Examples with solutions for All Operations in Mixed Fractions: Dividing whole numbers by fractions

Exercise #1

2:23= 2:\frac{2}{3}=

Video Solution

Step-by-Step Solution

To solve the expression 2:232:\frac{2}{3}, follow these steps:

  • Step 1: Rewrite the expression as a division problem:
    This means 2÷232 \div \frac{2}{3}.
  • Step 2: Convert the division to a multiplication by using the reciprocal:
    The reciprocal of 23\frac{2}{3} is 32\frac{3}{2}.
  • Step 3: Multiply by the reciprocal:
    2×32=2321=62=32 \times \frac{3}{2} = \frac{2 \cdot 3}{2 \cdot 1} = \frac{6}{2} = 3.

Therefore, the solution to the problem is 3 3 .

Answer

3 3

Exercise #2

1:14= 1:\frac{1}{4}=

Video Solution

Step-by-Step Solution

To solve the division problem 1:14 1 : \frac{1}{4} , we will follow these steps:

  • Step 1: Express the division as a fraction operation: 1÷14 1 \div \frac{1}{4} .
  • Step 2: Use the invert-and-multiply rule. Find the reciprocal of 14\frac{1}{4}, which is 44.
  • Step 3: Multiply the whole number by the reciprocal: 1×4=4 1 \times 4 = 4 .

Thus, after performing these operations, we find that the result of the division 1:14 1 : \frac{1}{4} is 4 4 .

Answer

4 4

Exercise #3

4:35= 4:\frac{3}{5}=

Video Solution

Step-by-Step Solution

To solve the problem 4:35 4:\frac{3}{5} , follow these steps:

  • Step 1: Convert the division into a multiplication by using the reciprocal of 35\frac{3}{5}. The reciprocal is 53\frac{5}{3}.
  • Step 2: Multiply the whole number 4 by 53\frac{5}{3}:
    4×53=4×53=203 4 \times \frac{5}{3} = \frac{4 \times 5}{3} = \frac{20}{3}
  • Step 3: Simplify 203\frac{20}{3} to a mixed number:
    Perform the division 20÷320 \div 3, which gives 6 as a whole number and leaves a remainder of 2.
  • Thus, 203\frac{20}{3} as a mixed number is 623 6\frac{2}{3} .

Therefore, the solution to the problem is 623 6\frac{2}{3} .

Answer

623 6\frac{2}{3}

Exercise #4

3:34= 3:\frac{3}{4}=

Video Solution

Step-by-Step Solution

To solve the problem 3:34 3:\frac{3}{4} , we must perform division of the whole number 3 by the fraction 34\frac{3}{4}. Here are the steps:

  • Step 1: Recall the rule for dividing by a fraction. Dividing by 34\frac{3}{4} is the same as multiplying by its reciprocal, 43\frac{4}{3}.
  • Step 2: Rewrite the expression as a multiplication problem: 3×433 \times \frac{4}{3}.
  • Step 3: Perform the multiplication: 3×43=3×43=1233 \times \frac{4}{3} = \frac{3 \times 4}{3} = \frac{12}{3}.
  • Step 4: Simplify the fraction: 123=4\frac{12}{3} = 4.

The solution to the division 3:34 3:\frac{3}{4} is 4 4 .

Answer

4 4

Exercise #5

5:25= 5:\frac{2}{5}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll perform the division 5÷25 5 \div \frac{2}{5} by converting it into multiplication:

  • Step 1: Recognize that dividing by a fraction is the same as multiplying by its reciprocal.
  • Step 2: Convert the division into multiplication: 5÷25=5×52 5 \div \frac{2}{5} = 5 \times \frac{5}{2} .
  • Step 3: Multiply the whole number by the reciprocal of the fraction: 5×52=252 5 \times \frac{5}{2} = \frac{25}{2} .
  • Step 4: Convert the improper fraction to a mixed number: 252=1212 \frac{25}{2} = 12 \frac{1}{2} .

Through these steps, we find that the solution to the division problem is 1212 12 \frac{1}{2} .

Answer

1212 12\frac{1}{2}

Exercise #6

7:78= 7:\frac{7}{8}=

Video Solution

Step-by-Step Solution

To solve this problem, we will follow these steps:

  • Step 1: Rewrite the division as multiplication by the reciprocal of the fraction.
  • Step 2: Perform the multiplication calculation.

Now, let's work through each step:
Step 1: The given problem is 7:78 7:\frac{7}{8} , which means 7÷78 7 \div \frac{7}{8} .
Instead of dividing, multiply by the reciprocal:
7÷78=7×87 7 \div \frac{7}{8} = 7 \times \frac{8}{7} .

Step 2: Perform the multiplication:
7×87=7×87 7 \times \frac{8}{7} = \frac{7 \times 8}{7} .
The 77 in the numerator and denominator cancel each other out, resulting in:
567=8 \frac{56}{7} = 8 .

Therefore, the solution to the problem is 8 8 .

Answer

8 8

Exercise #7

3:12= 3:\frac{1}{2}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Identify the reciprocal of the divisor.
  • Step 2: Multiply the dividend by the reciprocal.

Now, let's work through each step:
Step 1: The divisor is 12 \frac{1}{2} . The reciprocal of 12 \frac{1}{2} is 2.

Step 2: Multiply the dividend, which is 3, by the reciprocal of the divisor:
3×2=6 3 \times 2 = 6

Therefore, the solution to the problem is 6 6 .

Answer

6 6

Exercise #8

3:23= 3:\frac{2}{3}=

Video Solution

Step-by-Step Solution

We need to find the value of 3:23 3:\frac{2}{3} , which means dividing 3 by 23\frac{2}{3}.

To solve this, follow these steps:

  • Step 1: Find the reciprocal of 23\frac{2}{3}. The reciprocal is obtained by swapping the numerator and the denominator, thus the reciprocal of 23\frac{2}{3} is 32\frac{3}{2}.
  • Step 2: Multiply 3 by the reciprocal 32\frac{3}{2}.
  • Step 3: Perform the multiplication: 3×323 \times \frac{3}{2}.

Let's execute these steps:

Step 2: Since multiplying a whole number by a fraction gives:

3×32=3×32=92 3 \times \frac{3}{2} = \frac{3 \times 3}{2} = \frac{9}{2}

Step 3: Convert the improper fraction 92\frac{9}{2} to a mixed number:

Divide 9 by 2 which gives 4 as the quotient and 1 as the remainder. Thus, the mixed number is 4124\frac{1}{2}.

Therefore, the solution to the ratio 3:233:\frac{2}{3} is 412\mathbf{4\frac{1}{2}}.

Answer

412 4\frac{1}{2}

Exercise #9

3:57= 3:\frac{5}{7}=

Video Solution

Step-by-Step Solution

To divide the whole number 3 by the fraction 57\frac{5}{7}, we follow these steps:

  • Step 1: Identify the reciprocal of the fraction. The reciprocal of 57\frac{5}{7} is 75\frac{7}{5}.
  • Step 2: Multiply the whole number 3 by this reciprocal.
  • Step 3: Perform the multiplication to find the result.

Let's calculate this:
Step 1: The reciprocal of 57\frac{5}{7} is 75\frac{7}{5}.
Step 2: Multiply: 3×75=3×75=2153 \times \frac{7}{5} = \frac{3 \times 7}{5} = \frac{21}{5}.
Step 3: Convert the improper fraction 215\frac{21}{5} to a mixed number:

  • Divide 21 by 5. It goes 4 times with a remainder of 1.
  • The quotient is 4, and the remainder is 1. Therefore, 215=415\frac{21}{5} = 4\frac{1}{5}.

Thus, the solution to 3:573 : \frac{5}{7} is 4154\frac{1}{5}.

The correct choice among the given answers is: 4154\frac{1}{5}.

Answer

415 4\frac{1}{5}

Exercise #10

3:56= 3:\frac{5}{6}=

Video Solution

Step-by-Step Solution

To solve this problem, let's carry out the following steps:

  • Step 1: Recognize that the expression 3:56 3 : \frac{5}{6} represents division, so it becomes 3÷56 3 \div \frac{5}{6} .
  • Step 2: Use the rule that dividing by a fraction is the same as multiplying by its reciprocal. Thus, convert this to 3×65 3 \times \frac{6}{5} .
  • Step 3: Multiply 3 (which can be written as 31\frac{3}{1}) by 65\frac{6}{5}:
    31×65=3×61×5=185\frac{3}{1} \times \frac{6}{5} = \frac{3 \times 6}{1 \times 5} = \frac{18}{5}.
  • Step 4: Convert 185\frac{18}{5} to a mixed number. Divide 18 by 5:
    - 5 goes into 18 three times with a remainder of 3.
    - Therefore, 185=335\frac{18}{5} = 3\frac{3}{5}.

Thus, the solution to the problem is 335 3\frac{3}{5} .

Answer

335 3\frac{3}{5}

Exercise #11

1:23= 1:\frac{2}{3}=

Video Solution

Step-by-Step Solution

We need to evaluate the expression 1÷23 1 \div \frac{2}{3} .

To do this, we use the principle that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, the expression becomes:

1×32 1 \times \frac{3}{2} .

Next, we multiply the whole number by the reciprocal:

1×32=32 1 \times \frac{3}{2} = \frac{3}{2} .

To express 32\frac{3}{2} as a mixed number, we write it as:

112 1\frac{1}{2} .

Thus, the solution to the problem is 112 1\frac{1}{2} , which matches choice 3 from the options provided.

Answer

112 1\frac{1}{2}

Exercise #12

4:68= 4:\frac{6}{8}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll proceed as follows:

  • Step 1: Simplify the fraction 68\frac{6}{8}.
  • Step 2: Use the formula for dividing by a fraction by multiplying by its reciprocal.
  • Step 3: Simplify the resulting fraction or convert it to a mixed number.

Let's work through these steps:

Step 1: Simplify 68\frac{6}{8}.
68\frac{6}{8} simplifies to 34\frac{3}{4} by dividing the numerator and the denominator by 2 (the greatest common divisor).

Step 2: Find the reciprocal of 34\frac{3}{4} and multiply it by 4.
The reciprocal of 34\frac{3}{4} is 43\frac{4}{3}.
So, 4÷34=4×43=1634 \div \frac{3}{4} = 4 \times \frac{4}{3} = \frac{16}{3}.

Step 3: Simplify 163\frac{16}{3} to a mixed number.
163\frac{16}{3} can be expressed as 5135\frac{1}{3} since 16 divided by 3 is 5 with a remainder of 1.

Therefore, the solution to the problem is 5135\frac{1}{3}.

Answer

513 5\frac{1}{3}

Exercise #13

4:47= 4:\frac{4}{7}=

Video Solution

Step-by-Step Solution

To solve the problem 4:47 4 : \frac{4}{7} , we'll apply the concept of dividing by a fraction:

Step 1: Convert the division to multiplication by the reciprocal of the fraction. The reciprocal of 47 \frac{4}{7} is 74 \frac{7}{4} .

Thus, the expression 4:47 4 : \frac{4}{7} is equivalent to 4×74 4 \times \frac{7}{4} .

Step 2: Perform the multiplication:

4×74=41×74 4 \times \frac{7}{4} = \frac{4}{1} \times \frac{7}{4}

Step 3: Multiply the numerators and the denominators:

4×71×4=284 \frac{4 \times 7}{1 \times 4} = \frac{28}{4}

Step 4: Simplify the fraction:

284=7 \frac{28}{4} = 7

Therefore, the result of dividing 4 4 by 47 \frac{4}{7} is 7 7 .

Answer

7 7

Exercise #14

2:25= 2:\frac{2}{5}=

Video Solution

Step-by-Step Solution

To solve this problem, we will follow these steps:

  • Step 1: Find the reciprocal of the fraction 25\frac{2}{5}.
  • Step 2: Multiply the whole number 2 by this reciprocal.
  • Step 3: Simplify the result, if necessary.

Now, let's work through each step:
Step 1: The reciprocal of the fraction 25\frac{2}{5} is 52\frac{5}{2}.
Step 2: Multiply the whole number 2 by 52\frac{5}{2}:

2×52=2×52=102 2 \times \frac{5}{2} = \frac{2 \times 5}{2} = \frac{10}{2}

Step 3: Simplify 102\frac{10}{2}:
Divide 10 by 2, which gives us 5.

Therefore, the solution to the problem is 5 5 .

Answer

5 5

Exercise #15

1:34= 1:\frac{3}{4}=

Video Solution

Step-by-Step Solution

To solve this problem, let's divide 11 by 34\frac{3}{4}. The solution involves converting the division into a multiplication:

  • Step 1: Recognize 1:34\,1:\frac{3}{4}\, as the division 134\frac{1}{\frac{3}{4}}.

  • Step 2: Convert division into multiplication: 134=1×43\frac{1}{\frac{3}{4}} = 1 \times \frac{4}{3}.

  • Step 3: Compute the multiplication: 1×43=431 \times \frac{4}{3} = \frac{4}{3}.

  • Step 4: Convert 43\frac{4}{3} into a mixed number: 1131\frac{1}{3}.

Therefore, the solution to the division 1:341 : \frac{3}{4} is 113 1\frac{1}{3}

The correct answer is (113)(1 \frac{1}{3}).

Answer

113 1\frac{1}{3}