Converting Decimals & Fractions - Accuplacer Arithmetic

1. What is the product of 0.6 and $\frac{5}{8}$?
   Answer: $\frac{3}{8}$ To find the product, convert the decimal to a fraction. $0.6 = \frac{6}{10} = \frac{3}{5}$. Now multiply the fractions: $\frac{3}{5} \times \frac{5}{8}$. The 5s in the numerator and denominator cancel out, leaving $\frac{3}{8}$. Alternatively, convert $\frac{5}{8}$ to 0.625 and multiply $0.6 \times 0.625 = 0.375$, which is equal to $\frac{3}{8}$.
2. A piece of lumber is 4.8 meters long. A carpenter cuts off a piece that is $2 \frac{1}{4}$ meters long. What is the length, in meters, of the remaining piece of lumber?
   Answer: 2.55 First, convert the mixed number $2 \frac{1}{4}$ to a decimal. $\frac{1}{4}$ is 0.25, so $2 \frac{1}{4}$ is 2.25. To find the length of the remaining piece, subtract the length of the cut piece from the original length: $4.8 - 2.25$. Aligning the decimals gives $4.80 - 2.25 = 2.55$. The remaining piece is 2.55 meters long.
3. What is the difference between $2 \frac{3}{8}$ and 2.3?
   Answer: 0.075 First, convert the mixed number $2 \frac{3}{8}$ to a decimal. Since $\frac{1}{8} = 0.125$, then $\frac{3}{8} = 3 \times 0.125 = 0.375$. So, $2 \frac{3}{8} = 2.375$. The difference is found by subtracting the smaller number from the larger number: $2.375 - 2.300 = 0.075$.
4. A list of numbers is shown: $\frac{5}{8}, 0.6, \frac{2}{3}, 0.65$. Which number is the second smallest in the list?
   Answer: $\frac{5}{8}$ To order the numbers, convert them all to decimals with the same number of places. $\frac{5}{8} = 0.625$, $0.6 = 0.600$, $\frac{2}{3} \approx 0.667$, and $0.65 = 0.650$. Ordering from least to greatest gives: 0.600, 0.625, 0.650, 0.667. The corresponding original numbers are 0.6, $\frac{5}{8}$, 0.65, $\frac{2}{3}$. The second smallest number is $\frac{5}{8}$.
5. What is the value of the expression $(\frac{1}{5} + 0.4) \times \frac{1}{2}$?
   Answer: 0.3 First, evaluate the expression inside the parentheses. Convert $\frac{1}{5}$ to a decimal, which is 0.2. Then, add: $0.2 + 0.4 = 0.6$. Next, multiply the result by $\frac{1}{2}$, which is the same as multiplying by 0.5. $0.6 \times 0.5 = 0.3$.
6. A student incorrectly converts $\frac{7}{12}$ to a decimal by calculating $\frac{12}{7}$ instead. By how much does the student's incorrect answer exceed the correct decimal value?
   Answer: 1.129 The correct conversion: $\frac{7}{12} = 0.583\overline{3}$ The student's incorrect calculation: $\frac{12}{7} = 1.714285...$ The difference is $1.714285... - 0.583\overline{3} = 1.130952...$ Rounded to three decimal places, this is approximately 1.129. Choice A represents a smaller error. Choice C is close to $\frac{12}{7}$ but not the difference. Choice D is approximately $\frac{12}{7}$ itself.
7. What is the sum of $\frac{2}{3}$ and 0.25? Express your answer as a fraction.
   Answer: $\frac{11}{12}$ To add a fraction and a decimal, first convert the decimal to a fraction. $0.25 = \frac{25}{100} = \frac{1}{4}$. Now, add the two fractions: $\frac{2}{3} + \frac{1}{4}$. Find a common denominator, which is 12. Convert the fractions: $\frac{2}{3} = \frac{8}{12}$ and $\frac{1}{4} = \frac{3}{12}$. Add them: $\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$.
8. What is the result of $\frac{3}{2} \times 0.2$?
   Answer: $\frac{3}{10}$ To solve this, you can convert both numbers to the same format. As fractions: $0.2 = \frac{2}{10} = \frac{1}{5}$. Then multiply: $\frac{3}{2} \times \frac{1}{5} = \frac{3 \times 1}{2 \times 5} = \frac{3}{10}$. Alternatively, as decimals: $\frac{3}{2} = 1.5$. Then multiply: $1.5 \times 0.2 = 0.3$. The fraction $\frac{3}{10}$ is equivalent to the decimal 0.3.
9. Two fractions $\frac{5}{8}$ and $\frac{7}{12}$ are converted to decimals. What is the positive difference between these decimal values, expressed as a fraction in lowest terms?
   Answer: $\frac{1}{24}$ Convert to decimals: $\frac{5}{8} = 0.625$ and $\frac{7}{12} = 0.58\overline{3}$. Since $0.625 > 0.58\overline{3}$, the difference is $0.625 - 0.58\overline{3}$. Working with fractions: $\frac{5}{8} - \frac{7}{12} = \frac{15}{24} - \frac{14}{24} = \frac{1}{24}$. Choice B simplifies to $\frac{1}{10}$. Choice C equals 0.12. Choice D equals $\frac{2}{24}$, which would result from calculation errors.
10. What is the value of $\frac{0.75 + \frac{1}{8}}{2}$, expressed as a decimal?
    Answer: 0.4375 First, evaluate the numerator. Convert $\frac{1}{8}$ to a decimal, which is 0.125. Add the values in the numerator: $0.75 + 0.125 = 0.875$. Now, divide the result by 2: $0.875 \div 2 = 0.4375$.
11. What is the value of $2.5 - (\frac{3}{5} \times 2)$?
    Answer: 1.3 Following the order of operations, first calculate the value inside the parentheses. Convert $\frac{3}{5}$ to a decimal, which is 0.6. Then multiply: $0.6 \times 2 = 1.2$. Now, perform the subtraction: $2.5 - 1.2 = 1.3$.
12. A rectangular garden is $3 \frac{1}{2}$ meters long and 2.2 meters wide. What is its area in square meters, expressed as a decimal?
    Answer: 7.7 The area of a rectangle is length times width. First, convert the length, $3 \frac{1}{2}$ meters, to a decimal, which is 3.5 meters. Then, multiply the length and width: $3.5 \times 2.2$. The calculation is $3.5 \times 2.2 = 7.7$. The area is 7.7 square meters.
13. The decimal 0.64 can be written as $\frac{a}{b}$ where $a$ and $b$ are positive integers with no common factors other than 1. If this same fraction $\frac{a}{b}$ is used to represent a percentage, what percentage does it represent?
    Answer: 64% First convert 0.64 to a fraction: $0.64 = \frac{64}{100} = \frac{16}{25}$ (dividing by $\gcd(64,100) = 4$). So $a = 16$ and $b = 25$. When $\frac{16}{25}$ represents a percentage, we calculate $\frac{16}{25} \times 100\% = 64\%$. Choice A would result from incorrectly using just the numerator 16. Choice B would result from calculating $\frac{25}{100}$. Choice D would result from using $\frac{4}{5}$ instead of $\frac{16}{25}$.
14. Which of the following is equivalent to $\frac{7}{16}$?
    Answer: 0.4375 To convert the fraction $\frac{7}{16}$ to a decimal, divide the numerator (7) by the denominator (16). $7 \div 16 = 0.4375$.
15. Which of the following numbers has a value between $\frac{2}{5}$ and $\frac{1}{2}$?
    Answer: 0.45 To compare the numbers, convert the fractions to decimals. $\frac{2}{5} = 0.4$ and $\frac{1}{2} = 0.5$. The question asks for a number between 0.4 and 0.5. The number 0.45 is between 0.4 and 0.5. The other options are $\frac{3}{8} = 0.375$ (too small), 0.39 (too small), and $\frac{5}{9} \approx 0.556$ (too large).
16. Which of the following fractions is closest to the decimal 0.285?
    Answer: $\frac{2}{7}$ Convert each fraction to decimal and find the closest to 0.285. $\frac{2}{7} = 0.285714...$ $\frac{3}{11} = 0.272727...$ $\frac{4}{13} = 0.307692...$ $\frac{5}{17} = 0.294117...$ The differences from 0.285 are: |0.285714 - 0.285| ≈ 0.000714, |0.272727 - 0.285| ≈ 0.012273, |0.307692 - 0.285| ≈ 0.022692, |0.294117 - 0.285| ≈ 0.009117. Choice A has the smallest difference.
17. A recipe calls for $\frac{3}{8}$ cup of sugar, but Maria only has a measuring cup marked in decimal increments of 0.05 cups. What is the smallest amount she can measure that will give her at least the required amount of sugar?
    Answer: 0.40 cups First convert $\frac{3}{8}$ to decimal: $3 \div 8 = 0.375$. Since Maria's measuring cup only measures in increments of 0.05, she needs the smallest multiple of 0.05 that is greater than or equal to 0.375. Checking: 0.35 < 0.375, so this is too little. 0.40 > 0.375, and 0.40 = 8 × 0.05, so this works. Choice A (0.35) is less than needed. Choice B (0.375) is exact but not measurable with her cup. Choice C (0.38) is not a multiple of 0.05.
18. Which of the following fractions is closest in value to 0.58?
    Answer: $\frac{4}{7}$ To find the closest fraction, convert each fraction to a decimal and compare its distance from 0.58. $\frac{1}{2} = 0.50$, which is $0.58 - 0.50 = 0.08$ away. $\frac{4}{7} \approx 0.571$, which is $0.58 - 0.571 = 0.009$ away. $\frac{3}{5} = 0.60$, which is $0.60 - 0.58 = 0.02$ away. $\frac{5}{8} = 0.625$, which is $0.625 - 0.58 = 0.045$ away. The smallest difference is 0.009, so $\frac{4}{7}$ is the closest.
19. A measurement is recorded as 0.875 inches. Which of the following fractions, in simplest form, is equivalent to this measurement?
    Answer: $\frac{7}{8}$ To convert the decimal 0.875 to a fraction, write it over its place value: $\frac{875}{1000}$. To simplify, find the greatest common divisor of 875 and 1000. Both are divisible by 25: $875 \div 25 = 35$ and $1000 \div 25 = 40$, resulting in $\frac{35}{40}$. This can be simplified further by dividing by 5: $35 \div 5 = 7$ and $40 \div 5 = 8$. The simplest form is $\frac{7}{8}$.
20. A recipe calls for $\frac{9}{4}$ cups of sugar. Which of the following decimal measurements is the same amount?
    Answer: 2.25 To convert the improper fraction $\frac{9}{4}$ to a decimal, divide the numerator (9) by the denominator (4). $9 \div 4 = 2.25$. Alternatively, convert it to a mixed number: $\frac{9}{4} = 2 \frac{1}{4}$. Since $\frac{1}{4} = 0.25$, the decimal is 2.25.

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