Converting Fractions & Decimals - Pre Algebra

Elena writes the fraction $\frac{5}{8}$ as a decimal and gets 0.625. She then rounds this decimal to the nearest tenth to get 0.6, and converts 0.6 back to a fraction in lowest terms. What is the difference between her original fraction and her final fraction, expressed as a decimal?
Answer: 0.025 Starting with 5/8 = 0.625. Rounding to the nearest tenth: 0.625 rounds to 0.6. Converting 0.6 back to a fraction: 0.6 = 6/10 = 3/5 in lowest terms. The difference between the original and final fractions: 5/8 - 3/5. To subtract, find a common denominator: 5/8 - 3/5 = 25/40 - 24/40 = 1/40 = 0.025. Choice A is correct. Choice B (0.1) might result from incorrectly calculating 0.625 - 0.6 without considering the fraction conversion. Choice C (0.075) could come from calculation errors. Choice D (0.05) might result from errors in finding the common denominator.
A recipe calls for $2.75$ cups of flour. Sarah only has a measuring cup marked in fractions. She needs to convert $2.75$ to a mixed number. However, she wants the fractional part to have a denominator that is a power of 2. What mixed number should she use?
Answer: $2\frac{3}{4}$ First, convert 2.75 to a fraction: 2.75 = 2 + 0.75 = 2 + 75/100 = 2 + 3/4 (after reducing 75/100). So 2.75 = 2¾. We need to check which answer has a denominator that is a power of 2. Powers of 2 include: 1, 2, 4, 8, 16, 32... Choice A: 2¾ has denominator 4 = 2², which is a power of 2. This is correct. Choice B: 2⁷⁄₁₀ has denominator 10, which is not a power of 2. Choice C: 2¹¹⁄₁₆ has denominator 16 = 2⁴, which is a power of 2, but 11/16 = 0.6875 ≠ 0.75. Choice D: 2¹⁵⁄₂₀ has denominator 20, which is not a power of 2.
A student claims that $0.45$ is equivalent to $\frac{9}{20}$ . To verify this, she cross-multiplies and checks if $0.45 \times 20 = 9 \times 1$ . She calculates the left side as $9.0$ and the right side as $9$ . Based on this verification method, what can she conclude?
Answer: The fractions are equivalent because both sides equal 9 When comparing decimals and fractions, you can use cross-multiplication to check if they're equivalent. The student is testing whether $0.45 = \frac{9}{20}$ by treating the decimal as $\frac{0.45}{1}$ and cross-multiplying: $0.45 \times 20$ versus $9 \times 1$ . Her calculation is correct: $0.45 \times 20 = 9.0$ and $9 \times 1 = 9$ . Since $9.0$ and $9$ represent the same value (the decimal point and zero don't change the numerical value), the cross-products are equal, confirming that $0.45$ and $\frac{9}{20}$ are equivalent. This makes choice D correct. Choice A is wrong because the cross-multiplication provides sufficient information to determine equivalence. Choice B represents a common misconception about decimal notation— $9.0$ and $9$ are mathematically identical values, just written in different forms. The presence of a decimal point and trailing zero doesn't make the numbers unequal. Choice C is incorrect because cross-multiplication is a valid method for comparing any two fractions, including when one is written in decimal form. The verification method works perfectly fine for decimal comparisons. Remember that $9$ , $9.0$ , $9.00$ , and $\frac{9}{1}$ all represent the exact same number. When checking your work with cross-multiplication, focus on the numerical value, not the format. Trailing zeros after a decimal point don't change a number's value—they're just different ways of writing the same thing.
Three students convert $\frac{7}{15}$ to a decimal. Alex gets $0.4666...$ , Bella gets $0.47$ , and Carlos gets $0.4\overline{6}$ . Which student(s) correctly converted the fraction?
Answer: Both Alex and Carlos correctly converted the fraction To convert 7/15 to a decimal, we perform long division: 7 ÷ 15 = 0.4666... where the 6 repeats infinitely. Alex wrote 0.4666..., which correctly shows the repeating pattern using ellipsis notation. Carlos wrote 0.4̄6̄, which correctly shows the repeating 6 using bar notation. Both representations are mathematically correct ways to express the same repeating decimal. Bella wrote 0.47, which is a rounded approximation but not the exact decimal equivalent of 7/15. Choice C is correct because both Alex and Carlos gave exact representations of the decimal, just using different notation conventions.
When converting $\frac{3}{11}$ to a decimal, the result is $0.\overline{272}$ . If someone mistakenly writes this as the non-repeating decimal $0.272$ and then converts it back to a fraction in lowest terms, what fraction do they get?
Answer: $\frac{34}{125}$ The person incorrectly treats 0.272̄ as 0.272 (non-repeating). Converting 0.272 to a fraction: 0.272 = 272/1000. To reduce to lowest terms, find GCD(272, 1000). 272 = 8 × 34 = 2³ × 34, and 1000 = 8 × 125 = 2³ × 125. The GCD is 8. Dividing both by 8: 272 ÷ 8 = 34 and 1000 ÷ 8 = 125. So 272/1000 = 34/125 in lowest terms. Choice B is correct. Choice A gives the unreduced fraction. Choice C (136/500) is what you get if you divide by 2 instead of 8. Choice D (68/250) is what you get if you divide by 4 instead of 8. Both C and D are not in lowest terms.
A decimal $0.875$ needs to be written as a fraction. After converting to $\frac{875}{1000}$ , the next step is to reduce to lowest terms. How many times must you divide both the numerator and denominator by their greatest common factor to reach the fraction in lowest terms?
Answer: You must divide once by 125 When you need to reduce a fraction to lowest terms, you're looking for the greatest common factor (GCF) of the numerator and denominator, then dividing both by that number in one step. To find the GCF of 875 and 1000, let's find their prime factorizations: - $875 = 5^3 \times 7 = 125 \times 7$ - $1000 = 2^3 \times 5^3 = 8 \times 125$ The GCF is $5^3 = 125$ since that's the highest power of the common factor 5. Dividing both numerator and denominator by 125 gives us $\frac{875 ÷ 125}{1000 ÷ 125} = \frac{7}{8}$ , which is in lowest terms since 7 and 8 share no common factors. Answer choice A incorrectly divides by 25 instead of 125, giving $\frac{35}{40}$ , which isn't fully reduced since both 35 and 40 are divisible by 5. Answer choice B suggests dividing twice (by 25, then by 5), which would work mathematically but isn't the most efficient approach since you're not using the greatest common factor in one step. Answer choice C suggests dividing by 5 three separate times, which again works but misses the point of finding the GCF. The key insight is that reducing to lowest terms means finding the greatest common factor and dividing once, not repeatedly dividing by smaller common factors. Always look for the largest number that divides evenly into both parts of your fraction.
Jamie is comparing two fractions: $\frac{11}{16}$ and $0.6875$ . She converts the first fraction to a decimal and notices something interesting about the relationship between these two numbers. What is the value of $\frac{11}{16} - 0.6875$ ?
Answer: $0$ First, convert 11/16 to a decimal. 11 ÷ 16 = 0.6875. So 11/16 = 0.6875. Therefore, 11/16 - 0.6875 = 0.6875 - 0.6875 = 0. The 'something interesting' Jamie notices is that the fraction and decimal are exactly equal. Choice A is correct. Choice B (0.0625) might result from a student who incorrectly converts 11/16 (perhaps getting 0.75 and then calculating 0.75 - 0.6875). Choice C (-0.0625) could result from calculation errors or switching the order of subtraction. Choice D (0.125) might come from various computational mistakes in the decimal conversion.

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