Mixed Operations with Fractions Worksheet | Fun and Engaging 6th ... - Free Printable

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

Section A

2) $\frac{1}{2} + \frac{1}{3}$
* Common denominator is 6.
* $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$.
* Top row boxes: 3, 2
* Sum: $3+2=5$.
* Bottom row boxes: 5, 6

3) $\frac{3}{4} - \frac{1}{3}$
* Common denominator is 12.
* $\frac{3}{4} = \frac{9}{12}$ and $\frac{1}{3} = \frac{4}{12}$.
* Top row boxes: 9, 12, 4, 12
* Difference: $9-4=5$.
* Bottom row boxes: 5, 12

4) $\frac{5}{11} + \text{[box]}$ resulting in $\frac{101}{77}$
* The final denominator is 77, so the common denominator is 77 ($11 \times 7$).
* First fraction: $\frac{5}{11} = \frac{35}{77}$. So the second fraction must have a denominator of 7.
* Numerator check: $101 - 35 = 66$. So the second fraction is $\frac{66}{77}$, which simplifies to $\frac{?}{7}$. Wait, looking at the layout: $\frac{5}{11} + \frac{\square}{\square} = \frac{\square}{\square} + \frac{\square}{\square}$.
* Let's find the missing addend first. Total is $\frac{101}{77}$. First part is $\frac{5}{11} = \frac{35}{77}$. Missing part is $\frac{101-35}{77} = \frac{66}{77} = \frac{6}{7}$.
* So the second term is $\frac{6}{7}$.
* First line boxes (left): 6, 7
* First line boxes (right - converted): $\frac{35}{77} + \frac{66}{77}$. Boxes: 35, 77, 66, 77.

5) $\frac{7}{13} - \text{[box]}$ resulting in $\frac{19}{65}$
* Final denominator is 65 ($13 \times 5$).
* First fraction: $\frac{7}{13} = \frac{35}{65}$.
* Equation: $\frac{35}{65} - \frac{x}{65} = \frac{19}{65}$.
* $35 - x = 19 \rightarrow x = 16$.
* The subtracted fraction is $\frac{16}{65}$. Since 16 and 65 share no factors, the original fraction was likely $\frac{16}{65}$ or derived from a simpler denominator? Let's look at the box structure: $\frac{7}{13} - \frac{\square}{\square}$. If the result is $\frac{19}{65}$, the second fraction converts to denominator 65.
* Let's assume the second fraction has a denominator of 5 (since $13 \times 5 = 65$).
* $\frac{7}{13} - \frac{y}{5} = \frac{35}{65} - \frac{13y}{65} = \frac{19}{65}$.
* $35 - 13y = 19 \rightarrow 13y = 16$. This doesn't yield an integer.
* Let's re-read the structure. It might just be finding equivalent fractions.
* $\frac{7}{13} - \frac{\square}{\square} = \frac{\square}{\square} - \frac{\square}{\square} = \frac{19}{65}$.
* Maybe the second fraction is $\frac{16}{65}$ directly? No, usually these simplify.
* Let's try denominator 1? No.
* Let's look at problem 6 for a clue. Problem 6 ends in 48.
* Let's re-evaluate Problem 5. Is it possible the second fraction is $\frac{16}{65}$? If so, boxes are 16, 65. Then right side: $\frac{35}{65} - \frac{16}{65}$.
* However, typically the "missing box" fraction is simpler. What if the denominator is not 5?
* Actually, let's look at the result $\frac{19}{65}$.
* $\frac{7}{13} = \frac{35}{65}$.
* $\frac{35}{65} - \frac{16}{65} = \frac{19}{65}$.
* The fraction being subtracted is $\frac{16}{65}$. It cannot be simplified further.
* Left boxes: 16, 65
* Right boxes: 35, 65, 16, 65

6) $\text{[box]} - \text{[box]} = \text{[box]} - \frac{28}{48} = -\frac{1}{48}$
* Work backward. Result is $-\frac{1}{48}$.
* Equation: $X - \frac{28}{48} = -\frac{1}{48}$.
* $X = \frac{28}{48} - \frac{1}{48} = \frac{27}{48}$.
* So the first part of the equation equals $\frac{27}{48}$.
* Simplify $\frac{27}{48}$: Divide by 3 $\rightarrow \frac{9}{16}$.
* So we need two fractions that subtract to $\frac{9}{16}$.
* Looking at the denominators on the right side ($\frac{28}{48}$), the common denominator used is 48.
* $\frac{27}{48}$ comes from a fraction with a denominator that divides 48. 16 divides 48.
* So the first fraction is likely $\frac{9}{16}$.
* What is the second fraction? We established the calculation is $\frac{27}{48} - \frac{28}{48}$.
* The prompt shows: $\frac{\square}{\square} - \frac{\square}{\square} = \frac{\square}{\square} - \frac{28}{48}$.
* This implies the first two boxes are the unsimplified versions or the original problem.
* If the result is negative, the second number is larger.
* Let's assume the original problem was $\frac{9}{16} - \frac{7}{12}$?
* $\frac{9}{16} = \frac{27}{48}$.
* $\frac{7}{12} = \frac{28}{48}$.
* This fits perfectly.
* Left boxes: 9, 16, 7, 12
* Right boxes (first fraction converted): 27, 48

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Section B

2) $5\frac{2}{3} + 2\frac{1}{4}$
* Convert to improper fractions:
* $5\frac{2}{3} = \frac{17}{3}$
* $2\frac{1}{4} = \frac{9}{4}$
* Top row boxes: 17, 9
* Common denominator is 12.
* $\frac{17}{3} = \frac{68}{12}$
* $\frac{9}{4} = \frac{27}{12}$
* Middle row boxes: 68, 27
* Sum: $68 + 27 = 95$.
* Bottom left box: 95
* Convert back to mixed number: $95 \div 12 = 7$ with remainder $11$.
* Bottom right boxes: 7, 11

3) $3\frac{\square}{\square} - \frac{1}{6} = \dots = 3\frac{19}{30}$
* Work backward from the answer $3\frac{19}{30}$.
* $3\frac{19}{30} = \frac{109}{30}$.
* The equation is $X - \frac{1}{6} = \frac{109}{30}$.
* $X = \frac{109}{30} + \frac{1}{6} = \frac{109}{30} + \frac{5}{30} = \frac{114}{30}$.
* Simplify $\frac{114}{30}$: Divide by 6 $\rightarrow \frac{19}{5}$.
* $\frac{19}{5} = 3\frac{4}{5}$.
* So the starting mixed number is $3\frac{4}{5}$.
* Top left boxes: 4, 5
* Convert to improper fraction for the next step: $3\frac{4}{5} = \frac{19}{5}$.
* Top right box (numerator): 19 (Denominator is already 5? No, the image shows $\frac{\square}{\square} - \frac{1}{6}$ then below $\frac{\square}{30} - \frac{\square}{30}$).
* Let's trace the boxes carefully.
* Line 1: $3\frac{4}{5} - \frac{1}{6} = \frac{19}{5} - \frac{1}{6}$. Box above 5 is 19.
* Line 2: Common denom 30. $\frac{19}{5} = \frac{114}{30}$. $\frac{1}{6} = \frac{5}{30}$.
* Boxes: 114, 5.
* Line 3: $114 - 5 = 109$. Result $\frac{109}{30}$.
* Box: 109.

4) $\text{[box]} - 1\frac{5}{7} = \dots = \frac{97}{21}$
* Answer is $\frac{97}{21}$.
* $1\frac{5}{7} = \frac{12}{7} = \frac{36}{21}$.
* $X - \frac{36}{21} = \frac{97}{21}$.
* $X = \frac{97+36}{21} = \frac{133}{21}$.
* Simplify $\frac{133}{21}$: Both divisible by 7. $133 \div 7 = 19$. $21 \div 7 = 3$.
* $X = \frac{19}{3} = 6\frac{1}{3}$.
* Top left boxes (mixed number): 6, 1, 3
* Convert to improper: $\frac{19}{3}$.
* Line 2 left boxes: 19, 3
* Line 2 right boxes (converted subtrahend): The subtrahend is $1\frac{5}{7} = \frac{12}{7}$. To get denominator 21, multiply by 3. $\frac{36}{21}$. Boxes: 36, 21.
* Wait, the template shows: $\frac{\square}{\square} - \frac{\square}{21}$. The denominator 21 is pre-filled for the second fraction? No, looking at crop 4, it shows $\frac{\square}{\square} - \frac{\square}{7}$ on line 1? No, it shows $\square \frac{\square}{\square} - 1\frac{5}{7}$.
* Line 2: $= \frac{\square}{\square} - \frac{\square}{\square}$.
* Line 3: $= \frac{\square}{\square} - \frac{\square}{\square}$. (This is the common denom step).
* Line 4: $= \frac{97}{21}$.
* So, Line 3 must be $\frac{133}{21} - \frac{36}{21}$.
* Line 2 must be the improper conversions: $\frac{19}{3} - \frac{12}{7}$.
* Top boxes: 6, 1, 3
* Line 2 boxes: 19, 3, 12, 7
* Line 3 boxes: 133, 21, 36, 21
* Final mixed number: $\frac{97}{21} = 4\frac{13}{21}$.
* Bottom boxes: 4, 13, 21

5) $4\frac{\square}{\square} + \square\frac{1}{8} = \dots = 6\frac{25}{72}$
* Final answer: $6\frac{25}{72} = \frac{457}{72}$.
* One term is $\square\frac{1}{8}$. Let's call the whole number part $B$. So $B\frac{1}{8} = \frac{8B+1}{8}$.
* Other term is $4\frac{A}{C}$.
* The intermediate step shows $\frac{304}{\square} + \frac{\square}{\square}$.
* Denominator is 72.
* $\frac{304}{72}$ simplifies? $304 \div 8 = 38$. $72 \div 8 = 9$. So $\frac{38}{9}$.
* $\frac{38}{9} = 4\frac{2}{9}$.
* So the first mixed number is $4\frac{2}{9}$.
* Top left boxes: 2, 9
* Now find the second number.
* Total sum numerator is 457. First part numerator is 304.
* Second part numerator: $457 - 304 = 153$.
* So the second fraction is $\frac{153}{72}$.
* Simplify $\frac{153}{72}$: Divide by 9 $\rightarrow \frac{17}{8}$.
* $\frac{17}{8} = 2\frac{1}{8}$.
* So the second mixed number is $2\frac{1}{8}$.
* Top middle box (whole number): 2
* Line 2 boxes: Denom is 72 (from 304/__). Second fraction is $\frac{153}{72}$. Boxes: 72, 153, 72.
* Line 3 box (sum numerator): 457.

6) $\frac{9}{\square} - 3\frac{\square}{5} = \dots = -2\frac{46}{65}$
* Final answer: $-2\frac{46}{65} = -\frac{176}{65}$.
* Equation: $A - B = -\frac{176}{65}$.
* Look at the intermediate step: $\frac{9}{\square} - \frac{\square}{5} = \frac{\square}{65} - \frac{\square}{65}$.
* The first term is $\frac{9}{\text{something}}$. The common denominator is 65.
* 65 is $5 \times 13$. So the denominator of the first fraction is likely 13.
* If first fraction is $\frac{9}{13}$, then $\frac{9}{13} = \frac{45}{65}$.
* So $\frac{45}{65} - \frac{Y}{65} = -\frac{176}{65}$.
* $45 - Y = -176 \rightarrow Y = 221$.
* So the second fraction (improper) is $\frac{221}{65}$.
* Check if this matches $3\frac{\square}{5}$.
* $\frac{221}{65}$ simplifies? $221 \div 13 = 17$. $65 \div 13 = 5$. So $\frac{17}{5}$.
* $\frac{17}{5} = 3\frac{2}{5}$.
* This fits the format $3\frac{\square}{5}$. The numerator is 2.
* Top left box (denom): 13
* Top right box (numerator): 2
* Line 2 boxes: First fraction becomes $\frac{45}{65}$. Second fraction becomes $\frac{221}{65}$.
* Left side: $\frac{9}{13} - \frac{17}{5}$.
* Right side boxes: 45, 221.
* Line 3 box (result numerator): $45 - 221 = -176$. Box: -176 (or just 176 if the negative sign is outside, but the final answer has a negative sign. The box is just a square. Usually, you put the absolute value if the sign is pre-printed, but here the final answer is $-2\dots$. The intermediate line is $= \frac{\square}{65}$. I will provide -176 to be safe, or note the magnitude). Given the final answer is negative, the numerator is -176.

Final Answer:
Section A
2) Top: 3, 2 | Bottom: 5, 6
3) Top: 9, 12, 4, 12 | Bottom: 5, 12
4) Left: 6, 7 | Right: 35, 77, 66, 77
5) Left: 16, 65 | Right: 35, 65, 16, 65
6) Left: 9, 16, 7, 12 | Right: 27, 48

Section B
2) Top: 17, 9 | Middle: 68, 27 | Bottom Left: 95 | Bottom Right: 7, 11
3) Top Left: 4, 5 | Top Right: 19 | Middle: 114, 5 | Bottom: 109
4) Top: 6, 1, 3 | Row 2: 19, 3, 12, 7 | Row 3: 133, 21, 36, 21 | Bottom: 4, 13, 21
5) Top Left: 2, 9 | Top Mid: 2 | Row 2: 72, 153, 72 | Row 3: 457
6) Top Left: 13 | Top Right: 2 | Row 2: 45, 221 | Row 3: -176