Decompose Fractions Multiple Ways - 4th Grade Math

1. Chen wants to break apart $\frac{6}{8}$ into a sum of fractions with denominator 8. Choose the option that shows two different correct decompositions.
   Answer: $\frac{6}{8}=\frac{3}{8}+\frac{3}{8}$ and $\frac{6}{8}=\frac{2}{8}+\frac{4}{8}$ 6/8 breaks apart correctly into 3/8 + 3/8 and 2/8 + 4/8 because both pairs of eighths add up to 6/8. Choice B's pairs add to 5/8 and 7/8 instead of 6/8. Choice C's pairs both add to 7/8. Choice D's pairs both add to 8/8, one whole.
2. Look at the equation: $\frac{9}{14} = \frac{a}{14} + \frac{b}{14} + \frac{c}{14}$, where $a$, $b$, and $c$ are three whole numbers that are all different from each other. Which set of values for $a$, $b$, and $c$ makes this equation true?
   Answer: $a = 2, b = 3, c = 4$ The correct answer is D because $2 + 3 + 4 = 9$ and all three values are different from one another. Choice A also adds to 9 but repeats the value 2, so the numbers aren't all different. Choice B adds to 10, not 9. Choice C adds to 9 but uses the same value three times.
3. Write $6/12$ as a sum of fractions with denominator 12 in two different ways. Choose the answer that shows two different correct decompositions.
   Answer: $6/12=3/12+3/12$ and $6/12=1/12+5/12$ Both parts of Choice C add correctly to $6/12$, and the two decompositions use genuinely different pairs of twelfths rather than just reordering the same pair. Choice A has one part that adds to only $5/12$, so that decomposition is not correct. Choice B uses the same two fractions in both decompositions, just written in a different order, so it isn't really two different ways. Choice D has one part that adds to only $5/12$, so it isn't fully correct either. Choice C is the only option with two decompositions that are both correct and genuinely different.
4. Ava found that $\frac{12}{16}$ can be decomposed as $\frac{12}{16} = \frac{8}{16} + \frac{4}{16}$. She wants to create a new decomposition by breaking apart the $\frac{8}{16}$ term into two equal parts while keeping the $\frac{4}{16}$ term unchanged. What new equation will she write?
   Answer: $\frac{12}{16} = \frac{4}{16} + \frac{4}{16} + \frac{4}{16}$ The correct answer is D because splitting $\frac{8}{16}$ into two equal parts gives $\frac{4}{16} + \frac{4}{16}$, so the full decomposition is $\frac{4}{16} + \frac{4}{16} + \frac{4}{16}$. Choice A splits $\frac{8}{16}$ unevenly into $\frac{2}{16}$ and $\frac{6}{16}$. Choice B changes the total, since $\frac{4}{16} + \frac{4}{16} + \frac{3}{16}$ adds up to $\frac{11}{16}$, not $\frac{12}{16}$. Choice C also splits $\frac{8}{16}$ unevenly, into $\frac{3}{16}$ and $\frac{5}{16}$.
5. Write $\frac{7}{12}$ as a sum of fractions with the same denominator in two different ways. Choose the answer that is correct.
   Answer: $\frac{7}{12}=\frac{3}{12}+\frac{4}{12}$ and $\frac{7}{12}=\frac{2}{12}+\frac{5}{12}$ The first pair is correct because both equations use twelfths and both sums equal $\frac{7}{12}$. The second pair is incorrect because it mixes sixths with twelfths. The third pair is incorrect because $\frac{1}{12}+\frac{4}{12}$ equals $\frac{5}{12}$, not $\frac{7}{12}$. The fourth pair is incorrect because both of its equations sum to $\frac{6}{12}$, not $\frac{7}{12}$.
6. Use the number line to determine which decomposition equation correctly represents the fraction shown by the marked segments.
   Answer: $\frac{7}{9} = \frac{4}{9} + \frac{2}{9} + \frac{1}{9}$ The number line shows three marked segments representing $\frac{7}{9}$ total: the first segment spans $\frac{4}{9}$, the second segment spans $\frac{2}{9}$, and the third segment spans $\frac{1}{9}$, giving $\frac{4}{9} + \frac{2}{9} + \frac{1}{9} = \frac{7}{9}$. The other choices represent different groupings that don't match the marked segments shown on the number line.
7. Emma wrote three different decompositions for $\frac{8}{10}$: Decomposition 1: $\frac{8}{10} = \frac{5}{10} + \frac{3}{10}$ Decomposition 2: $\frac{8}{10} = \frac{4}{10} + \frac{4}{10}$ Decomposition 3: $\frac{8}{10} = \frac{2}{10} + \frac{2}{10} + \frac{4}{10}$ Which statement about her decompositions is true?
   Answer: All three decompositions are correct The correct answer is B because each decomposition breaks $\frac{8}{10}$ into parts with the same denominator that add up to 8 tenths, whether split into two parts or three. Choice A wrongly excludes the other two valid decompositions. Choice C makes the same mistake, picking only one valid decomposition as if the others were wrong. Choice D does the same for the three-part decomposition, when in fact all three ways of splitting the eighths are equally valid.
8. Write $\tfrac{5}{8}$ as a sum of fractions with denominator 8 in two different ways. Choose the answer that shows two correct and different decompositions.
   Answer: Decomposition 1: $\tfrac{5}{8}=\tfrac{1}{8}+\tfrac{4}{8}$; Decomposition 2: $\tfrac{5}{8}=\tfrac{2}{8}+\tfrac{3}{8}$ Both parts of Choice D add correctly to $\tfrac{5}{8}$, and the two decompositions use genuinely different pairs of fractions rather than just reordering the same pair. Choice A uses the same two fractions in both decompositions, just written in a different order, so it isn't really two different ways. Choice B has both parts add up to less than $\tfrac{5}{8}$, so neither decomposition is correct. Choice C has one part that adds to $\tfrac{5}{8}$ correctly, but the other part adds to only $\tfrac{4}{8}$, so it isn't fully correct. Choice D is the only option with two decompositions that are both correct and genuinely different.
9. A recipe calls for $\frac{11}{15}$ cup of flour. Ben wants to measure this using three different measuring steps. Which equation shows a way Ben could NOT decompose $\frac{11}{15}$?
   Answer: $\frac{11}{15} = \frac{4}{15} + \frac{4}{15} + \frac{4}{15}$ Choice C is incorrect because $\frac{4}{15} + \frac{4}{15} + \frac{4}{15} = \frac{12}{15}$, which is greater than $\frac{11}{15}$. Choices A, B, and D all correctly sum to $\frac{11}{15}$: Choice A gives $\frac{5+3+3}{15} = \frac{11}{15}$, Choice B gives $\frac{6+3+2}{15} = \frac{11}{15}$, and Choice D gives $\frac{7+2+2}{15} = \frac{11}{15}$.
10. Decompose $\frac{4}{5}$ into a sum of fractions with the same denominator. Show two different ways. Choose the correct pair of equations.
    Answer: $\frac{4}{5}=\frac{1}{5}+\frac{3}{5}$ and $\frac{4}{5}=\frac{2}{5}+\frac{2}{5}$ The first pair is correct because both equations use fifths and both sums equal $\frac{4}{5}$. The second pair is incorrect because it mixes tenths with fifths. The third pair is incorrect because $\frac{1}{5}+\frac{2}{5}$ only equals $\frac{3}{5}$, not $\frac{4}{5}$. The fourth pair is incorrect because $\frac{2}{5}+\frac{3}{5}$ equals $\frac{5}{5}$, not $\frac{4}{5}$.
11. Jamal decomposed $\frac{5}{6}$ in two different ways: $\frac{5}{6} = \frac{2}{6} + \frac{3}{6}$ and $\frac{5}{6} = \frac{1}{6} + \frac{4}{6}$. He wants to create a third decomposition using exactly four addends. Which equation shows a valid third way?
    Answer: $\frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{2}{6}$ Choice A correctly decomposes $\frac{5}{6}$ into four addends: $\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{2}{6} = \frac{5}{6}$. Choice B sums to $\frac{4}{6}$, not $\frac{5}{6}$. Choice C sums to $\frac{6}{6}$, which is greater than $\frac{5}{6}$. Choice D sums to $\frac{5}{6}$ but includes $\frac{0}{6}$, which is not a meaningful part in a decomposition.

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