Identifying Equivalent Values - Accuplacer Arithmetic

Which of the following values is NOT equivalent to $\frac{7}{4}$ ?
Answer: 17.5% To find the equivalent values of $\frac{7}{4}$ , first convert it to a decimal by dividing 7 by 4, which equals 1.75. To convert the decimal to a percentage, multiply by 100, which gives 175%. The decimal 1.75 is equivalent to the mixed number $1\frac{3}{4}$ . The value that is NOT equivalent is 17.5%, which is equal to 0.175.
The length of a machined part must be between 0.85 inches and $\frac{7}{8}$ inches. Which of the following lengths is acceptable?
Answer: 0.86 First, convert all values to a common format, like decimals. The fraction $\frac{7}{8}$ is equal to $7 \div 8 = 0.875$ . The acceptable range is between 0.85 and 0.875 inches. Now, evaluate the options: A) 80% = 0.80 (too small). B) 0.86 is between 0.85 and 0.875 (acceptable). C) 0.88 is too large. D) $\frac{9}{10}$ = 0.90 (too large).
A machine operates at 87.5% efficiency. The maintenance manual requires this to be expressed as a simplified fraction for gear ratio calculations and as a decimal for computer input. What fraction and decimal represent this efficiency?
Answer: $\frac{7}{8}$ and 0.875 When you encounter efficiency percentages that need to be converted to both fractions and decimals, you're working with percentage-to-fraction conversion and decimal equivalents. The key is converting the percentage to a fraction in lowest terms while maintaining the exact decimal value. Starting with 87.5%, you first convert this to a fraction by writing it as $\frac{87.5}{100}$ . To eliminate the decimal in the numerator, multiply both numerator and denominator by 10: $\frac{875}{1000}$ . Now simplify by finding the greatest common divisor. Both 875 and 1000 are divisible by 125, giving you $\frac{7}{8}$ . For the decimal, 87.5% equals 0.875 (move the decimal point two places left). Let's examine why the other answers fall short. Choice A gives $\frac{35}{40}$ , which simplifies to $\frac{7}{8}$ but represents 87.5%, so the fraction is actually correct—however, this fraction isn't in its simplest form as required. Choice B shows $\frac{875}{1000}$ , which is mathematically equivalent to 87.5% but isn't simplified to lowest terms as the problem requests. Choice C has the correct simplified fraction $\frac{7}{8}$ but shows 0.88 as the decimal, which is incorrect—0.88 would represent 88%, not 87.5%. Remember that when a problem asks for a "simplified fraction," always reduce to lowest terms by dividing both numerator and denominator by their greatest common divisor. Also, be precise with decimal conversions—rounding can lead to wrong answers when exact values are required.
An investment grew by 125% over five years. An investor wants to know what fraction this growth represents (as an improper fraction in lowest terms) and what decimal multiplier to use for calculating the final amount from the initial investment. What are these equivalent values?
Answer: $\frac{5}{4}$ growth, with a 2.25 multiplier for final amount When you encounter percentage growth problems, you need to distinguish between the growth itself and the multiplier used to calculate the final amount. These are related but different concepts. A 125% growth means the investment increased by an amount equal to 125% of its original value. To convert this percentage to a fraction: $125\% = \frac{125}{100} = \frac{5}{4}$ (after reducing to lowest terms by dividing both numerator and denominator by 25). However, to find the final amount, you need a different calculation. If something grows by 125%, the final amount equals the original amount plus the growth: 100% + 125% = 225% of the original. Converting to decimal: $225\% = 2.25$ . So you multiply the initial investment by 2.25 to get the final amount. Looking at the wrong answers: Choice A incorrectly calculates the growth fraction as $\frac{9}{4}$ , which would represent 225% growth rather than 125%. Choice B has the correct growth fraction but uses 1.25 as the multiplier—this would only account for 25% growth, not 125%. Choice C shows $\frac{1}{4}$ for growth, which represents just 25% growth, completely misunderstanding the problem. Choice D correctly shows $\frac{5}{4}$ for the growth and 2.25 as the multiplier for calculating the final amount. Remember this pattern: for any percentage growth, convert the percentage to a fraction for the growth amount, but add 100% to that percentage and convert to decimal for the final amount multiplier.
A basketball player's free throw percentage is 0.72, but the sports announcer wants to express this as a simplified fraction and the scoreboard shows percentages. What fraction represents this success rate, and what percentage should appear on the scoreboard?
Answer: $\frac{18}{25}$ and 72% 0.72 = $\frac{72}{100} = \frac{18}{25}$ when simplified, and 0.72 = 72%. Choice B shows the unsimplified fraction. Choice C incorrectly shows 0.72% instead of 72%. Choice D shows $\frac{36}{50}$ which equals $\frac{18}{25}$ but is not fully simplified.
Which of the following has a value between 4% and $\frac{1}{20}$ ?
Answer: 0.045 First, establish the range in a common format, such as decimals. 4% is equal to 0.04. The fraction $\frac{1}{20}$ is equal to 0.05. The question asks for a value between 0.04 and 0.05. Now, check the options: A) $\frac{1}{25}$ = 0.04, which is not between the values. B) 0.045 is between 0.04 and 0.05. C) 5.1% = 0.051, which is greater than 0.05. D) 0.052 is greater than 0.05.
In a chemistry lab, a solution contains 37.5% acid by volume. A student needs to express this concentration as both a fraction in lowest terms and as a decimal for two different calculation methods. What is the equivalent fraction and decimal?
Answer: $\frac{3}{8}$ and 0.375 37.5% = $\frac{37.5}{100} = \frac{375}{1000} = \frac{3}{8}$ in lowest terms, and 37.5% = 0.375 as a decimal. Choice B shows the unreduced fraction. Choice C rounds the decimal incorrectly. Choice D shows $\frac{15}{40}$ which equals $\frac{3}{8}$ but is not in lowest terms as specified.
A survey found that $\frac{7}{12}$ of students prefer online learning. The school administration needs this data as both a decimal (rounded to three decimal places) and as a percentage (rounded to the nearest tenth) for their report. What are these equivalent values?
Answer: 0.583 and 58.3% Converting fractions to decimals and percentages is a fundamental skill that requires careful attention to rounding rules. When you encounter problems like this, you need to perform the division accurately and apply the specified rounding to each form separately. To convert $\frac{7}{12}$ to a decimal, divide 7 by 12: $7 \div 12 = 0.58333...$ Since you need three decimal places, look at the fourth decimal place (3) to determine rounding. Because 3 < 5, you round down, giving you 0.583. For the percentage, multiply the unrounded decimal by 100: $0.58333... \times 100 = 58.333...\%$ Rounding to the nearest tenth means looking at the hundredths place (3). Since 3 < 5, round down to 58.3%. Let's examine why the other options are incorrect. Choice A gives 0.584, which incorrectly rounds up the decimal—the fourth decimal place is 3, so you should round down, not up. Choice C shows 0.583 (correct) but 58.4% (incorrect)—this makes the same upward rounding error for the percentage. Choice D provides 0.58, which only shows two decimal places instead of the required three, and also has the wrong percentage. Remember that rounding decisions must be made independently for each form requested. The decimal and percentage may round in different directions even though they represent the same value. Always check the specific number of decimal places or precision required for each part of the answer.
Four measurements are recorded: 0.08 meters, 15% of a meter, $\frac{1}{12}$ of a meter, and 0.1 meters. Which measurement is the largest?
Answer: 15% of a meter To compare the values, convert them all to decimals. 0.08 is already a decimal. 15% is $15 \div 100 = 0.15$ . $\frac{1}{12}$ is $1 \div 12 \approx 0.0833$ . 0.1 is already a decimal. The four values are 0.08, 0.15, 0.0833, and 0.1. The largest of these is 0.15, which is 15% of a meter.
A number is equal to 3 divided by 8. What is 200% of this number?
Answer: 0.75 First, find the initial number: $3 \div 8 = \frac{3}{8} = 0.375$ . Next, find 200% of this number. 200% is equivalent to multiplying by 2. So, $0.375 \times 2 = 0.75$ . Option A is the original number, not 200% of it.
What is 20% of $\frac{1}{8}$ , expressed as a decimal?
Answer: 0.025 This is a multi-step problem. First, convert the numbers to a common format. 20% is 0.20, and $\frac{1}{8}$ is 0.125. The problem asks for "20% of $\frac{1}{8}$ ", which means multiplication: $0.20 \times 0.125 = 0.025$ . Option A is the correct answer. The other options result from calculation errors.
Which of the following is equivalent to $\frac{1/5}{2}$ ?
Answer: 0.1 The expression $\frac{1/5}{2}$ means $\frac{1}{5} \div 2$ . To divide by a whole number, you can multiply by its reciprocal: $\frac{1}{5} \times \frac{1}{2} = \frac{1}{10}$ . As a decimal, $\frac{1}{10}$ is equal to 0.1.
A recipe calls for $2\frac{1}{4}$ cups of flour. A baker wants to make only 50% of the recipe. Which value represents the amount of flour needed?
Answer: 1.125 cups First, convert $2\frac{1}{4}$ to a decimal: 2.25. Making 50% of the recipe means finding 50% of 2.25. 50% is equivalent to 0.5. So, calculate $2.25 \times 0.5 = 1.125$ . The baker needs 1.125 cups of flour. Option B, 1.25, is half of 2.5. Option D, $1\frac{3}{8}$ , is 1.375.
What is the sum of $\frac{1}{2}$ , 0.4, and 30%, expressed as a fraction?
Answer: $\frac{6}{5}$ First, convert all numbers to a common format. Using fractions: $\frac{1}{2}$ is already a fraction. 0.4 is $\frac{4}{10}$ , which simplifies to $\frac{2}{5}$ . 30% is $\frac{30}{100}$ , which simplifies to $\frac{3}{10}$ . The sum is $\frac{1}{2} + \frac{2}{5} + \frac{3}{10}$ . Find a common denominator, which is 10. The sum becomes $\frac{5}{10} + \frac{4}{10} + \frac{3}{10} = \frac{12}{10}$ . This simplifies to $\frac{6}{5}$ .
A recipe calls for $\frac{3}{8}$ cup of sugar, but Maria only has a measuring cup marked in decimal increments. She also needs to calculate what percent of a full cup this represents for her nutrition tracking app. Which decimal amount should she measure, and what percent of a cup is this?
Answer: 0.375 cups, which is 37.5% of a cup To convert $\frac{3}{8}$ to decimal: $3 \div 8 = 0.375$ . To convert to percent: $0.375 \times 100\% = 37.5\%$ . Choice B uses rounded decimal but incorrect percent. Choice C confuses $\frac{3}{8}$ with $\frac{13}{40}$ . Choice D incorrectly calculates $\frac{5}{16}$ instead of $\frac{3}{8}$ .
A container holds $\frac{4}{5}$ of a liter of water. If 25% of the water is removed, what fraction of a liter of water remains?
Answer: $\frac{3}{5}$ If 25% of the water is removed, then 75% of the water remains (100% - 25% = 75%). The problem is to find 75% of $\frac{4}{5}$ . Convert 75% to a fraction, which is $\frac{75}{100} = \frac{3}{4}$ . Now, multiply the fractions: $\frac{3}{4} \times \frac{4}{5} = \frac{12}{20}$ . This simplifies to $\frac{3}{5}$ . So, $\frac{3}{5}$ of a liter remains.
A store offers a discount that reduces the original price by $\frac{2}{5}$ . If this discount is equivalent to 0.4 off the original price, what fraction represents the sale price as a portion of the original price, and what is this as a percent?
Answer: $\frac{3}{5}$ of the original price, which is 60% If the discount is $\frac{2}{5}$ off, then the sale price is $1 - \frac{2}{5} = \frac{3}{5}$ of the original price. Converting: $\frac{3}{5} = 0.6 = 60\%$ . Choice B gives the discount amount, not sale price. Choice C has correct fraction but wrong percent calculation. Choice D incorrectly adds the discount to the original price.
Which of the following values is closest to $\frac{5}{6}$ ?
Answer: 83% First, convert $\frac{5}{6}$ to a decimal or percent. $5 \div 6 = 0.8333...$ , which is $83.33...\%$ . Now compare this value to the options. A) 0.8 is 0.0333... away. B) 83% or 0.83 is 0.0033... away. C) 0.85 is 0.0167... away. D) 86% or 0.86 is 0.0267... away. The closest value is 83%.
The fraction $\frac{18}{48}$ is equivalent to which of the following?
Answer: 0.375 First, simplify the fraction $\frac{18}{48}$ . Both the numerator and the denominator are divisible by 6. $18 \div 6 = 3$ and $48 \div 6 = 8$ . So the simplified fraction is $\frac{3}{8}$ . To convert $\frac{3}{8}$ to a decimal, divide 3 by 8, which equals 0.375. This is equivalent to 37.5%.
Which fraction is equivalent to 1.125?
Answer: $\frac{9}{8}$ The decimal 1.125 can be written as the mixed number $1\frac{125}{1000}$ . The fraction $\frac{125}{1000}$ can be simplified. Dividing both numerator and denominator by 125 gives $\frac{1}{8}$ . So, the mixed number is $1\frac{1}{8}$ . Converting this to an improper fraction gives $\frac{(1 \times 8) + 1}{8} = \frac{9}{8}$ .

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