Mixed Operations with Fractions worksheet for practicing fraction arithmetic with missing values.

Here are the solutions to fill in the missing boxes for the worksheet.

Section A

*Fill in the missing boxes to help you find the sums.*

2) $\frac{1}{2} + \frac{1}{3}$
* To get a denominator of 6: multiply $\frac{1}{2}$ by 3 ($\frac{3}{6}$) and $\frac{1}{3}$ by 2 ($\frac{2}{6}$).
* Add them: $3 + 2 = 5$.
* Boxes: Top row: 3, 2. Bottom box: 5, 6.

3) $\frac{3}{4} - \frac{1}{3}$
* Common denominator is 12.
* $\frac{3}{4}$ becomes $\frac{9}{12}$ (multiply by 3).
* $\frac{1}{3}$ becomes $\frac{4}{12}$ (multiply by 4).
* Subtract: $9 - 4 = 5$.
* Boxes: Top row: 9, 12, 4, 12. Bottom box: 5, 12.

4) $\frac{5}{11} + \dots = \frac{101}{77}$
* The final denominator is 77, so we convert $\frac{5}{11}$ to $\frac{35}{77}$ (multiply by 7).
* We need to find what adds to 35 to get 101. $101 - 35 = 66$.
* So the second fraction is $\frac{66}{77}$. This simplifies to $\frac{6}{7}$.
* Boxes: First fraction bottom: 7. Middle fractions: 35, 77, 66, 77.

5) $\frac{7}{13} - \dots = \frac{19}{65}$
* The final denominator is 65, so we convert $\frac{7}{13}$ to $\frac{35}{65}$ (multiply by 5).
* We need to find what subtracts from 35 to get 19. $35 - 19 = 16$.
* So the second fraction is $\frac{16}{65}$.
* Boxes: First fraction bottom: 5. Middle fractions: 35, 65, 16, 65.

6) $\dots - \dots = \frac{1}{48}$
* The result is negative, meaning the second number is bigger.
* We see the second part ends with $\frac{28}{48}$.
* To get a result of $-\frac{1}{48}$, the first numerator must be 27 (since $27 - 28 = -1$).
* The first fraction is $\frac{27}{48}$, which simplifies to $\frac{9}{16}$.
* Boxes: First fraction: 9, 16. Second fraction top: 27. Third fraction top: 27.

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Section B

*Convert mixed numbers to improper fractions.*

2) $5\frac{2}{3} + 2\frac{1}{4}$
* Convert to improper: $5\frac{2}{3} = \frac{17}{3}$ and $2\frac{1}{4} = \frac{9}{4}$.
* Common denominator is 12. $\frac{17}{3} = \frac{68}{12}$ and $\frac{9}{4} = \frac{27}{12}$.
* Add: $68 + 27 = 95$. Result: $\frac{95}{12}$.
* Mixed number: $12 \times 7 = 84$, remainder 11. So, $7\frac{11}{12}$.
* Boxes: Top row: 17, 9. Middle row: 68, 27. Bottom improper: 95. Mixed number: 7, 11.

3) $3\dots - \frac{1}{6} = 3\frac{19}{30}$
* The answer has a denominator of 30. The common denominator used was 30.
* Working backward from the answer $\frac{109}{30}$ ($3\frac{19}{30}$):
* The subtraction line shows $\frac{\square}{30} - \frac{5}{30}$.
* To get 109 on top, the first numerator must be 114 ($114 - 5 = 109$).
* $\frac{114}{30}$ simplifies to $\frac{19}{5}$ or $3\frac{4}{5}$.
* Boxes: First mixed number fraction: 4, 5. Improper fraction top: 19. Subtraction numerators: 114, 5. Final improper: 109.

4) $\dots - 1\frac{5}{7} = 4\frac{13}{21}$
* Final answer $\frac{97}{21}$. Denominator is 21.
* Convert $1\frac{5}{7}$ to $\frac{12}{7}$, then to $\frac{36}{21}$.
* Working backward: $\square - 36 = 97$. The first numerator is 133.
* $\frac{133}{21}$ simplifies to $\frac{19}{3}$ or $6\frac{1}{3}$.
* Boxes: First mixed number: 6, 1, 3. Improper: 19. Subtraction: 133, 21, 36, 21. Final mixed: 4, 13, 21.

5) $4\dots + \dots\frac{1}{8} = 6\frac{25}{72}$
* Final answer $\frac{457}{72}$ ($6\frac{25}{72}$). Denominator is 72.
* The problem gives one sum as $\frac{304}{72}$.
* Find the other part: $457 - 304 = 153$. So the second fraction is $\frac{153}{72}$.
* $\frac{304}{72}$ simplifies to $\frac{38}{9}$ or $4\frac{2}{9}$.
* $\frac{153}{72}$ simplifies to $\frac{17}{8}$ or $2\frac{1}{8}$.
* Boxes: First mixed fraction: 2, 9. Improper: 38. Second mixed whole: 2. Improper: 17. Sum improper: 153, 72. Total improper: 457.

6) $\frac{9}{\square} - 3\dots = -2\frac{46}{65}$
* Final answer is $-\frac{176}{65}$. Denominator is 65.
* The subtraction shows $\frac{\square}{65} - \frac{\square}{65}$.
* The second term is a mixed number $3\frac{\square}{5}$. Let's assume the first term is a simple fraction over 13 (since $13 \times 5 = 65$).
* If the first term is $\frac{9}{13}$, it becomes $\frac{45}{65}$.
* Equation: $45 - \text{numerator} = -176$. Numerator $= 221$.
* $\frac{221}{65}$ simplifies to $\frac{17}{5}$ or $3\frac{2}{5}$.
* Boxes: First denominator: 13. Mixed fraction: 2, 5. First improper top: 45. Second improper top: 221. Result improper: -176.

Final Answer:
Section A
2) 3, 2, 5/6
3) 9/12, 4/12, 5/12
4) 7, 35/77, 66/77
5) 5, 35/65, 16/65
6) 9/16, 27/48, 27

Section B
2) 17, 9, 68, 27, 95, 7 11/12
3) 4/5, 19, 114, 5, 109
4) 6 1/3, 19, 133/21, 36/21, 4 13/21
5) 2/9, 38, 2, 17, 153/72, 457
6) 13, 2/5, 45, 221, -176