Examples with solutions for Division of Fractions: In combination with other operations

Exercise #1

Complete the following exercise:

14:12+14=? \frac{1}{4}:\frac{1}{2}+\frac{1}{4}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem 14:12+14 \frac{1}{4} : \frac{1}{2} + \frac{1}{4} , follow these steps:

Step 1: Perform the division 14:12 \frac{1}{4} : \frac{1}{2} .

  • When dividing by a fraction, multiply by the reciprocal. Hence, 14:12 \frac{1}{4} : \frac{1}{2} becomes 14×21 \frac{1}{4} \times \frac{2}{1} .
  • Multiply the numerators: 1×2=2 1 \times 2 = 2 .
  • Multiply the denominators: 4×1=4 4 \times 1 = 4 .
  • So, 14×21=24=12 \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2} after simplification.

Step 2: Now add the result from Step 1 to 14\frac{1}{4}.

  • We need to add 12+14 \frac{1}{2} + \frac{1}{4} .
  • Convert 12\frac{1}{2} to have a common denominator with 14\frac{1}{4}. 12=24\frac{1}{2} = \frac{2}{4}.
  • Add the fractions: 24+14=34\frac{2}{4} + \frac{1}{4} = \frac{3}{4}.

Therefore, the solution to the problem is 34 \frac{3}{4} .

Answer

34 \frac{3}{4}

Exercise #2

Complete the following exercise:

23:34+19=? \frac{2}{3}:\frac{3}{4}+\frac{1}{9}=\text{?}

Video Solution

Step-by-Step Solution

To solve this fraction problem, follow these steps:

  • Step 1: Evaluate the Division:
    The expression starts with 23÷34 \frac{2}{3} \div \frac{3}{4} . To divide fractions, multiply by the reciprocal:
    23×43=2×43×3=89 \frac{2}{3} \times \frac{4}{3} = \frac{2 \times 4}{3 \times 3} = \frac{8}{9}
  • Step 2: Add Fractions:
    Now add 89 \frac{8}{9} to 19 \frac{1}{9} . Since the denominators are the same, add the numerators directly:
    89+19=8+19=99=1 \frac{8}{9} + \frac{1}{9} = \frac{8 + 1}{9} = \frac{9}{9} = 1

Thus, the solution to the expression 23:34+19 \frac{2}{3}:\frac{3}{4} + \frac{1}{9} is 1 1 .

Answer

1 1

Exercise #3

Complete the following exercise:

12:1214=? \frac{1}{2}:\frac{1}{2}-\frac{1}{4}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem, 12:1214\frac{1}{2}:\frac{1}{2}-\frac{1}{4}, follow these steps:

  • Step 1: First, interpret the division 12:12\frac{1}{2} : \frac{1}{2} as multiplication by the reciprocal. This becomes 12×21\frac{1}{2} \times \frac{2}{1}.
  • Step 2: Perform the multiplication: 12×21=1×22×1=22=1. \frac{1}{2} \times \frac{2}{1} = \frac{1 \times 2}{2 \times 1} = \frac{2}{2} = 1.
  • Step 3: Next, subtract 14\frac{1}{4} from 1: 114=4414=34. 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}.

Therefore, the solution to the problem is 34\frac{3}{4}.

Answer

34 \frac{3}{4}

Exercise #4

Solve the following exercise:

35:56+15=? \frac{3}{5}:\frac{5}{6}+\frac{1}{5}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem, we'll follow the outlined steps:

  • Step 1: Calculate the division 35:56 \frac{3}{5} : \frac{5}{6} .
  • Step 2: Add 15 \frac{1}{5} to the result from Step 1.
  • Step 3: Simplify the resulting fraction.

Now, let's work through each step:

Step 1: To divide 35 \frac{3}{5} by 56 \frac{5}{6} , we multiply by the reciprocal of 56 \frac{5}{6} . This gives us:

35×65=3×65×5=1825 \frac{3}{5} \times \frac{6}{5} = \frac{3 \times 6}{5 \times 5} = \frac{18}{25}

Step 2: Now, add 15 \frac{1}{5} to 1825 \frac{18}{25} . First, we convert 15 \frac{1}{5} to the same denominator as 1825 \frac{18}{25} :

15=525 \frac{1}{5} = \frac{5}{25}

Step 3: Add 1825 \frac{18}{25} and 525 \frac{5}{25} :

1825+525=18+525=2325 \frac{18}{25} + \frac{5}{25} = \frac{18 + 5}{25} = \frac{23}{25}

Thus, the solution to the problem is 2325\frac{23}{25}.

Answer

2325 \frac{23}{25}

Exercise #5

Solve the following exercise:

34:54+12=? \frac{3}{4}:\frac{5}{4}+\frac{1}{2}=\text{?}

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Calculate the division 34:54 \frac{3}{4} : \frac{5}{4} .
  • Step 2: Use the formula ab:cd=ab×dc\frac{a}{b} : \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.
  • Step 3: Simplify the division result.
  • Step 4: Add 12 \frac{1}{2} to the result of the division.
  • Step 5: Simplify the addition result to get the final answer.

Now, let's work through each step:

Step 1: We need to calculate 34:54 \frac{3}{4} : \frac{5}{4} . Using the division of fractions formula, this becomes:

34×45=3×44×5=1220 \frac{3}{4} \times \frac{4}{5} = \frac{3 \times 4}{4 \times 5} = \frac{12}{20} .

Step 2: Simplify 1220 \frac{12}{20} . Divide the numerator and the denominator by their greatest common divisor, which is 4:

12÷420÷4=35 \frac{12 \div 4}{20 \div 4} = \frac{3}{5} .

Step 3: Add 12 \frac{1}{2} to the result 35\frac{3}{5}:

The common denominator for addition is 10. Therefore:

35=3×25×2=610 \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} and 12=1×52×5=510 \frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10} .

Add these two fractions:

610+510=6+510=1110 \frac{6}{10} + \frac{5}{10} = \frac{6 + 5}{10} = \frac{11}{10} .

Therefore, the solution to the problem is 1110 \frac{11}{10} .

Answer

1110 \frac{11}{10}

Exercise #6

5+472= ? 5+\frac{\frac{4}{7}}{2}=\text{ ?}

Video Solution

Step-by-Step Solution

To simplify the fraction exercise, we will multiply 47 \frac{4}{7} by 12 \frac{1}{2} .

We will then rearrange the exercise accordingly and following the order of operations rules, we will first solve the multiplication exercise:

5+47×12= 5+\frac{4}{7}\times\frac{1}{2}=

Note that in the multiplication exercise, we can reduce 4 in the numerator and 2 in the denominator by 2:

5+27×11=5+27+1 5+\frac{2}{7}\times\frac{1}{1}=5+\frac{2}{7}+1

Finally we will combine the whole numbers to get:

5+1+27=627 5+1+\frac{2}{7}=6\frac{2}{7}

Answer

627 6\frac{2}{7}

Exercise #7

1072+278= \frac{\frac{10}{7}}{2}+\frac{2}{\frac{7}{8}}=

Video Solution

Step-by-Step Solution

To solve the expression 1072+278 \frac{\frac{10}{7}}{2}+\frac{2}{\frac{7}{8}} , we need to perform operations in the correct order as per the rules of the order of operations (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Step 1: Simplify the complex fraction 1072 \frac{\frac{10}{7}}{2}
A complex fraction can be simplified by multiplying the numerator by the reciprocal of the denominator. In this case, the numerator is 107 \frac{10}{7} and the denominator is 2 (which means 21 \frac{2}{1} ).


1072=107×12=10172=1014 \frac{\frac{10}{7}}{2} = \frac{10}{7} \times \frac{1}{2} = \frac{10 \cdot 1}{7 \cdot 2} = \frac{10}{14}


Simplify 1014 \frac{10}{14} by dividing both the numerator and the denominator by their greatest common divisor (2):


1014=57 \frac{10}{14} = \frac{5}{7}

Step 2: Simplify the complex fraction 278 \frac{2}{\frac{7}{8}}
Again, multiply the numerator by the reciprocal of the denominator:
The reciprocal of 78 \frac{7}{8} is 87 \frac{8}{7} .


278=2×87=287=167 \frac{2}{\frac{7}{8}} = 2 \times \frac{8}{7} = \frac{2 \cdot 8}{7} = \frac{16}{7}

Step 3: Add the simplified fractions 57+167 \frac{5}{7} + \frac{16}{7}
Since the fractions have like denominators, we can add the numerators directly:


57+167=5+167=217 \frac{5}{7} + \frac{16}{7} = \frac{5 + 16}{7} = \frac{21}{7}


Simplify 217 \frac{21}{7} by dividing the numerator by the denominator:


217=3 \frac{21}{7} = 3

Thus, the solution to the expression is 3 3 .

Answer

3 3

Exercise #8

35910+7913= \frac{\frac{3}{5}}{\frac{9}{10}}+\frac{\frac{7}{9}}{\frac{1}{3}}=

Video Solution

Step-by-Step Solution

To solve the expression 35910+7913 \frac{\frac{3}{5}}{\frac{9}{10}}+\frac{\frac{7}{9}}{\frac{1}{3}} , we need to apply the division of fractions and simplify the resulting expressions.

First, consider the expression 35910 \frac{\frac{3}{5}}{\frac{9}{10}} :

  • When dividing by a fraction, multiply by its reciprocal. The reciprocal of 910 \frac{9}{10} is 109 \frac{10}{9} .
  • Therefore, 35910=35×109 \frac{\frac{3}{5}}{\frac{9}{10}} = \frac{3}{5} \times \frac{10}{9} .
  • Multiplying the numerators and the denominators, we get 3×105×9=3045 \frac{3 \times 10}{5 \times 9} = \frac{30}{45} .
  • Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 15: 30÷1545÷15=23 \frac{30 \div 15}{45 \div 15} = \frac{2}{3} .

Next, consider the expression 7913 \frac{\frac{7}{9}}{\frac{1}{3}} :

  • The reciprocal of 13 \frac{1}{3} is 31 \frac{3}{1} .
  • Therefore, 7913=79×31 \frac{\frac{7}{9}}{\frac{1}{3}} = \frac{7}{9} \times \frac{3}{1} .
  • Multiplying the numerators and the denominators, we get 7×39×1=219 \frac{7 \times 3}{9 \times 1} = \frac{21}{9} .
  • Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3: 21÷39÷3=73 \frac{21 \div 3}{9 \div 3} = \frac{7}{3} .

Now add the simplified fractions: 23+73 \frac{2}{3} + \frac{7}{3} .

  • The fractions have a common denominator, 3, so we can simply add the numerators: 2+73=93 \frac{2 + 7}{3} = \frac{9}{3} .
  • Simplify 93 \frac{9}{3} by dividing both the numerator and the denominator by 3: 9÷33÷3=3 \frac{9 \div 3}{3 \div 3} = 3 .

Therefore, the final solution to the expression is 3 3 .

Answer

3 3

Exercise #9

Solve the following problem:

3121316= 3\frac{1}{2}-\frac{\frac{1}{3}}{\frac{1}{6}}=

Video Solution

Step-by-Step Solution

When we are presented with a fraction over a fraction (in this case one-third over one-sixth) We can convert it into a more manageable form.

1/3:1/6 1/3 : 1/6

It's important to remember that a fraction is actually another sign of division, hence the given exercise is in fact equivalent to one-third divided by one-sixth.
When dealing with division of fractions, the easiest method for solving them is by performing "multiplication by the reciprocal" as shown below:

1/3×6/1 1/3\times6/1

Multiply the numerator by the numerator and the denominator by the denominator to obtain the following result:

63 \frac{6}{3}

Which when reduced equals

21 \frac{2}{1}

Now let's return to the original exercise. In order to solve it we need to take the mixed fraction and convert it to an improper fraction.
We can achieve this by simply moving the whole numbers back to the numerator.

To do this we'll multiply the whole number by the denominator and then proceed to add it to the numerator

3×2=6 3\times2=6

6+1=7 6+1=7

Therefore the resulting fraction is:

72 \frac{7}{2}

We want to proceed to perform the subtraction exercise.
When both fractions have the same denominator we subtract them.
Therefore in order to achieve this we'll expand the fraction 21 \frac{2}{1} to a denominator of 2, and obtain the following:

42 \frac{4}{2}

We can now proceed to perform subtraction -

7242=32 \frac{7}{2}-\frac{4}{2}=\frac{3}{2}

Convert this back to a mixed fraction in order to obtain the following result:

Answer

112 1\frac{1}{2}