Decompose Fractions Multiple Ways - 4th Grade Math

1. Which equation correctly decomposes $\frac{5}{6}$ into three fractions with denominator $6$?

   $\frac{5}{6}=\frac{1}{6}+\frac{2}{6}+\frac{2}{6}$. Correctly splits $5$ into $1+2+2$ over denominator $6$.

2. Find and correct the equation: $\frac{5}{10}=\frac{2}{10}+\frac{4}{10}$

   Correct: $\frac{5}{10}=\frac{2}{10}+\frac{3}{10}$. The sum $2+4=6 \neq 5$, so it should be $2+3=5$.

3. What is the missing numerator in $\frac{9}{12}=\frac{5}{12}+\frac{\square}{12}$?

   $4$. Since $9-5=4$, the missing numerator is $4$.

4. What is the missing numerator? $\frac{7}{9}=\frac{3}{9}+\frac{?}{9}$

   $4$. Since $3+4=7$, the missing numerator is $4$.

5. What equation shows decomposing $\frac{10}{12}$ into $\frac{7}{12}$ plus another fraction with denominator $12$?

   $\frac{10}{12}=\frac{7}{12}+\frac{3}{12}$. The numerators $7$ and $3$ sum to $10$.

6. Which equation is correct? A: $\frac{3}{8}=\frac{1}{8}+\frac{2}{8}$ or B: $\frac{3}{8}=\frac{1}{8}+\frac{2}{7}$

   A: $\frac{3}{8}=\frac{1}{8}+\frac{2}{8}$. Option A keeps the same denominator; B changes it incorrectly.

7. Find the missing fraction: $\frac{10}{11}=\frac{4}{11}+\frac{?}{11}$

   $\frac{6}{11}$. Since $4+6=10$, the missing fraction is $\frac{6}{11}$.

8. What equation shows decomposing $\frac{7}{10}$ into three fractions with denominator $10$: $\frac{2}{10}$, $\frac{2}{10}$, and what?

   $\frac{7}{10}=\frac{2}{10}+\frac{2}{10}+\frac{3}{10}$. The numerators $2+2+3=7$.

9. Identify the missing fraction: $\frac{15}{16}=\frac{7}{16}+\frac{5}{16}+\square$.

   $\frac{3}{16}$. Since $15-7-5=3$, the missing fraction is $\frac{3}{16}$.

10. What is the missing numerator in $\frac{8}{9}=\frac{3}{9}+\frac{\square}{9}$?

    $5$. Since $8-3=5$, the missing numerator is $5$.

11. Identify the error: $\frac{4}{9}=\frac{1}{9}+\frac{3}{8}$. What should the second fraction be?

    $\frac{3}{9}$. The denominator must stay $9$, so $\frac{3}{9}$ is correct.

12. Find the missing fraction: $\frac{13}{15}=\frac{7}{15}+\frac{?}{15}$

    $\frac{6}{15}$. Since $7+6=13$, the missing fraction is $\frac{6}{15}$.

13. What equation shows decomposing $\frac{13}{16}$ into $\frac{1}{16}$ plus another fraction with denominator $16$?

    $\frac{13}{16}=\frac{1}{16}+\frac{12}{16}$. The numerators $1$ and $12$ sum to $13$.

14. What does it mean to decompose a fraction using the same denominator?

    Write $\frac{a}{b}$ as a sum of fractions with denominator $b$. Break the numerator into parts while keeping the same denominator.

15. What equation shows decomposing $\frac{9}{10}$ into four fractions with denominator $10$: $\frac{2}{10}+\frac{2}{10}+\frac{2}{10}+\;$what?

    $\frac{9}{10}=\frac{2}{10}+\frac{2}{10}+\frac{2}{10}+\frac{3}{10}$. The numerators $2+2+2+3=9$.

16. What equation shows decomposing $\frac{7}{10}$ into $\frac{1}{10}$ plus another fraction with denominator $10$?

    $\frac{7}{10}=\frac{1}{10}+\frac{6}{10}$. The numerators $1$ and $6$ sum to $7$.

17. Which equation correctly decomposes $\frac{6}{11}$: A) $\frac{1}{11}+\frac{5}{11}$ or B) $\frac{1}{11}+\frac{6}{11}$?

    A) $\frac{1}{11}+\frac{5}{11}$. Option A sums to $\frac{6}{11}$; B exceeds it.

18. What equation shows decomposing $\frac{10}{12}$ into $\frac{5}{12}$ plus another fraction with denominator $12$?

    $\frac{10}{12}=\frac{5}{12}+\frac{5}{12}$. The numerators $5$ and $5$ sum to $10$.

19. What equation matches a bar divided into $8$ equal parts with $3$ shaded and $2$ shaded (same denominator)?

    $\frac{5}{8}=\frac{3}{8}+\frac{2}{8}$. Shows $5$ eighths as $3$ eighths plus $2$ eighths.

20. Which equation shows two different decompositions of the same fraction $\frac{6}{10}$?

    $\frac{6}{10}=\frac{1}{10}+\frac{5}{10}=\frac{3}{10}+\frac{3}{10}$. Shows two ways: $1+5$ and $3+3$ both equal $6$.

21. What equation matches splitting a bar into $7$ equal parts and shading $2$ parts and $5$ parts?

    $\frac{7}{7}=\frac{2}{7}+\frac{5}{7}$. Shows $7$ parts split into groups of $2$ and $5$.

22. What is the missing numerator in $\frac{11}{15}=\frac{\square}{15}+\frac{6}{15}$?

    $5$. Since $11-6=5$, the missing numerator is $5$.

23. What equation rule shows how to split $\frac{a}{b}$ into two parts with denominator $b$?

    $\frac{a}{b}=\frac{m}{b}+\frac{a-m}{b}$. Split numerator $a$ into $m$ and $(a-m)$ over the same denominator.

24. What is a correct decomposition of $\frac{5}{8}$ into unit fractions with denominator $8$?

    $\frac{5}{8}=\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}$. Decomposes into five unit fractions (each $\frac{1}{8}$).

25. What is the missing numerator? $\frac{11}{12}=\frac{5}{12}+\frac{?}{12}$

    $6$. Since $5+6=11$, the missing numerator is $6$.

26. What is one correct decomposition of $\frac{8}{10}$ into four fractions with denominator $10$?

    $\frac{8}{10}=\frac{2}{10}+\frac{2}{10}+\frac{2}{10}+\frac{2}{10}$. Splits $8$ into four equal parts of $2$ each.

27. Find and correct the error: $\frac{4}{9}=\frac{1}{9}+\frac{1}{9}+\frac{1}{9}$.

    Correct: $\frac{4}{9}=\frac{1}{9}+\frac{1}{9}+\frac{2}{9}$. The original sums to $\frac{3}{9}$, not $\frac{4}{9}$.

28. What is the missing numerator in $\frac{14}{20}=\frac{9}{20}+\frac{\square}{20}$?

    $5$. Since $14-9=5$, the missing numerator is $5$.

29. Which equation equals $\frac{12}{13}$: A) $\frac{5}{13}+\frac{7}{13}$ or B) $\frac{6}{13}+\frac{7}{12}$?

    A) $\frac{5}{13}+\frac{7}{13}$. Option A sums to $\frac{12}{13}$; B has different denominators.

30. What equation shows decomposing $\frac{13}{16}$ into $\frac{8}{16}$ plus another fraction with denominator $16$?

    $\frac{13}{16}=\frac{8}{16}+\frac{5}{16}$. The numerators $8$ and $5$ sum to $13$.

31. Which equation correctly decomposes $\frac{5}{8}$: A) $\frac{2}{8}+\frac{3}{8}$ or B) $\frac{2}{8}+\frac{3}{7}$?

    A) $\frac{2}{8}+\frac{3}{8}$. Option A keeps the same denominator; B changes it.

32. What is a correct decomposition of $\frac{3}{4}$ into unit fractions with denominator $4$?

    $\frac{3}{4}=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}$. Decomposes into three unit fractions (each $\frac{1}{4}$).

33. What equation shows decomposing $\frac{7}{10}$ into $\frac{3}{10}$ plus another fraction with denominator $10$?

    $\frac{7}{10}=\frac{3}{10}+\frac{4}{10}$. The numerators $3$ and $4$ sum to $7$.

34. What equation justifies that $\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$ for same denominators?

    $\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$. Add numerators when denominators are the same.

35. What is the missing numerator in $\frac{17}{18}=\frac{10}{18}+\frac{\square}{18}$?

    $7$. Since $17-10=7$, the missing numerator is $7$.

36. What is the missing numerator? $\frac{9}{10}=\frac{1}{10}+\frac{?}{10}$

    $8$. Since $1+8=9$, the missing numerator is $8$.

37. What does it mean to decompose a fraction like $\frac{a}{b}$ into a sum with the same denominator?

    Write $\frac{a}{b}$ as a sum of fractions with denominator $b$. Break the numerator into smaller parts, keeping the same denominator.

38. Which equation correctly decomposes $\frac{6}{8}$ into two fractions with denominator $8$?

    $\frac{6}{8}=\frac{2}{8}+\frac{4}{8}$. Correctly splits $6$ into $2+4$ over denominator $8$.

39. What equation shows decomposing $\frac{10}{12}$ into three fractions with denominator $12$: $\frac{4}{12}$, $\frac{3}{12}$, and what?

    $\frac{10}{12}=\frac{4}{12}+\frac{3}{12}+\frac{3}{12}$. The numerators $4+3+3=10$.

40. What is the missing numerator? $\frac{12}{13}=\frac{9}{13}+\frac{?}{13}$

    $3$. Since $9+3=12$, the missing numerator is $3$.

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