Fractions, Decimals, and Percents - SSAT Upper Level Quantitative

1. The expression $\frac{0.125 + \frac{1}{8}}{62.5\% - 0.5}$ equals which of the following?
   Answer: $2$ Convert everything to decimals: $0.125 = \frac{1}{8}$, so the numerator is $0.125 + 0.125 = 0.25$. For the denominator: $62.5\% = 0.625$ and $0.5 = 0.5$, so $0.625 - 0.5 = 0.125$. Therefore, $\frac{0.25}{0.125} = 2$. Choice A ($\frac{8}{5} = 1.6$) might result from calculation errors, choice C ($\frac{5}{2} = 2.5$) could come from mishandling the denominator subtraction, and choice D (20) might result from inverting the fraction incorrectly.
2. If $\frac{5}{8}$ of a number is equal to $125\%$ of $0.4$, what percent of this number is $\frac{2}{5}$?
   Answer: $50\%$ First find the number: $\frac{5}{8} \cdot n = 125\% \cdot 0.4 = 1.25 \cdot 0.4 = 0.5$. So $n = 0.5 \cdot \frac{8}{5} = 0.8$. Now find what percent $\frac{2}{5}$ is of this number: $\frac{\frac{2}{5}}{0.8} = \frac{0.4}{0.8} = 0.5 = 50\%$. Choice B (62.5%) equals $\frac{5}{8}$, choice C (40%) equals $\frac{2}{5}$ itself, and choice D (80%) is the number we found expressed as a percentage.
3. In a survey, $0.375$ of respondents preferred coffee, $\frac{2}{5}$ preferred tea, and the remaining $22.5\%$ preferred other beverages. What fraction of coffee and tea drinkers combined preferred coffee?
   Answer: $\frac{15}{31}$ First verify the percentages add to 100%: Coffee = $0.375 = 37.5\%$, Tea = $\frac{2}{5} = 40\%$, Other = $22.5\%$. Total: $37.5\% + 40\% + 22.5\% = 100\%$ ✓. Coffee and tea drinkers combined represent $37.5\% + 40\% = 77.5\%$ of all respondents. Among just coffee and tea drinkers, the fraction who preferred coffee is: $\frac{37.5\%}{77.5\%} = \frac{0.375}{0.775} = \frac{375}{775} = \frac{15}{31}$. Choice B ($\frac{3}{8}$) is the original coffee fraction, choice C ($\frac{24}{31}$) would be the tea fraction among coffee and tea drinkers, and choice D ($\frac{31}{40}$) represents the combined coffee and tea fraction of all respondents.
4. A student converts $\frac{7}{12}$ to a percentage and rounds to the nearest tenth of a percent. What percentage does the student report?
   Answer: $58.3\%$ $\frac{7}{12} = 0.58\overline{3} = 0.583333...$ Converting to percentage: $0.583333... \times 100\% = 58.3333...\%$ Rounding to the nearest tenth of a percent means rounding to one decimal place: $58.3333...\% \approx 58.3\%$. Choice B shows two decimal places (not rounded to nearest tenth), choice C would require the third decimal digit to be 5 or higher (but it's 3), and choice D rounds to the nearest whole percent rather than tenth of a percent.

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