Convert between forms (fraction, decimal, percent) as needed - TACHS Math

Which decimal is exactly equal to $\dfrac{9}{40}$ ?
Answer: 0.225 When you need to convert a fraction to a decimal, you have two main approaches: long division or finding an equivalent fraction with a power of 10 in the denominator. Let's use the second method with $\frac{9}{40}$ . To convert this to a decimal, we need to get a denominator that's a power of 10. Since $40 = 8 \times 5$ , we can multiply both numerator and denominator by 25 to get a denominator of 1000: $\frac{9}{40} = \frac{9 \times 25}{40 \times 25} = \frac{225}{1000} = 0.225$ You can verify this with long division: $9 ÷ 40 = 0.225$ . Looking at the wrong answers: Choice A (0.18) represents $\frac{18}{100}$ or $\frac{9}{50}$ , which is close but uses 50 instead of 40 in the denominator. Choice C (0.23) might result from rounding 0.225 incorrectly or miscalculating during division. Choice D (0.24) could come from confusing this with $\frac{6}{25}$ or making an arithmetic error in the conversion process. The correct answer is B) 0.225. Study tip: When converting fractions to decimals, look for ways to make the denominator a power of 10 (10, 100, 1000, etc.) by strategic multiplication. This often gives you the exact decimal faster than long division and helps you avoid rounding errors that create trap answers.
The repeating decimal $0.\overline{3}$ is equal to which fraction?
Answer: $\dfrac{1}{3}$ When you encounter repeating decimals, you need to convert them to fractions using algebraic manipulation. The notation $0.\overline{3}$ means the digit 3 repeats infinitely: 0.333333... To convert this to a fraction, let $x = 0.\overline{3} = 0.333333...$ Since one digit repeats, multiply both sides by 10: $10x = 3.333333...$ Now subtract the original equation: $10x - x = 3.333... - 0.333...$ This gives you $9x = 3$ , so $x = \frac{3}{9} = \frac{1}{3}$ . Therefore, $0.\overline{3} = \frac{1}{3}$ , making B correct. Let's examine why the other options are wrong. Choice A, $\frac{1}{30}$ , equals approximately 0.033, which is far smaller than our repeating decimal. Choice C, $\frac{3}{10}$ , equals exactly 0.3 (terminating), not the infinitely repeating 0.333... Choice D, $\frac{33}{100}$ , equals 0.33 (also terminating), which stops after two decimal places rather than continuing forever. The key insight is that $\frac{1}{3}$ cannot be expressed exactly as a terminating decimal because 3 doesn't divide evenly into powers of 10, so it must repeat infinitely. Strategy tip: For single-digit repeating decimals, use the pattern: multiply by 10, subtract the original, then solve. For repeating decimals like $0.\overline{27}$ (two digits repeat), you'd multiply by 100 instead. The number of zeros in your multiplier should match the number of repeating digits.
Maria completed $\dfrac{3}{5}$ of her homework before dinner. What percent of her homework had she finished?
Answer: 60\% When you see a question asking for a fraction to be converted to a percentage, you're working with one of the most fundamental math conversions. The key is remembering that "percent" means "out of 100." To convert $\frac{3}{5}$ to a percentage, you need to find an equivalent fraction with 100 in the denominator. Since $5 \times 20 = 100$ , you multiply both the numerator and denominator by 20: $\frac{3}{5} = \frac{3 \times 20}{5 \times 20} = \frac{60}{100} = 60\%$ . Alternatively, you can divide: $3 \div 5 = 0.6$ , then multiply by 100 to get 60%. Looking at the wrong answers: Choice B (55%) might come from a calculation error or confusing $\frac{3}{5}$ with $\frac{11}{20}$ . Choice C (65%) could result from accidentally adding 5 to the correct answer of 60%, perhaps from misremembering fraction-to-percent conversions. Choice D (75%) equals $\frac{3}{4}$ , which suggests confusing the original fraction $\frac{3}{5}$ with $\frac{3}{4}$ - a common mistake when working quickly. The correct answer is A) 60%. Study tip: Memorize these common fraction-to-percent conversions: $\frac{1}{4} = 25\%$ , $\frac{1}{2} = 50\%$ , $\frac{3}{4} = 75\%$ , $\frac{1}{5} = 20\%$ , $\frac{2}{5} = 40\%$ , $\frac{3}{5} = 60\%$ , and $\frac{4}{5} = 80\%$ . These appear frequently on standardized tests and will save you calculation time.
Which percent is equivalent to $\dfrac{11}{8}$ ?
Answer: 137.5\% When you need to convert a fraction to a percentage, you're finding what portion the fraction represents out of 100. This is a fundamental skill that appears frequently on standardized tests. To convert $\frac{11}{8}$ to a percentage, multiply the fraction by 100%. First, divide 11 by 8: $11 ÷ 8 = 1.375$ . Then multiply by 100% to get $1.375 × 100\% = 137.5\%$ . This makes sense because $\frac{11}{8}$ is an improper fraction (numerator larger than denominator), so the result should be greater than 100%. Looking at the incorrect choices reveals common calculation errors. Choice B (125%) likely comes from incorrectly thinking $\frac{11}{8} = \frac{10}{8} = 1.25$ , or from a simple arithmetic mistake when dividing. Choice C (87.5%) represents the trap of flipping the fraction—this is what you'd get if you calculated $\frac{8}{11}$ instead of $\frac{11}{8}$ . Choice D (13.75%) comes from forgetting to multiply by 100 after finding the decimal 1.375, leaving off the crucial step that converts a decimal to a percentage. The correct answer is A) 137.5%. Study tip: When converting fractions to percentages, always check if your answer makes logical sense. If the numerator is larger than the denominator, your percentage must be greater than 100%. Also, remember that converting to a percentage always requires multiplying by 100—don't forget this final step.
Which fraction below is equivalent to $62.5\%$ in simplest form?
Answer: $\dfrac{5}{8}$ When you encounter percentage-to-fraction conversion problems, you need to convert the percentage to a fraction and then simplify it to lowest terms. To convert $62.5\%$ to a fraction, first write it as $\frac{62.5}{100}$ . Since we have a decimal in the numerator, multiply both numerator and denominator by 10 to eliminate it: $\frac{625}{1000}$ . Now find the greatest common factor (GCF) of 625 and 1000. Both numbers are divisible by 125: $625 ÷ 125 = 5$ and $1000 ÷ 125 = 8$ . Therefore, $62.5\% = \frac{5}{8}$ , making A correct. Let's verify why the other answers are wrong by converting each fraction to a percentage. Choice B: $\frac{3}{4} = 0.75 = 75\%$ , which is too large. Choice C: $\frac{2}{5} = 0.4 = 40\%$ , which is too small. Choice D: $\frac{7}{16} = 0.4375 = 43.75\%$ , also too small. You can double-check your work by converting $\frac{5}{8}$ back to a percentage: $5 ÷ 8 = 0.625 = 62.5\%$ ✓ Strategy tip: When converting percentages with decimals to fractions, always multiply to eliminate the decimal first, then find the GCF to simplify. Also remember that $62.5\% = \frac{5}{8}$ is a common fraction-percentage pair worth memorizing, along with others like $\frac{1}{8} = 12.5\%$ and $\frac{3}{8} = 37.5\%$ .
Convert $2\dfrac{1}{4}$ to a percent.
Answer: 225\% Converting mixed numbers to percents requires two key steps: first convert to a decimal, then multiply by 100 to get the percentage. Start by converting $2\frac{1}{4}$ to a decimal. The whole number part is 2, and you need to convert the fraction $\frac{1}{4}$ to decimal form. Since $\frac{1}{4} = 0.25$ , the mixed number becomes $2 + 0.25 = 2.25$ . To convert any decimal to a percent, multiply by 100: $2.25 \times 100 = 225\%$ . This confirms that choice B is correct. Now let's examine why the other answers are wrong. Choice A (22.5%) represents a common error where students mistakenly treat the mixed number as $\frac{1}{4}$ plus some confusion about place value, arriving at a number that's 10 times too small. Choice C (2.25%) is the decimal form without converting to percent—students who choose this forgot the crucial step of multiplying by 100. Choice D (202.5%) suggests students incorrectly added 200% (thinking 2 = 200%) plus 2.5% (mishandling $\frac{1}{4}$ ), rather than properly converting the entire mixed number. Remember this pattern: when converting mixed numbers to percents, always convert to decimal form first, then multiply by 100. Don't try to convert the whole number and fraction parts separately, as this leads to calculation errors. Mixed numbers greater than 1 will always yield percentages greater than 100%.
Which fraction is equivalent to $0.375$ in simplest form?
Answer: $\dfrac{3}{8}$ When you encounter a decimal that needs to be converted to a fraction, you're working with place value relationships. The key is recognizing what each decimal place represents and then simplifying the resulting fraction. To convert $0.375$ to a fraction, start by reading the decimal: "three hundred seventy-five thousandths." This tells you the fraction is $\frac{375}{1000}$ . Now you need to simplify by finding the greatest common factor (GCF) of 375 and 1000. Finding the GCF: Both numbers are divisible by 5, then by 5 again, then by 5 once more. So $375 ÷ 125 = 3$ and $1000 ÷ 125 = 8$ , giving you $\frac{3}{8}$ . You can verify this is correct: $3 ÷ 8 = 0.375$ . Looking at the wrong answers: Choice B ( $\frac{37}{100}$ ) represents $0.37$ , not $0.375$ - this would be the result if you misread the decimal as having only two decimal places. Choice C ( $\frac{15}{32}$ ) equals $0.46875$ when you divide - this is a completely different decimal. Choice D ( $\frac{375}{1000}$ ) is technically equivalent to $0.375$ , but it's not in simplest form since both numerator and denominator share common factors. Study tip: When converting decimals to fractions, always check if your final answer can be simplified further. The TACHS often includes the unsimplified version as a distractor, so don't stop at your first correct fraction - reduce it to lowest terms.
Which decimal is equal to $125\%$ ?
Answer: 1.25 When you encounter percentage-to-decimal conversion problems, remember that "percent" literally means "per hundred," so you're converting a fraction with denominator 100 into decimal form. To convert $125\%$ to a decimal, you can use the standard method: divide by 100 (or equivalently, move the decimal point two places to the left). Starting with $125\%$ , think of this as $125.0\%$ . Moving the decimal point two places left gives you $1.25$ . You can verify this makes sense because $125\% = \frac{125}{100} = 1.25$ . Choice B (1.25) is correct because it properly represents $125\%$ as a decimal greater than 1, which makes sense since $125\%$ means "125 out of 100" or "more than the whole." Choice A (0.0125) results from moving the decimal point too far left—this would be $1.25\%$ as a decimal. Choice C (12.5) comes from moving the decimal point the wrong direction (right instead of left), giving you a value 10 times too large. Choice D (0.125) represents $12.5\%$ as a decimal—you'd get this by mistakenly thinking $125\% = 12.5\%$ or by incorrectly applying the conversion rule. For percentage conversions, always remember the two-step check: does your decimal make logical sense compared to the original percentage, and did you move the decimal point exactly two places to the left? Percentages over $100\%$ should always convert to decimals greater than 1.
What percent is equivalent to $\dfrac{7}{20}$ ?
Answer: 35\% When you need to convert a fraction to a percentage, you're finding what portion that fraction represents out of 100. The most reliable approach is to convert the fraction to a decimal first, then multiply by 100. To convert $\frac{7}{20}$ to a percentage, divide the numerator by the denominator: $7 ÷ 20 = 0.35$ . Then multiply by 100 to get the percentage: $0.35 × 100 = 35\%$ . Alternatively, you can find an equivalent fraction with 100 in the denominator. Since $20 × 5 = 100$ , multiply both numerator and denominator by 5: $\frac{7}{20} = \frac{7 × 5}{20 × 5} = \frac{35}{100} = 35\%$ . Looking at the wrong answers: Choice B (28%) might result from incorrectly calculating $7 × 4 = 28$ when trying to scale up to 100, but $20 × 4 = 80$ , not 100. Choice C (40%) could come from mistakenly thinking $\frac{8}{20} = 40\%$ and confusing the numerator. Choice D (70%) represents a common error where students might flip the fraction or miscalculate $7 × 10 = 70$ , forgetting that $20 × 10 = 200$ , not 100. The correct answer is A) 35%. Study tip: When converting fractions to percentages, always check if the denominator divides evenly into 100. Common denominators like 20, 25, and 50 make this conversion straightforward. If not, use decimal division as your backup method.
What fraction in simplest form is equivalent to $260\%$ ?
Answer: $\dfrac{13}{5}$ When you encounter percentage-to-fraction conversion problems, remember that percent means "per hundred," so you're essentially working with a fraction that has 100 in the denominator initially. To convert $260\%$ to a fraction, start by writing it as $\frac{260}{100}$ . Now you need to simplify this fraction to its lowest terms by finding the greatest common factor (GCF) of 260 and 100. Find the factors: 260 = 4 × 65 = 4 × 5 × 13, and 100 = 4 × 25 = 4 × 5². The GCF is 20. Dividing both numerator and denominator by 20 gives you $\frac{260 ÷ 20}{100 ÷ 20} = \frac{13}{5}$ , which matches choice A. Let's examine why the other answers are incorrect. Choice B, $\frac{26}{100}$ , represents 26%, not 260% - this is what you'd get if you mistakenly dropped a zero. Choice C, $\frac{52}{5}$ , equals $\frac{520}{50}$ or 1040%, which is far too large - this might result from incorrectly doubling somewhere in your calculation. Choice D, $\frac{5}{13}$ , is the reciprocal of the correct answer and would represent approximately 38%, likely from flipping the numerator and denominator. Remember this pattern: percentages over 100% will always convert to improper fractions (where the numerator is larger than the denominator). If you get a proper fraction when converting a percentage over 100%, double-check your work.
Which decimal is closest to $\dfrac{5}{6}$ when the answer is rounded to the nearest thousandth?
Answer: 0.833 When you need to convert a fraction to a decimal and round to a specific place value, you're testing your division skills and understanding of place value precision. To find the decimal equivalent of $\frac{5}{6}$ , divide 5 by 6 using long division. When you perform this division, you get 0.8333..., where the 3 repeats infinitely. To round to the nearest thousandth, you look at the fourth decimal place. Since you have 0.8333... and the fourth decimal place is 3 (which is less than 5), you round down, giving you 0.833. Looking at the answer choices: A) 0.833 is correct—it's exactly $\frac{5}{6}$ rounded to the nearest thousandth. B) 0.875 represents $\frac{7}{8}$ , which is a different fraction entirely and larger than $\frac{5}{6}$ . C) 0.667 is approximately $\frac{2}{3}$ , which is significantly smaller than $\frac{5}{6}$ . D) 0.785 doesn't correspond to any common fraction and is also smaller than the actual value of $\frac{5}{6}$ . Remember that when converting fractions to decimals on standardized tests, it's helpful to memorize common fraction-decimal equivalents. Know that $\frac{5}{6} = 0.833...$ and $\frac{1}{6} = 0.166...$ These sixths appear frequently on math exams. Also, always check your rounding carefully—the digit in the place value immediately after your target determines whether you round up or down.
What percent is equal to $0.004$ ?
Answer: 0.4\% Converting between decimals and percentages is a fundamental skill that appears frequently on standardized tests. When you see a decimal that needs to be expressed as a percentage, remember that "percent" literally means "per hundred." To convert $0.004$ to a percentage, multiply by 100 (or equivalently, move the decimal point two places to the right). Here's why: $0.004 \times 100 = 0.4$ , so $0.004 = 0.4\%$ . Looking at the wrong answers reveals common conversion mistakes. Choice B (4%) results from moving the decimal point too far right—this would be correct if you were converting $0.04$ to a percentage, not $0.004$ . Choice C (0.04%) happens when students move the decimal point in the wrong direction, essentially dividing by 100 instead of multiplying. This is the opposite of what you should do. Choice D (40%) represents an even more dramatic error, moving the decimal point three places right instead of two. The correct answer is A (0.4%) because $0.004 \times 100 = 0.4\%$ . Study tip: Always remember the conversion rule—to go from decimal to percentage, multiply by 100 (move decimal right two places). To go from percentage to decimal, divide by 100 (move decimal left two places). Practice with small decimals like these since they're especially tricky and commonly tested.
Which fraction is equal to $0.072$ in simplest form?
Answer: $\dfrac{9}{125}$ When converting a decimal to a fraction, you need to understand place value and then simplify to lowest terms. The decimal $0.072$ has three decimal places, so it equals $\frac{72}{1000}$ (72 thousandths). To find the simplest form, you must find the greatest common divisor (GCD) of 72 and 1000. Breaking this down: 72 = 8 × 9 = 2³ × 3², and 1000 = 10³ = (2 × 5)³ = 2³ × 5³. The GCD is 2³ = 8. Dividing both numerator and denominator by 8: $\frac{72}{1000} = \frac{72 ÷ 8}{1000 ÷ 8} = \frac{9}{125}$ Since 9 = 3² and 125 = 5³ share no common factors, this fraction is in simplest form. Choice A ( $\frac{9}{125}$ ) is correct - this is our simplified result. Choice B ( $\frac{72}{1000}$ ) represents the decimal correctly but isn't simplified. Both 72 and 1000 are divisible by 8. Choice C ( $\frac{18}{250}$ ) is partially simplified from $\frac{72}{1000}$ (divided by 4), but it can be reduced further. $\frac{18}{250} = \frac{9}{125}$ when you divide by 2. Choice D ( $\frac{7}{96}$ ) equals approximately 0.073, which is close to 0.072 but not equal. This could trap students who estimate rather than calculate precisely. Remember: always check if your fraction can be simplified further by finding common factors. Don't stop at the first conversion - reduce to lowest terms for the final answer.

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