Converting Rational Number Forms - Accuplacer Qas

1. If $x$ is a number such that $x \div 0.8 = \frac{5}{4}$, what is $x$ expressed as a percentage?
   Answer: 100% First, solve for $x$. The equation is $\frac{x}{0.8} = \frac{5}{4}$. It is helpful to convert both numbers to the same format. As decimals, $\frac{5}{4} = 1.25$. The equation becomes $\frac{x}{0.8} = 1.25$. Multiply both sides by 0.8 to isolate $x$: $x = 1.25 \times 0.8 = 1$. To express the number 1 as a percentage, multiply by 100: $1 \times 100\% = 100\%$.
2. A recipe calls for $\frac{3}{8}$ cup of flour, but Maria's measuring cup only shows decimal markings. She needs to convert this fraction to a decimal and then determine what percentage of a full cup this represents. What is the percentage?
   Answer: 37.5% First convert $\frac{3}{8}$ to decimal: $3 \div 8 = 0.375$. Then convert to percentage: $0.375 \times 100\% = 37.5\%$. Choice B results from rounding $\frac{3}{8}$ incorrectly to 0.383. Choice C comes from miscalculating $3 \div 8$ as 0.358. Choice D results from confusing $\frac{3}{8}$ with $\frac{2}{5}$.
3. A stock price decreased from $\frac{5}{8}$ of its original value to 56.25% of its original value. What is the amount of decrease expressed as a fraction in lowest terms?
   Answer: $\frac{1}{16}$ First convert both values to the same form. $\frac{5}{8} = 0.625 = 62.5\%$ and the final value is 56.25%. The decrease is $62.5\% - 56.25\% = 6.25\%$. Converting back to a fraction: $6.25\% = 0.0625 = \frac{625}{10000} = \frac{1}{16}$. Choice A results from calculation error. Choice C comes from incorrectly computing the decrease as $\frac{3}{32}$. Choice D represents the original difference calculation error.
4. Which of the following represents the same rational number as $2.4\overline{6}$ when written as an improper fraction in lowest terms?
   Answer: $\frac{37}{15}$ Let $x = 2.4\overline{6} = 2.4666...$ Then $10x = 24.666...$ and $100x = 246.666...$ Subtracting: $100x - 10x = 246.666... - 24.666...$ gives $90x = 222$, so $x = \frac{222}{90} = \frac{37}{15}$ in lowest terms. Choice A doesn't reduce to the same decimal. Choice C is the unreduced form. Choice D reduces to $\frac{37}{15}$ but isn't in lowest terms since $\gcd(74,30) = 2$.
5. A team wins 6 of its 11 games. What decimal represents the fraction of games the team won?
   Answer: 0.$\overline{54}$ To convert the fraction $\frac{6}{11}$ to a decimal, divide the numerator by the denominator: $6 \div 11$. The result of the division is $0.545454...$. In this decimal, the block of digits '54' repeats indefinitely. The correct notation for this repeating decimal is $0.\overline{54}$.
6. Express $0.\overline{234}$ as a fraction, then determine which of the following fractions is equivalent to half of that result.
   Answer: $\frac{234}{1998}$ Let $x = 0.\overline{234}$. Then $1000x = 234.\overline{234}$. Subtracting: $1000x - x = 234$, so $999x = 234$ and $x = \frac{234}{999}$. Half of this is $\frac{1}{2} \times \frac{234}{999} = \frac{234}{1998}$. Choice A is the original fraction, not half. Choice B simplifies incorrectly. Choice C doubles the denominator but keeps the same numerator, which would give one-half, but uses wrong values.
7. A student incorrectly converts the fraction $\frac{7}{12}$ to a percentage by first converting to the decimal 0.58, then multiplying by 100 to get 58%. What was the student's error, and what is the correct percentage?
   Answer: The student rounded too early; the correct answer is 58.33% $\frac{7}{12} = 7 \div 12 = 0.58333...$ The student rounded 0.58333... to 0.58 too early in the process. The correct percentage is $0.58333... \times 100\% = 58.33\%$ (or $58\frac{1}{3}\%$). Choice B gives an incorrect decimal conversion. Choice C would result from calculating $\frac{12}{7}$ instead, but that doesn't give 58.17%. Choice D is incorrect since the student did make an error.
8. Multiplying a positive number by 1.15 is equivalent to increasing the number by what fraction? Give your answer in simplest form.
   Answer: $\frac{3}{20}$ Multiplying a number by 1.15 can be thought of as $1 \times \text{number} + 0.15 \times \text{number}$. This represents the original number (100%) plus an increase of 0.15 (15%). The question asks for the value of the increase, which is 0.15. To convert the decimal 0.15 to a fraction, write it as $\frac{15}{100}$ and simplify by dividing the numerator and denominator by 5, which gives $\frac{3}{20}$.
9. A stock valued at $48 per share loses $\frac{3}{16}$ of its value in one day. What is the new value of the stock, expressed as a decimal?
   Answer: $39.00 First, calculate the amount of the loss: $\frac{3}{16} \times 48 = \frac{144}{16} = 9$. The stock lost $9 in value. To find the new value, subtract the loss from the original value: $48 - 9 = 39$. Expressed as a decimal, the new value is $39.00.
10. A value is increased by 150%. This is equivalent to multiplying the original value by what number, expressed as a fraction?
    Answer: $\frac{5}{2}$ An increase of 150% means the new value is the original 100% plus the 150% increase, which totals 250% of the original value. To convert a percentage to a multiplier, convert it to a decimal: $250\% = 2.5$. To express 2.5 as a fraction, write it as $2\frac{1}{2}$, which is equivalent to the improper fraction $\frac{5}{2}$.
11. A scientist measures a quantity as 87.5% of a standard unit. She needs to express this measurement as a fraction with a denominator that is a power of 2. Which fraction in lowest terms satisfies this requirement?
    Answer: $\frac{7}{8}$ When you encounter percentage problems requiring specific fraction formats, start by converting the percentage to a fraction, then manipulate it to meet the given constraints. Converting 87.5% to a fraction: $87.5\% = \frac{87.5}{100} = \frac{875}{1000}$. To eliminate the decimal, multiply both numerator and denominator by 10, giving us $\frac{875}{1000}$. Now reduce this fraction to lowest terms by finding the greatest common factor. Since $875 = 7 \times 125$ and $1000 = 8 \times 125$, we get $\frac{875}{1000} = \frac{7}{8}$. Notice that 8 is $2^3$, which is indeed a power of 2, and the fraction is in lowest terms since 7 and 8 share no common factors. Looking at the wrong answers: Choice A, $\frac{175}{200}$, reduces to $\frac{7}{8}$ but isn't in lowest terms as presented, and 200 isn't a power of 2. Choice B, $\frac{14}{16}$, also reduces to $\frac{7}{8}$ but isn't in lowest terms—you can divide both numerator and denominator by 2. Choice D, $\frac{35}{40}$, reduces to $\frac{7}{8}$ as well, but 40 isn't a power of 2, and it's not in lowest terms. The answer is C: $\frac{7}{8}$. Strategy tip: When working with percentages involving decimals like 87.5%, look for patterns—87.5% is exactly $\frac{7}{8}$, just as 12.5% is $\frac{1}{8}$, 25% is $\frac{1}{4}$, etc. Memorizing these common fraction-percentage equivalents saves time on standardized tests.
12. A baker uses 0.625 of a bag of flour for a recipe. What percentage of the flour is left in the bag?
    Answer: 37.5% The entire bag of flour represents 100%, or 1. If 0.625 of the flour is used, the amount remaining is $1 - 0.625 = 0.375$. To express this decimal as a percentage, multiply by 100: $0.375 \times 100\% = 37.5\%$.
13. In a survey, 6.25% of respondents chose "other." What fraction of respondents chose "other"?
    Answer: $\frac{1}{16}$ To convert a percentage to a fraction, first write it as a decimal: $6.25\% = 0.0625$. Then, write the decimal as a fraction based on its place value: $0.0625 = \frac{625}{10000}$. To simplify, divide the numerator and denominator by their greatest common divisor. Both are divisible by 625. $625 \div 625 = 1$ and $10000 \div 625 = 16$. So, the simplified fraction is $\frac{1}{16}$.
14. When $\frac{11}{6}$ is written in decimal form and rounded to three decimal places, the result is 1.833. If this rounded decimal is then converted back to a fraction in lowest terms, what fraction is obtained?
    Answer: $\frac{1833}{1000}$ This question tests your understanding of decimal-to-fraction conversion and the distinction between exact values and rounded approximations. When you convert the rounded decimal 1.833 back to a fraction, you need to work with what you actually have: the three-decimal-place number 1.833, not the original fraction $\frac{11}{6}$. To convert 1.833 to a fraction, recognize that this decimal has three places after the decimal point, so it equals $\frac{1833}{1000}$. You can verify this isn't in lowest terms by checking if 1833 and 1000 share common factors. Since 1833 = 3 × 611 and 1000 = 8 × 125, they share no common factors, so $\frac{1833}{1000}$ is already in lowest terms. Looking at the wrong answers: Choice A ($\frac{11}{6}$) represents the original fraction before rounding, but the question asks what you get when converting the rounded decimal back. Choice B ($\frac{611}{333}$) appears to come from incorrectly factoring or manipulating the numbers. Choice D ($\frac{1833}{999}$) uses the wrong denominator—999 would be used if you were dealing with a repeating decimal, but 1.833 is a terminating decimal. The key strategy here is to read carefully: you're not finding an equivalent to the original fraction, but rather converting a specific rounded decimal. When converting terminating decimals to fractions, the denominator is always a power of 10 based on the number of decimal places.
15. In a group of people, $\frac{3}{8}$ are students. Of the remaining people, 40% are teachers. What decimal represents the fraction of the whole group that are teachers?
    Answer: 0.25 First, find the fraction of the group that are not students. If $\frac{3}{8}$ are students, then $1 - \frac{3}{8} = \frac{5}{8}$ are remaining people. Next, find the fraction of the group that are teachers by taking 40% of the remaining people. Convert 40% to a fraction: $40\% = \frac{40}{100} = \frac{2}{5}$. Now, multiply this by the fraction of remaining people: $\frac{2}{5} \times \frac{5}{8} = \frac{10}{40} = \frac{1}{4}$. Finally, convert the fraction $\frac{1}{4}$ to a decimal, which is 0.25.
16. A recipe requires $2\frac{1}{4}$ cups of sugar. A baker has a bag containing 3.5 cups of sugar. After making the recipe, what percentage of the sugar originally in the bag is left?
    Answer: 35.7% First, convert all numbers to decimals. The recipe requires $2\frac{1}{4} = 2.25$ cups. The bag contains 3.5 cups. Calculate the amount of sugar left after making the recipe: $3.5 - 2.25 = 1.25$ cups. To find what percentage of the original amount is left, divide the remaining amount by the original amount and multiply by 100: $\left(\frac{1.25}{3.5}\right) \times 100\%$. $\frac{1.25}{3.5} \approx 0.35714$. Multiplying by 100 gives approximately 35.7%.
17. Which of the following lists numbers in order from least to greatest?
    Answer: $0.6, \frac{5}{8}, 65\%$ To compare the numbers, convert them all to the same format, such as decimals. $\frac{5}{8} = 5 \div 8 = 0.625$. $65\% = 0.65$. The number 0.6 is already a decimal. Now, order the decimals from least to greatest: 0.6, 0.625, 0.65. This corresponds to the original list: $0.6, \frac{5}{8}, 65\%$.
18. A wire is 10.4 meters long. A piece measuring 3.9 meters is cut from it. What fraction of the original wire is left?
    Answer: $\frac{5}{8}$ First, calculate the length of the remaining wire: $10.4 - 3.9 = 6.5$ meters. Next, express this remaining length as a fraction of the original length: $\frac{6.5}{10.4}$. To simplify this fraction, multiply the numerator and denominator by 10 to remove the decimals, resulting in $\frac{65}{104}$. The greatest common divisor of 65 and 104 is 13. Simplifying the fraction gives $\frac{65 \div 13}{104 \div 13} = \frac{5}{8}$.
19. A calculator displays the result 0.1428571429 when computing $\frac{1}{7}$. If this decimal pattern continues indefinitely, what is $\frac{3}{7}$ as a percentage, rounded to the nearest tenth of a percent?
    Answer: 42.9% When you encounter repeating decimals, you're working with fractions that don't convert to terminating decimals. The key insight here is recognizing the repeating pattern and using it to find related fractions. The calculator shows $\frac{1}{7} = 0.1428571429...$ where the digits 142857 repeat infinitely. This is written as $0.\overline{142857}$. To find $\frac{3}{7}$, you can multiply this repeating decimal by 3: $\frac{3}{7} = 3 \times \frac{1}{7} = 3 \times 0.\overline{142857}$. Multiplying each digit in the repeating block by 3 gives us 3×142857 = 428571. However, we need to handle the carrying properly. Working through the multiplication: $3 \times 0.\overline{142857} = 0.\overline{428571}$, which equals approximately 0.4285714286. Converting to a percentage: 0.4285714286 × 100% = 42.85714286%. Rounded to the nearest tenth of a percent, this gives us 42.9%. Looking at the wrong answers: Choice A (42.7%) rounds too far down, missing the precision needed. Choice B (43.0%) rounds up too aggressively, likely from imprecise decimal work. Choice C (42.8%) stops at the hundredths place before applying the rounding rule correctly to the tenths place. For repeating decimal problems, remember that you can often work with the pattern arithmetically rather than converting to exact fractions. Also, pay careful attention to rounding instructions—"nearest tenth of a percent" means one decimal place in your final percentage.
20. Three of the following numbers can be written as terminating decimals. Which number results in a non-terminating, repeating decimal?
    Answer: $\frac{5}{12}$ A fraction can be written as a terminating decimal if, when the fraction is in its simplest form, the prime factorization of the denominator consists only of 2s and 5s. A) $\frac{13}{20}$: Denominator is $20 = 2^2 \times 5$. It will terminate. B) $\frac{7}{16}$: Denominator is $16 = 2^4$. It will terminate. C) $\frac{9}{15}$: First, simplify to $\frac{3}{5}$. The denominator is 5. It will terminate. D) $\frac{5}{12}$: The denominator is $12 = 2^2 \times 3$. Because of the prime factor of 3, this fraction will result in a non-terminating, repeating decimal.

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