Convert Fractions And Percents - Praxis Core Math

1. A survey found that $\frac{7}{16}$ of respondents prefer Brand A. What percentage of respondents do NOT prefer Brand A?
   Answer: 56.25% First, find the fraction of respondents who do not prefer Brand A. This is $1 - \frac{7}{16} = \frac{16}{16} - \frac{7}{16} = \frac{9}{16}$. To convert this fraction to a percentage, divide 9 by 16 and multiply by 100. $9 \div 16 = 0.5625$. $0.5625 \times 100 = 56.25$.
2. What is 37.5% of the value that is $\frac{5}{2}$ of 16?
   Answer: 15 This is a two-step problem. First, find the value that is $\frac{5}{2}$ of 16: $\frac{5}{2} \times 16 = 5 \times 8 = 40$. Second, calculate 37.5% of this value. It's helpful to know the fraction equivalent of 37.5%, which is $\frac{3}{8}$. Now, calculate $\frac{3}{8}$ of 40: $\frac{3}{8} \times 40 = 3 \times 5 = 15$.
3. In a bag of marbles, 37.5% are red, $\frac{2}{5}$ are blue, and the rest are green. If there are 80 marbles in total, how many are green?
   Answer: 18 First, find the fraction of green marbles. Convert 37.5% to its fraction equivalent: 37.5% = 37.5/100 = 3/8. The fraction of blue marbles is 2/5. To add these fractions, find a common denominator of 40: 3/8 = 15/40 and 2/5 = 16/40. The combined fraction of red and blue marbles is 15/40 + 16/40 = 31/40. The fraction of green marbles is 1 - 31/40 = 9/40. To find the number of green marbles, multiply this fraction by the total: (9/40) × 80 = 9 × 2 = 18 green marbles.
4. A recipe calls for $2\frac{1}{2}$ cups of flour. A baker has already added 60% of the required flour. How much more flour, in cups, does the baker need to add? (Express your answer as a decimal)
   Answer: 1.0 First, convert the total amount of flour to a decimal: $2\frac{1}{2} = 2.5$ cups. The baker has added 60% of this amount. The remaining amount needed is $100$. Convert 40% to a decimal, which is 0.40. Calculate the remaining flour needed: $0.40 \times 2.5 = 1.0$ cup.
5. An item is on sale for 20% off. The sales tax is 8% of the sale price. The total cost of the item is what percentage of the original price?
   Answer: 86.4% Let the original price be P. After a 20% discount, the sale price is $100$ of the original price, or $0.80P$. The sales tax is 8% of this sale price: $0.08 \times 0.80P = 0.064P$. The total cost is the sale price plus the tax: $0.80P + 0.064P = 0.864P$. As a percentage of the original price, this is 86.4%.
6. The number 18 is $\frac{3}{4}\%$ of which number?
   Answer: 2400 Let the unknown number be x. The problem can be written as an equation: $18 = (\frac{3}{4}\%) \times x$. First, convert the percentage to a decimal. $\frac{3}{4}\% = 0.75\%$. To convert a percent to a decimal, divide by 100: $0.75 \div 100 = 0.0075$. So, the equation is $18 = 0.0075x$. To solve for x, divide both sides by 0.0075: $x = \frac{18}{0.0075} = 2400$. A common error is to use 0.75 instead of 0.0075, which would yield 24.
7. The decimal 0.004 is equivalent to what fraction of 20%?
   Answer: $\frac{1}{50}$ The question asks for the value of the ratio $\frac{0.004}{20\%}$. First, convert both numbers to the same format. Let's use decimals. 20% is equal to 0.20. Now, calculate the ratio: $\frac{0.004}{0.20}$. To simplify, multiply the numerator and denominator by 1000 to remove decimals: $\frac{4}{200}$. Simplify this fraction by dividing both parts by 4: $\frac{1}{50}$.
8. In a school election, Candidate A received $0.35$ of the votes, Candidate B received 42% of the votes, and Candidate C received the remaining 69 votes. How many more votes did Candidate B receive than Candidate A?
   Answer: 21 First, find the total percentage of votes for Candidates A and B. Convert 0.35 to a percentage: 35%. Total percentage for A and B = $35\% + 42\% = 77\%$. The remaining percentage for Candidate C is $100\% - 77\% = 23\%$. We know that these 23% of the votes correspond to 69 votes. Let T be the total number of votes. Then $0.23 \times T = 69$, so $T = \frac{69}{0.23} = 300$. Now find the number of votes for A and B. A: $0.35 \times 300 = 105$ votes. B: $0.42 \times 300 = 126$ votes. The difference is $126 - 105 = 21$ votes. Alternatively, the difference in percentage is $42\% - 35\% = 7\%$. The difference in votes is $0.07 \times T = 0.07 \times 300 = 21$.
9. A student's grade is based on three tests, weighted 20%, 30%, and 50% respectively. The student scored $\frac{4}{5}$ on the first test and 70% on the second test. What score, as a percentage, must the student get on the third test to have an overall grade of 80%?
   Answer: 86% First, convert all scores and weights to decimals. The first test score is $\frac{4}{5} = 0.80$. The second test score is 70% = 0.70. The weights are 0.20, 0.30, and 0.50. Let T be the score on the third test. The overall grade must be 80% or 0.80. The equation for the weighted average is: $(0.20 \times 0.80) + (0.30 \times 0.70) + (0.50 \times T) = 0.80$. Calculate the known parts: $0.16 + 0.21 + 0.50T = 0.80$. Combine the terms: $0.37 + 0.50T = 0.80$. Subtract 0.37 from both sides: $0.50T = 0.43$. Solve for T: $T = \frac{0.43}{0.50} = 0.86$. As a percentage, the student must score 86% on the third test.
10. The fraction $\frac{2}{11}$ is equivalent to a repeating decimal. What is the digit in the 50th decimal place of this decimal representation?
    Answer: 8 First, convert $\frac{2}{11}$ to a decimal by dividing 2 by 11. This gives $0.181818...$, which can be written as $0.\overline{18}$. The repeating block of digits is '18', which has a length of 2. To find the 50th decimal digit, we need to determine if the position is odd or even. Since 50 is an even number, the digit will be the second digit in the repeating block, which is 8.
11. Arrange the following numbers from least to greatest: $0.55, \frac{5}{9}, 5.5$
    Answer: $5.5$ To compare the numbers, convert them all to decimals. $0.55$ is already a decimal. $\frac{5}{9} = 5 \div 9 = 0.555...$. $5.5$. Now, order the decimals from least to greatest: 0.055, 0.55, 0.555... . This corresponds to the original numbers in the order: $5.5$.
12. A mixture is 40% acid. If $\frac{1}{4}$ of the mixture is removed and replaced with pure water, what is the new concentration of acid, expressed as a simplified fraction?
    Answer: $\frac{3}{10}$ Let the initial volume of the mixture be V. The initial amount of acid is $0.40V$. When $\frac{1}{4}$ of the mixture is removed, $\frac{3}{4}$ of the mixture remains. The amount of acid remaining is $\frac{3}{4} \times (0.40V) = 0.30V$. The problem states that the removed portion is replaced with pure water, so the total volume is restored to V. The new concentration of acid is the amount of acid divided by the total volume: $\frac{0.30V}{V} = 0.30$. As a simplified fraction, 0.30 is $\frac{3}{10}$.
13. A company has 120 employees. Of these, $\frac{2}{3}$ are full-time. Of the full-time employees, 37.5% have been with the company for over 10 years. What percentage of ALL employees are full-time and have been with the company for over 10 years?
    Answer: 25% First, find the number of full-time employees: $\frac{2}{3} \times 120 = 2 \times 40 = 80$ full-time employees. Next, find the number of these employees who have been with the company for over 10 years. Convert 37.5% to a fraction, which is $\frac{3}{8}$. Now, calculate $\frac{3}{8}$ of 80: $\frac{3}{8} \times 80 = 3 \times 10 = 30$ employees. Finally, express this number as a percentage of the total number of employees (120): $\frac{30}{120} = \frac{1}{4}$. Convert the fraction $\frac{1}{4}$ to a percentage: $\frac{1}{4} = 25\%$.
14. Which of the following values is greater than 45% but less than $\frac{3}{5}$?
    Answer: 0.58 To compare the values, convert them to decimals. 45% is 0.45, and $\frac{3}{5}$ is 0.60. The question asks for a value between 0.45 and 0.60. A) $\frac{2}{5} = 0.40$, which is too small. B) 0.58 is between 0.45 and 0.60. C) 61% is 0.61, which is too large. D) $\frac{9}{20} = 0.45$, which is equal to the lower bound, not greater than it.
15. A restaurant bill is $45. A tip of $\frac{1}{6}$ of the bill amount is added. To the nearest whole percent, what percentage of the total cost (bill plus tip) is the tip?
    Answer: 14% First, calculate the tip amount: $\frac{1}{6} \times \$45 = \$7.50$. Next, calculate the total cost: $\$45 + \$7.50 = \$52.50$. The question asks for the percentage of the total cost that is the tip. This is $\frac{\text{tip}}{\text{total cost}} = \frac{7.50}{52.50}$. To convert this to a percentage, perform the division and multiply by 100: $(\frac{7.5}{52.5}) \times 100 \approx 0.1428 \times 100 \approx 14.28$. Rounded to the nearest whole percent, this is 14%. A common mistake is to simply convert $\frac{1}{6}$ to a percent ($\approx 17$), which is the tip as a percentage of the bill, not the total cost.
16. The sum of three numbers is 2.4. One number is $\frac{3}{4}$. Another number is 80% of the first number. What is the third number, expressed as a decimal?
    Answer: 1.05 First, convert all numbers to decimals. The first number is $\frac{3}{4} = 0.75$. The second number is 80% of the first, so it is $0.80 \times 0.75 = 0.60$. The sum of these two numbers is $0.75 + 0.60 = 1.35$. The sum of all three numbers is 2.4. To find the third number, subtract the sum of the first two from the total sum: $2.4 - 1.35 = 1.05$.
17. A map scale is 1 inch = 25 miles. The distance between two cities on the map is $3\frac{2}{5}$ inches. This distance represents what percentage of 850 miles?
    Answer: 10% First, find the actual distance between the cities. Convert $3\frac{2}{5}$ to a decimal: 3.4 inches. The actual distance is $3.4 \text{ inches} \times 25 \frac{\text{miles}}{\text{inch}} = 85$ miles. Next, find what percentage of 850 miles this distance is. The ratio is $\frac{85}{850} = \frac{1}{10}$. To convert $\frac{1}{10}$ to a percentage, multiply by 100: $\frac{1}{10} \times 100\% = 10\%$.
18. What is the value of the expression $(\frac{7}{8} - 0.8) \div 0.5$ expressed as a percentage?
    Answer: 15% First, convert the fraction to a decimal to perform the subtraction in the parentheses. $\frac{7}{8} = 0.875$. So, the expression in the parentheses is $0.875 - 0.8 = 0.075$. Next, divide this result by 0.5: $0.075 \div 0.5 = 0.15$. Finally, convert this decimal to a percentage by multiplying by 100: $0.15 \times 100 = 15\%$.
19. Which of the following fractions is equivalent to 1.25%?
    Answer: $\frac{1}{80}$ To convert a percentage to a fraction, first write it as a fraction over 100: $\frac{1.25}{100}$. To eliminate the decimal in the numerator, multiply both the numerator and the denominator by 100: $\frac{1.25 \times 100}{100 \times 100} = \frac{125}{10000}$. Now, simplify the fraction. Both are divisible by 125. $125 \div 125 = 1$ and $10000 \div 125 = 80$. So the simplified fraction is $\frac{1}{80}$. Distractor A, $\frac{5}{4}$, is 1.25, not 1.25%.
20. If 80 is increased by 45%, the result is decreased by $\frac{3}{4}$ of the new value. What is the final result?
    Answer: 29 First, increase 80 by 45%. The increase is $0.45 \times 80 = 36$. The new value is $80 + 36 = 116$. Next, this new value is decreased by $\frac{3}{4}$ of itself. The decrease amount is $\frac{3}{4} \times 116 = 3 \times 29 = 87$. The final result is the new value minus the decrease: $116 - 87 = 29$. Alternatively, if a value is decreased by $\frac{3}{4}$, then $\frac{1}{4}$ of it remains. So the final result is $\frac{1}{4} \times 116 = 29$.

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