Fractions, Decimals, and Percents - SHSAT Math

1. A recipe calls for $1\frac{3}{8}$ cups of flour, and Maria wants to measure it as a decimal. What exact decimal amount should she measure?
   Answer: 1.375 cups Write the mixed number as a whole number plus a fraction: $1\frac{3}{8} = 1 + \frac{3}{8}$. To change $\frac{3}{8}$ to a decimal, divide 3 by 8: 3 ÷ 8 = 0.375. So $1\frac{3}{8}$ = 1.375 cups exactly. You can check: 0.375 = 375/1000 = 3/8. The value 1.3 comes from reading the numerator 3 as the tenths digit. The value 1.35 comes from treating 3/8 as about 35 hundredths. The value 1.38 is 1.375 rounded to the nearest hundredth, which is close but not exact.
2. Which fraction equals $87.5\%$?
   Answer: $\dfrac{7}{8}$ When you encounter percentage-to-fraction conversion problems, the key is to convert the percentage to a decimal first, then find the equivalent fraction in simplest form. To solve this, start by converting $87.5\%$ to a decimal: $87.5\% = 87.5 ÷ 100 = 0.875$. Now you need to express $0.875$ as a fraction. Since there are three decimal places, write it as $\frac{875}{1000}$. To simplify, find the greatest common factor of 875 and 1000. Both are divisible by 125: $\frac{875 ÷ 125}{1000 ÷ 125} = \frac{7}{8}$. You can verify this by dividing: $7 ÷ 8 = 0.875 = 87.5\%$. Looking at the wrong answers: Choice B, $\frac{5}{6}$, equals approximately $0.833$ or $83.3\%$ - close to our target but not exact. Choice C, $\frac{13}{16}$, equals $0.8125$ or $81.25\%$ - this is exactly $6.25$ percentage points too low. Choice D, $\frac{4}{5}$, equals $0.8$ or $80\%$ - significantly lower than $87.5\%$. Study tip: Memorize common fraction-percentage equivalents like $\frac{1}{8} = 12.5\%$, $\frac{1}{4} = 25\%$, $\frac{3}{8} = 37.5\%$, $\frac{1}{2} = 50\%$, $\frac{5}{8} = 62.5\%$, $\frac{3}{4} = 75\%$, and $\frac{7}{8} = 87.5\%$. Recognizing these patterns will save you valuable time on the SHSAT.
3. Express $4.6\%$ as a decimal.
   Answer: 0.046 When you encounter a percentage-to-decimal conversion problem, remember that "percent" literally means "per hundred," so you're working with parts out of 100. To convert any percentage to a decimal, divide by 100 (or equivalently, move the decimal point two places to the left). For $4.6\%$, you have $4.6 ÷ 100 = 0.046$. Think of it this way: $4.6\%$ means $\frac{4.6}{100}$, which equals $0.046$. Looking at the answer choices, option A gives us $0.046$, which is correct. Option B ($0.46$) represents what you'd get if you only moved the decimal point one place to the left instead of two—this is the result of dividing by 10 instead of 100. Option C ($4.6$) is just the original number without any conversion; this would be correct if the question asked you to remove the percent symbol but not convert to decimal form. Option D ($46$) comes from multiplying by 10 instead of dividing by 100, which reverses the operation entirely. You can verify this makes sense: $0.046$ as a percentage would be $0.046 × 100 = 4.6\%$, confirming our answer. Study tip: Always remember the "move two places left" rule for percent-to-decimal conversion. If you're unsure, think of common examples: $50\% = 0.50$ and $25\% = 0.25$. This pattern will help you avoid the common trap of only moving one decimal place.
4. A survey shows that $0.83$ of students prefer option A. What is this as a percent, rounded to the nearest whole percent?
   Answer: 83\% When you see a decimal that needs to be converted to a percentage, remember that "percent" literally means "per hundred." To convert any decimal to a percentage, you multiply by 100 and add the percent symbol. Starting with $0.83$, you multiply by 100: $0.83 \times 100 = 83$. This gives you 83%, which rounds to 83% since there are no additional decimal places to consider. Looking at the wrong answers: Choice A (8%) represents a major conceptual error—perhaps confusing 0.83 with 0.08 or misunderstanding place value entirely. Choice B (17%) might come from incorrectly thinking 0.83 means "83 out of 100 are NOT option A" and calculating 100 - 83 = 17, but that's not what the decimal represents. Choice D (83.3%) suggests confusion about the rounding process—since 0.83 converted directly gives exactly 83%, there's no need for decimal places, and certainly no reason to add a ".3" that doesn't exist in the original number. The key insight is that 0.83 already represents 83 hundredths, so when you convert to percentage form (which is also based on hundredths), you get exactly 83%. Study tip: When converting decimals to percentages on the SHSAT, always multiply by 100 first, then apply any rounding. Don't overthink it—if you see 0.83, that's 83 hundredths, which is 83%.
5. The value $\frac{5}{8}$ is what percent greater than $\frac{1}{2}$?
   Answer: 25% To find percent increase, use: (new value - old value)/old value × 100%. Here: (5/8 - 1/2)/(1/2) × 100%. Convert to common denominator: 5/8 - 4/8 = 1/8. So: (1/8)/(1/2) × 100% = (1/8) × (2/1) × 100% = 1/4 × 100% = 25%. Choice A might come from calculating the absolute difference 1/8 = 0.125 = 12.5% but forgetting to divide by the original value. Choice B could result from incorrect fraction arithmetic. Choice D gives the ratio 5/8 ÷ 1/2 = 5/4 = 1.25, mistaking this for 80% somehow.
6. Express $0.007$ as a percent.
   Answer: 0.7\% When converting decimals to percentages, you need to understand that "percent" means "per hundred" or "out of 100." To convert any decimal to a percent, you multiply by 100 and add the percent symbol. To convert $0.007$ to a percent, multiply by 100: $0.007 \times 100 = 0.7$. Therefore, $0.007 = 0.7\%$. You can think of this another way: moving the decimal point two places to the right accomplishes the same thing as multiplying by 100. Starting with $0.007$, move the decimal two places right to get $0.7$. Looking at the wrong answers: Choice A ($0.007\%$) represents the common error of simply adding the percent symbol without doing any conversion. This would actually equal $0.00007$ as a decimal, not $0.007$. Choice B ($0.07\%$) comes from moving the decimal point only one place to the right instead of two, which is the mistake of multiplying by 10 instead of 100. Choice D ($0.0007\%$) results from moving the decimal point in the wrong direction (left instead of right), which is equivalent to dividing by 100 instead of multiplying. Remember this key pattern: decimal to percent means multiply by 100 (or move decimal point two places right), while percent to decimal means divide by 100 (or move decimal point two places left). Many SHSAT questions test whether you can keep these conversions straight, so practice both directions.
7. Write $0.045$ as a fraction in lowest terms.
   Answer: $\dfrac{9}{200}$ When converting a decimal to a fraction, you need to understand what each decimal place represents. The decimal $0.045$ has three digits after the decimal point, which means it represents $\frac{45}{1000}$ (45 thousandths). To get this fraction in lowest terms, you must find the greatest common factor (GCF) of 45 and 1000, then divide both the numerator and denominator by it. The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 1000 are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000. The GCF is 5. Dividing both parts by 5: $\frac{45 ÷ 5}{1000 ÷ 5} = \frac{9}{200}$ Looking at the wrong answers: Choice B ($\frac{45}{1000}$) is the unreduced form of the decimal - this is a common trap for students who forget to simplify. Choice C ($\frac{45}{100}$) represents $0.45$, not $0.045$ - this catches students who misread the decimal or place the numerator over the wrong power of 10. Choice D ($\frac{9}{1000}$) represents $0.009$ - this occurs when students correctly find that 9 goes in the numerator after reducing but forget to reduce the denominator as well. Remember: always check that your final fraction is in lowest terms by ensuring the numerator and denominator share no common factors other than 1. Convert your answer back to a decimal to verify it matches the original.
8. A basketball player makes $\frac{3}{5}$ of her free throws in the first half and $66\frac{2}{3}\%$ in the second half. If she attempted the same number of free throws in each half, what fraction of all her free throws did she make?
   Answer: $\frac{19}{30}$ When you encounter problems involving fractions and percentages across different time periods or groups, you need to find a common way to express all the rates, then calculate the overall average. First, convert the percentage to a fraction to work with consistent units. $66\frac{2}{3}\% = 66\frac{2}{3} \div 100 = \frac{200}{3} \div 100 = \frac{200}{300} = \frac{2}{3}$ Since she attempted the same number of free throws in each half, you can use any convenient number. Let's say she attempted 15 free throws per half (chosen because it's divisible by both denominators). First half: $\frac{3}{5} \times 15 = 9$ free throws made Second half: $\frac{2}{3} \times 15 = 10$ free throws made Total made: $9 + 10 = 19$ Total attempted: $15 + 15 = 30$ Overall fraction: $\frac{19}{30}$ Choice A ($\frac{2}{3}$) incorrectly assumes you just average the two fractions: $\frac{\frac{3}{5} + \frac{2}{3}}{2}$, but this ignores that the fractions represent rates, not totals. Choice C ($\frac{7}{10}$) might result from incorrectly converting the percentage or adding the numerators and denominators separately. Choice D ($\frac{3}{4}$) could come from rounding $66\frac{2}{3}\%$ to $67\%$ and then making calculation errors. Remember: when combining rates or percentages from equal-sized groups, convert everything to the same format first, then calculate the total outcomes divided by total attempts rather than averaging the rates themselves.
9. Convert $\dfrac{125}{8}$\% to a decimal.
   Answer: 0.15625 When converting a percentage to a decimal, you need to divide by 100. However, this problem presents the percentage as a fraction, so you'll need to handle that first. Start by converting $\frac{125}{8}$% to its decimal form as a percentage. Divide 125 by 8: $125 ÷ 8 = 15.625$%. Now you have 15.625% in standard percentage form. To convert any percentage to a decimal, divide by 100 (or move the decimal point two places to the left). So $15.625$. Looking at the wrong answers: Choice A (1.25) represents what you'd get if you mistakenly converted $\frac{125}{8}$ directly to a decimal without recognizing it's a percentage - essentially treating it as $\frac{125}{100}$. Choice B (15.625) is the trap of stopping halfway through the problem - you correctly found $\frac{125}{8} = 15.625$, but forgot to divide by 100 to complete the percentage-to-decimal conversion. Choice D (12.5) might result from calculation errors or mishandling the fraction conversion. The correct answer is C: 0.15625. Strategy tip: When you see a fraction followed by a percent sign, always convert the fraction to a decimal first, then divide by 100. Many students forget that second step because they get distracted by the initial fraction work. Double-check that your final decimal makes sense - percentages greater than 100% should convert to decimals greater than 1.0.
10. What decimal is equivalent to $45\dfrac{1}{2}\%$?
    Answer: 0.455 When you encounter a mixed number percentage like $45\dfrac{1}{2}\%$, you need to convert it to decimal form through a two-step process: first convert the mixed number to an improper fraction or decimal, then convert the percentage to a decimal. Start by converting $45\dfrac{1}{2}\%$ to a decimal percentage. The fraction $\dfrac{1}{2} = 0.5$, so $45\dfrac{1}{2}\% = 45.5\%$. Now convert this percentage to a decimal by dividing by 100 (or moving the decimal point two places left): $45.5\% = \frac{45.5}{100} = 0.455$. Looking at the wrong answers: Choice B (4.55) represents a common error where you forgot to convert from percentage to decimal—this would be your answer if you stopped at 45.5% without dividing by 100. Choice C (0.452) might result from incorrectly converting $\dfrac{1}{2}$ to 0.02 instead of 0.5, perhaps confusing it with $\dfrac{1}{50}$. Choice D (0.4555) could come from misreading the mixed number or making a calculation error during conversion. The correct answer is A (0.455). Strategy tip: When converting mixed number percentages, always convert the fractional part to a decimal first, then treat it as a regular percentage. Remember that converting any percentage to decimal means dividing by 100—a step that's easy to forget when you're focused on the mixed number conversion.
11. Which of the following represents $0.333\ldots$ (the repetend 3) as a fraction?
    Answer: $\dfrac{1}{3}$ When you encounter a repeating decimal like $0.333\ldots$, you're dealing with a rational number that can be expressed as a fraction. The key is recognizing common decimal-to-fraction conversions or using algebraic methods to find the exact fraction. Let's use algebra to convert $0.333\ldots$ to a fraction. Let $x = 0.333\ldots$. Since the digit 3 repeats infinitely, we can multiply both sides by 10: $10x = 3.333\ldots$. Now subtract the original equation: $10x - x = 3.333\ldots - 0.333\ldots$, which gives us $9x = 3$. Solving for $x$: $x = \frac{3}{9} = \frac{1}{3}$. We can verify this by dividing: $1 ÷ 3 = 0.333\ldots$ Looking at the wrong answers: Choice A ($\frac{1}{2}$) equals 0.5, which is clearly different from our repeating decimal. Choice C ($\frac{3}{10}$) equals 0.3 exactly, not $0.333\ldots$ — this represents a terminating decimal with just one 3 after the decimal point. Choice D ($\frac{33}{100}$) equals 0.33, which has exactly two 3's after the decimal point but doesn't continue infinitely. The correct answer is B: $\frac{1}{3}$. Study tip: Memorize that $\frac{1}{3} = 0.333\ldots$ and $\frac{2}{3} = 0.666\ldots$ — these are among the most common repeating decimals on standardized tests. Also remember that any fraction with a denominator containing only factors of 2 and 5 will terminate, while others typically repeat.
12. If $p=0.005$, which of the following is equal to $p$ expressed as a fraction in simplest form?
    Answer: $\dfrac{1}{200}$ When you encounter a decimal that needs to be converted to a fraction, the key is understanding place value. Each decimal place represents a power of 10 in the denominator. The decimal $0.005$ has three decimal places, so it equals $\frac{5}{1000}$ (5 in the thousandths place). However, this fraction isn't in simplest form yet. To simplify, you need to find the greatest common factor of 5 and 1000. Since $1000 = 5 \times 200$, you can divide both numerator and denominator by 5: $\frac{5}{1000} = \frac{5 ÷ 5}{1000 ÷ 5} = \frac{1}{200}$. This confirms answer choice B is correct. Looking at the wrong answers: Choice A gives $\frac{1}{20} = 0.05$, which is ten times larger than our target value—a common error when miscounting decimal places. Choice C is $\frac{5}{1000}$, which equals $0.005$ but isn't simplified. While mathematically equivalent to the correct answer, it's not in "simplest form" as the question requires. Choice D gives $\frac{5}{100} = 0.05$, again ten times too large. Remember this pattern: to convert a decimal to a fraction, use the place value to determine your denominator (tenths = 10, hundredths = 100, thousandths = 1000), then simplify by finding common factors. Always check that your final fraction is fully reduced—the SHSAT often includes the unreduced form as a distractor.
13. Which of the following represents $250\%$ as a decimal?
    Answer: 2.5 When you encounter percentage-to-decimal conversion problems, remember that "percent" literally means "per hundred," so you're working with parts of 100. To convert any percentage to a decimal, divide by 100 (or equivalently, move the decimal point two places to the left). For $250\%$, you calculate: $250 ÷ 100 = 2.5$. You can also think of this as moving the decimal point in 250 two places left: from $250.$ to $2.50$ or simply $2.5$. Let's examine why the other choices represent common conversion errors. Choice B ($0.25$) would be correct if you were converting $25\%$ to a decimal, not $250\%$. This suggests confusion about place value or accidentally dropping a zero. Choice C ($25$) comes from the opposite mistake—multiplying by 100 instead of dividing, which would convert a decimal to a percentage rather than the reverse. Choice D ($0.025$) results from moving the decimal point too far left (three places instead of two), treating the conversion as if you were dividing by 1000. The correct answer is A ($2.5$). Remember this key insight: percentages greater than $100\%$ always convert to decimals greater than 1. Since $250\%$ represents "250 parts per 100," or "2.5 times the whole," your decimal answer should exceed 1. This reasoning can help you eliminate obviously incorrect choices and check your work on percentage conversion problems.
14. A recipe calls for $0.6$ cup of oil. What fraction of a cup is this amount? Give your answer in simplest form.
    Answer: $\dfrac{3}{5}$ When you encounter a decimal that needs to be converted to a fraction, you're working with place value concepts. The key is recognizing what each decimal place represents and then simplifying the resulting fraction. The decimal $0.6$ means "6 tenths" because the 6 is in the tenths place. This directly translates to the fraction $\frac{6}{10}$. However, fractions should always be expressed in lowest terms, so you need to find the greatest common factor of 6 and 10. Both numbers are divisible by 2: $\frac{6}{10} = \frac{6 ÷ 2}{10 ÷ 2} = \frac{3}{5}$. Therefore, choice A is correct. Let's examine why the other options are incorrect. Choice B gives $\frac{6}{10}$, which equals $0.6$ but isn't simplified to lowest terms. While mathematically equivalent to the correct answer, this isn't the best form. Choice C shows $\frac{2}{3}$, which equals approximately $0.667$ when converted to a decimal—close to $0.6$ but not exact. Choice D presents $\frac{5}{8}$, which equals $0.625$ as a decimal, again close but not equivalent to $0.6$. Remember that when converting decimals to fractions, always check if your answer can be simplified. The SHSAT often includes the unsimplified fraction as a distractor, so reducing to lowest terms is essential. Also, be careful not to choose fractions that are "close" to the decimal value—they must be exactly equivalent.
15. Which percent is equivalent to $\dfrac{11}{20}$?
    Answer: 55\% Converting fractions to percentages is a fundamental skill that appears frequently on the SHSAT. When you see a fraction that needs to be converted to a percent, you have two main approaches: convert to a decimal first, or find an equivalent fraction with 100 in the denominator. To convert $\frac{11}{20}$ to a percentage, let's use the decimal method. Divide 11 by 20: $11 ÷ 20 = 0.55$. To convert any decimal to a percentage, multiply by 100 (or move the decimal point two places right): $0.55 × 100 = 55\%$. Alternatively, you can find an equivalent fraction with 100 in the denominator. Since $20 × 5 = 100$, multiply both numerator and denominator by 5: $\frac{11}{20} = \frac{11 × 5}{20 × 5} = \frac{55}{100} = 55\%$. Looking at the wrong answers: Choice A (45%) would result from incorrectly calculating $\frac{9}{20}$ instead of $\frac{11}{20}$. Choice B (50%) represents $\frac{10}{20}$ or $\frac{1}{2}$, which is close but not our fraction. Choice D (60%) would come from $\frac{12}{20}$ or $\frac{3}{5}$, again close but incorrect. The correct answer is C) 55%. Strategy tip: Memorize common fraction-to-percent conversions like $\frac{1}{4} = 25\%$, $\frac{1}{2} = 50\%$, and $\frac{3}{4} = 75\%$. For fractions with denominators of 20, remember that each unit represents 5%, so $\frac{11}{20} = 11 × 5\% = 55\%$.
16. Write $\dfrac{9}{40}$ as a decimal.
    Answer: 0.225 When converting fractions to decimals, you have two main approaches: long division or finding an equivalent fraction with a power of 10 in the denominator. For $\frac{9}{40}$, let's use the second method since it's often faster. To convert $\frac{9}{40}$ to a decimal, notice that 40 can be transformed into a power of 10. Since $40 \times 2.5 = 100$, multiply both numerator and denominator by 2.5: $\frac{9}{40} = \frac{9 \times 2.5}{40 \times 2.5} = \frac{22.5}{100} = 0.225$ You can verify this with long division: $9 \div 40 = 0.225$. Looking at the wrong answers: Choice A (0.18) might come from incorrectly calculating $\frac{9}{50}$ instead of $\frac{9}{40}$, or from rounding errors during division. Choice C (0.23) could result from rounding 0.225 incorrectly to two decimal places. Choice D (0.24) might come from approximating $\frac{9}{40}$ as $\frac{6}{25}$ or making computational errors. The correct answer is B) 0.225. Strategy tip: When converting fractions to decimals on the SHSAT, look for denominators that are factors of powers of 10 (like 4, 5, 8, 20, 25, 40, 50). You can often avoid long division by multiplying both parts of the fraction to get 10, 100, or 1000 in the denominator, making the decimal conversion immediate.
17. Express $7.2$ as a percent.
    Answer: 720\% When you see a problem asking to express a number as a percent, you need to understand what "percent" means. Percent literally means "per hundred" or "out of 100." So expressing a number as a percent means showing how many parts out of 100 that number represents. To convert any number to a percent, multiply it by 100 and add the percent symbol. This is because you're finding how many hundredths the number contains. For $7.2$, we calculate: $7.2 \times 100 = 720$, so $7.2 = 720\%$. Think of it this way: $7.2$ means $7.2$ wholes, which is the same as $720$ hundredths, or $720\%$. Choice A ($0.72\%$) represents what you'd get if you divided $7.2$ by $100$ instead of multiplying - this is the opposite of the correct operation. Choice B ($7.2\%$) is the trap of simply adding a percent sign without converting. This would only be correct if the original number were $0.072$. Choice C ($72\%$) results from multiplying by $10$ instead of $100$, missing a factor of $10$. Remember this pattern: to convert a decimal or whole number to a percent, always multiply by $100$. Moving the decimal point two places to the right is the same operation. Numbers greater than $1$ will always give you percentages greater than $100\%$, which is perfectly normal - it just means more than one whole.
18. Order the following from least to greatest: $40\%,$ $0.38,$ $\dfrac{2}{5}$.
    Answer: $0.38 < 40\% = \dfrac{2}{5}$ To compare numbers written in different forms, convert them all to decimals. The value $0.38$ is already a decimal. For the percent, $40\% = \frac{40}{100} = 0.40$. For the fraction, $\frac{2}{5} = \frac{4}{10} = 0.40$. So $40\%$ and $\frac{2}{5}$ are the same number, and both are greater than $0.38$. The correct statement is $0.38 < 40\% = \dfrac{2}{5}$. The ordering $0.38 < 40\% < \dfrac{2}{5}$ gets $0.38$ right but claims $\frac{2}{5}$ is larger than $40\%$, and $0.38 < \dfrac{2}{5} < 40\%$ makes the opposite claim; neither is true, because the two values are equal. Placing $\frac{2}{5}$ below $0.38$ comes from comparing the digits 2 and 38 instead of converting.
19. A store offers a discount where customers pay $\frac{3}{4}$ of the original price. If the discounted price of an item is $18.75, what percent discount did the customer receive?
    Answer: 25% If customers pay 3/4 of the original price, then the discounted price is 3/4 of the original. Setting up the equation: (3/4) × original price = $18.75, so original price = $25. The discount amount is $25 - $18.75 = $6.25. The percent discount is ($6.25/$25) × 100% = 25%. Choice B incorrectly calculates 1/3 as the discount rate. Choice C incorrectly uses 2/3 instead of 1/4. Choice D confuses the percent paid (75%) with the percent discount.
20. A string is cut so that $12.5\%$ of its original length remains. Express this remaining fraction of the original length as a fraction in simplest form.
    Answer: $\dfrac{1}{8}$ When you encounter percentage-to-fraction conversion problems, you're working with the relationship between decimal representations and their fractional equivalents. The key is converting the percentage to a decimal first, then finding the simplest fractional form. To convert $12.5\%$ to a fraction, start by writing it as a decimal: $12.5\% = 0.125$. Now convert this decimal to a fraction by recognizing that $0.125 = \frac{125}{1000}$. To simplify, find the greatest common divisor of 125 and 1000. Since $125 = 5^3$ and $1000 = 2^3 \times 5^3$, the GCD is $5^3 = 125$. Dividing both numerator and denominator by 125 gives us $\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$. Looking at the wrong answers: Choice A ($\frac{1}{6}$) equals approximately $16.67\%$, which is significantly larger than $12.5\%$. Choice C ($\frac{1}{4}$) equals $25\%$, exactly double what we need. Choice D ($\frac{1}{2}$) equals $50\%$, which is four times too large. The correct answer is B: $\frac{1}{8}$. Remember that $12.5\%$ is exactly half of $25\%$ (which equals $\frac{1}{4}$), so the answer should be half of $\frac{1}{4}$, giving you $\frac{1}{8}$. This relationship can serve as a quick check: when you halve a percentage, you double the denominator of its fractional equivalent.

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