calculate this, convert 70% to a fraction ($\frac{70}{100}$ or $\frac{7}{10}$) and multiply: $\frac{1}{5} \times \frac{7}{10} = \frac{7}{50}$. This fraction is already in its simplest form.
2. What is 0.035 expressed as a percentage?
   Answer: 3.5% To convert a decimal to a percentage, multiply the decimal by 100. This is equivalent to moving the decimal point two places to the right. For 0.035, moving the decimal point two places to the right gives 3.5. Therefore, 0.035 is equal to 3.5%.
3. If 35% of a number is 70, what is $\frac{1}{4}$ of that number?
   Answer: 50 This is a two-step problem. First, find the original number. Let the number be $n$. The problem states that 35% of $n$ is 70, which can be written as the equation $0.35n = 70$. To find $n$, divide both sides by 0.35: $n = \frac{70}{0.35} = 200$. The original number is 200. The second step is to find $\frac{1}{4}$ of this number: $\frac{1}{4} \times 200 = 50$.
4. If $0.045$ is written as a percent, and then that percent is written as a fraction in lowest terms, what is the denominator of the resulting fraction?
   Answer: 200 This question tests your ability to convert between decimals, percents, and fractions—a fundamental skill that appears frequently on arithmetic assessments. To convert $0.045$ to a percent, multiply by 100 (or move the decimal point two places right): $0.045 \times 100 = 4.5\%$. Now you need to write $4.5\%$ as a fraction in lowest terms. Since $4.5\% = \frac{4.5}{100}$, you have a decimal in the numerator. To eliminate this, multiply both numerator and denominator by 10: $\frac{4.5 \times 10}{100 \times 10} = \frac{45}{1000}$. Now reduce to lowest terms by finding the greatest common divisor of 45 and 1000. Since $45 = 9 \times 5$ and $1000 = 8 \times 125 = 8 \times 5^3$, the GCD is 5. Dividing both by 5 gives $\frac{9}{200}$. The denominator is 200. Choice A (20) would result if you incorrectly converted $0.045$ to $45\%$ instead of $4.5\%$, then wrote $\frac{45}{100} = \frac{9}{20}$. Choice B (400) might come from doubling 200 through a calculation error. Choice D (1000) is the unreduced denominator before simplifying $\frac{45}{1000}$. When converting decimals to percents to fractions, always double-check your decimal-to-percent conversion first—this is where most errors occur. Remember that multiplying by 100 moves the decimal two places right, and always reduce fractions to lowest terms as the final step.
5. What is $\frac{7}{16}$ converted to a percentage?
   Answer: 43.75% To convert a fraction to a percentage, first convert the fraction to a decimal by dividing the numerator by the denominator. $7 \div 16 = 0.4375$. Then, to convert the decimal to a percentage, multiply by 100 (or move the decimal point two places to the right). $0.4375 \times 100 = 43.75$. So, $\frac{7}{16}$ is equal to 43.75%.
6. A number, $x$, is multiplied by 140%. The result is then divided by $\frac{7}{5}$. Which of the following represents the final value in terms of $x$?
   Answer: $x$ First, convert the percentage and fraction to decimals for easier calculation. 140% is equal to 1.4. The fraction $\frac{7}{5}$ is equal to 1.4. The problem states that $x$ is multiplied by 140%, which gives $1.4x$. Then, this result is divided by $\frac{7}{5}$, which is $1.4$. The calculation is $\frac{1.4x}{1.4}$. The 1.4 in the numerator and denominator cancel out, leaving just $x$.
7. If $\frac{2}{3}$ is equal to $x\%$ of $\frac{5}{6}$, what is the value of $x$ expressed as a decimal?
   Answer: 80 When you see "x% of" in a problem, you're dealing with percent equations. The key structure is: "A is x% of B" translates to the equation $A = \frac{x}{100} \times B$. Here, $\frac{2}{3}$ equals $x\%$ of $\frac{5}{6}$, so you can write: $\frac{2}{3} = \frac{x}{100} \times \frac{5}{6}$ To solve for $x$, multiply both sides by $\frac{100 \times 6}{5}$: $x = \frac{2}{3} \times \frac{100 \times 6}{5} = \frac{2 \times 100 \times 6}{3 \times 5} = \frac{1200}{15} = 80$ Since the question asks for $x$ as a decimal, the answer is 80.0, which matches choice D. Looking at the wrong answers: Choice A (8.0) likely comes from miscalculating $\frac{2}{3} \div \frac{5}{6}$ and getting $\frac{4}{5} = 0.8$, then confusing this with 8.0. Choice B (0.8) represents this same calculation error—finding $\frac{4}{5}$ but forgetting that percentages need to be multiplied by 100. Choice C (83.33) appears to come from converting $\frac{5}{6}$ to a percentage (approximately 83.33%) instead of solving for $x$. Remember: when solving "A is x% of B" problems, set up the equation $A = \frac{x}{100} \times B$ and solve for $x$. Don't forget that the final answer represents the percentage value, not a decimal that needs further conversion.
8. A basketball player's free throw percentage decreased from $\frac{4}{5}$ to $72\%$. By what percent did their free throw percentage decrease?
   Answer: 10% Convert $\frac{4}{5} = 80\%$. The decrease is from $80\%$ to $72\%$, which is $8$ percentage points. The percent decrease is $\frac{8}{80} \times 100\% = 10\%$. Choice A gives the absolute decrease in percentage points, not the percent decrease. Choice C results from using $72$ as the denominator instead of $80$. Choice D comes from calculation errors in the division.
9. What percent is equivalent to the fraction $\frac{7}{12}$ when rounded to the nearest tenth of a percent?
   Answer: 58.3% $\frac{7}{12} = 0.583\overline{3} = 58.333...\%$. Rounded to the nearest tenth of a percent, this is $58.3\%$. Choice B shows more precision than requested. Choice C results from incorrect rounding (looking only at the first 3 after decimal). Choice D results from rounding to the nearest whole percent instead.
10. What is the result of $\frac{3}{5}$ subtracted from 150%, expressed as a decimal?
    Answer: 0.9 First, convert both values to decimals. $\frac{3}{5} = 0.6$. 150% = 1.5. The question asks for $\frac{3}{5}$ to be subtracted from 150%, so the calculation is $1.5 - 0.6$. The result is 0.9.
11. Which of the following fractions is equivalent to $8\frac{1}{3}\%$?
    Answer: $\frac{1}{12}$ First, convert the mixed number to an improper fraction: $8\frac{1}{3} = \frac{(8 \times 3) + 1}{3} = \frac{25}{3}$. The expression is now $\frac{25}{3}\%$. A percent is a value per 100, so to convert it to a fraction, divide by 100: $\frac{25}{3} \div 100 = \frac{25}{3} \times \frac{1}{100} = \frac{25}{300}$. Finally, simplify the fraction by dividing the numerator and denominator by 25: $\frac{25 \div 25}{300 \div 25} = \frac{1}{12}$.
12. A shirt is discounted by 30%. The sale price is then taxed at 10%. The total tax paid is what fraction of the original price?
    Answer: $\frac{7}{100}$ Let the original price be $P$. A 30% discount means the sale price is 100% - 30% = 70% of the original price, or $0.70P$. The tax is 10% of this sale price: $0.10 \times 0.70P = 0.07P$. The tax paid is 0.07 of the original price $P$. To express 0.07 as a fraction, write it as $\frac{7}{100}$.
13. Which of the following fractions is equivalent to 0.4%?
    Answer: $\frac{1}{250}$ To convert a percent to a fraction, write it as a number over 100. So, 0.4% is $\frac{0.4}{100}$. To remove the decimal in the numerator, multiply both the numerator and the denominator by 10: $\frac{0.4 \times 10}{100 \times 10} = \frac{4}{1000}$. Finally, simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 4: $\frac{4 \div 4}{1000 \div 4} = \frac{1}{250}$.
14. What is 40% of $\frac{5}{8}$? Express the answer as a decimal.
    Answer: 0.25 To solve this, first convert the values to a common format, like decimals. 40% is equivalent to 0.40, and the fraction $\frac{5}{8}$ is equivalent to 0.625. The problem asks for '40% of $\frac{5}{8}$', which means you multiply the two values: $0.40 \times 0.625 = 0.25$.
15. A population increased from 500 to 625. The population increase is what value, expressed as a decimal?
    Answer: 0.25 First, find the amount of the increase: $625 - 500 = 125$. To express this increase as a value relative to the original population, create a fraction with the increase over the original amount: $\frac{125}{500}$. To convert this fraction to a decimal, simplify it first by dividing the numerator and denominator by 125: $\frac{125 \div 125}{500 \div 125} = \frac{1}{4}$. The fraction $\frac{1}{4}$ is equal to the decimal 0.25.
16. A solution contains $37.5\%$ alcohol by volume. If this percentage is first converted to a decimal, then that decimal is converted to a fraction and reduced to lowest terms, what is the sum of the numerator and denominator?
    Answer: 11 This question tests your ability to convert between percentages, decimals, and fractions while simplifying to lowest terms. When you see a problem asking for conversions through multiple forms, work systematically through each step. Start by converting the percentage to a decimal: $37.5\% = 37.5 ÷ 100 = 0.375$. Next, convert this decimal to a fraction. Since $0.375$ has three decimal places, write it as $\frac{375}{1000}$. Now you need to reduce this fraction to lowest terms by finding the greatest common factor (GCF) of 375 and 1000. To find the GCF, factor both numbers: - $375 = 3 × 125 = 3 × 5^3$ - $1000 = 8 × 125 = 2^3 × 5^3$ The GCF is $125$, so: $\frac{375}{1000} = \frac{375 ÷ 125}{1000 ÷ 125} = \frac{3}{8}$ The sum of numerator and denominator is $3 + 8 = 11$, which is answer choice D. Let's examine why the other answers are wrong: Choice A (1375) likely comes from adding the original numbers 375 + 1000 without reducing the fraction. Choice C (375) represents just the original numerator before reduction. Choice B (8) is only the denominator of the reduced fraction, missing the numerator entirely. Remember that "reduced to lowest terms" means you must find the GCF and divide both numerator and denominator by it. Always double-check that your final fraction cannot be simplified further.
17. Three of the following values are equivalent. Which value is NOT equivalent to the others?
    Answer: $\frac{4}{5}$ To find the value that is not equivalent, we can convert all options to the same format, such as a decimal. Option A is 0.08. Option B, 8%, is equivalent to $\frac{8}{100}$ or 0.08. Option C, $\frac{2}{25}$, can be converted to a decimal by dividing 2 by 25, which equals 0.08. Option D, $\frac{4}{5}$, is equivalent to 0.80. Since 0.80 is different from 0.08, $\frac{4}{5}$ is the value not equivalent to the others.
18. A recipe calls for $2\frac{3}{4}$ cups of flour. If Maria has already added $0.625$ cups and needs to add $87.5\%$ of the remaining amount, how many more cups should she add?
    Answer: 1.875 cups Convert $2\frac{3}{4}$ to decimal: $2.75$ cups total needed. Remaining after $0.625$ cups: $2.75 - 0.625 = 2.125$ cups. Convert $87.5\% = \frac{7}{8} = 0.875$. Amount to add: $0.875 \times 2.125 = 1.859375 \approx 1.875$ cups. Choice B is the remaining amount before applying the percentage. Choice C results from applying $87.5\%$ to the original recipe amount. Choice D reverses the digits of the correct answer.
19. A company's revenue this year is 165% of its revenue last year. Which of the following mixed numbers represents this year's revenue as a fraction of last year's revenue?
    Answer: $1\frac{13}{20}$ To convert a percentage to a fraction, place it over 100. So, 165% becomes $\frac{165}{100}$. This is an improper fraction. To convert it to a mixed number, divide 165 by 100. It goes in 1 time with a remainder of 65. This gives $1\frac{65}{100}$. The fraction part, $\frac{65}{100}$, can be simplified by dividing both the numerator and denominator by their greatest common factor, which is 5: $\frac{65 \div 5}{100 \div 5} = \frac{13}{20}$. So the final mixed number is $1\frac{13}{20}$.
20. A student needs a score of at least 70% on a test to pass. The student answered 45 out of 60 questions correctly. Which statement is true?
    Answer: The student's score was 5% above the passing score. First, calculate the student's score as a percentage. The score is the fraction $\frac{45}{60}$. Simplifying this fraction gives $\frac{3}{4}$. To convert $\frac{3}{4}$ to a percentage, convert it to a decimal (0.75) and multiply by 100, which is 75%. The passing score is 70%. The student's score of 75% is 5 percentage points higher than the passing score. Therefore, the student's score was 5% above the passing score.

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