Multiplication of Integers by a Fraction and a Mixed number: Multiplying fractions by whole numbers

Examples with solutions for Multiplication of Integers by a Fraction and a Mixed number: Multiplying fractions by whole numbers

Exercise #1

6×34= 6\times\frac{3}{4}=

Video Solution

Step-by-Step Solution

To solve the problem 6×346 \times \frac{3}{4}, we follow these steps:

  • Step 1: Express the integer 6 as a fraction: 61 \frac{6}{1} .
  • Step 2: Multiply the fractions: 61×34\frac{6}{1} \times \frac{3}{4} .
  • Step 3: Multiply the numerators: 6×3=186 \times 3 = 18.
  • Step 4: Multiply the denominators: 1×4=41 \times 4 = 4.
  • Step 5: Form the resulting fraction: 184\frac{18}{4}.
  • Step 6: Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2: 184÷22=92\frac{18}{4} \div \frac{2}{2} = \frac{9}{2}.
  • Step 7: Convert 92\frac{9}{2} to a mixed number: Divide 9 by 2 to get 4 with a remainder of 1, thus 92=412\frac{9}{2} = 4\frac{1}{2}.

Therefore, the solution to the problem is 412 4\frac{1}{2} .

Answer

412 4\frac{1}{2}

Exercise #2

2×57= 2\times\frac{5}{7}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll multiply the whole number 2 by the fraction 57\frac{5}{7}:

  • Step 1: Multiply the numerator 5 by the whole number 2:

2×5=10 2 \times 5 = 10

  • Step 2: Write the result over the original denominator 7:

107 \frac{10}{7}

  • Step 3: Convert 107\frac{10}{7} to a mixed number:

Since 10 divided by 7 is 1 with a remainder of 3, we can express this as:

137 1\frac{3}{7}

Therefore, the solution to the problem is 137\textbf{1}\frac{\textbf{3}}{\textbf{7}}.

Answer

137 1\frac{3}{7}

Exercise #3

4×23= 4\times\frac{2}{3}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll multiply the whole number 4 by the fraction 23 \frac{2}{3} as follows:

  • Step 1: Convert the whole number 4 into a fraction. This can be written as 41 \frac{4}{1} .
  • Step 2: Use fraction multiplication rules: multiply the numerators together and the denominators together.
  • Step 3: So, multiply the numerators: 4×2=8 4 \times 2 = 8 .
  • Step 4: Multiply the denominators: 1×3=3 1 \times 3 = 3 .
  • Step 5: The result is 83 \frac{8}{3} .
  • Step 6: Since 83 \frac{8}{3} is an improper fraction, convert it to a mixed number.
        8÷3=2 8 \div 3 = 2 with a remainder of 2.
        Thus, 83=223 \frac{8}{3} = 2\frac{2}{3} .

Therefore, the solution to the problem is 223 2\frac{2}{3} .

Answer

223 2\frac{2}{3}

Exercise #4

3×12= 3\times\frac{1}{2}=

Video Solution

Step-by-Step Solution

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

  • Multiply the numerator of the fraction by the integer.
  • Keep the denominator unchanged.
  • Convert the resulting improper fraction to a mixed number, if necessary.

Now, let's work through each step:
Step 1: Multiply the numerator of 12 \frac{1}{2} , which is 1 1 , by 3 3 :
1×3=3 1 \times 3 = 3 .

Step 2: Write the result over the original denominator:
32 \frac{3}{2} .

Step 3: Convert the improper fraction 32 \frac{3}{2} to a mixed number:
Divide 3 3 by 2 2 . This gives 1 1 as the quotient and 1 1 as the remainder, so:
32=112 \frac{3}{2} = 1\frac{1}{2} .

Therefore, the solution to the problem is 112 1\frac{1}{2} .

Answer

112 1\frac{1}{2}

Exercise #5

Solve:

7×38= 7\times\frac{3}{8}=

Video Solution

Step-by-Step Solution

To solve this problem, we will start by multiplying the whole number 7 by the fraction 38 \frac{3}{8} using the rule for multiplying a whole number by a fraction.

Calculate the product:

  • 7×38=7×38 7 \times \frac{3}{8} = \frac{7 \times 3}{8}
  • =218 = \frac{21}{8}

The fraction 218 \frac{21}{8} is an improper fraction, meaning the numerator is greater than the denominator. To convert it to a mixed number, we divide 21 by 8:

  • 21 divided by 8 equals 2 with a remainder of 5.
  • This gives us the mixed number: 258 2\frac{5}{8}

The remainder becomes the numerator of the fraction part, and the denominator remains the same as in the original fraction.

Therefore, the solution to the problem is 258 2\frac{5}{8} .

Answer

258 2\frac{5}{8}

Exercise #6

8×12= 8\times\frac{1}{2}=

Video Solution

Step-by-Step Solution

To solve this mathematical problem, we need to multiply a whole number, 8, with the fraction, 12\frac{1}{2}.

Here are the steps:

  • Step 1: Identify the given values. The integer is 8 and the fraction is 12\frac{1}{2}.
  • Step 2: Apply the multiplication formula for an integer and a fraction: a×bc=a×bc a \times \frac{b}{c} = \frac{a \times b}{c} .
  • Step 3: Multiply the integer by the numerator of the fraction: 8×1=8 8 \times 1 = 8 .
  • Step 4: Divide the result by the denominator of the fraction: 82=4\frac{8}{2} = 4.
  • Step 5: Simplify the fraction if necessary. In this case, 82\frac{8}{2} simplifies directly to 4.

Therefore, the multiplication of 8 by 12\frac{1}{2} is 4 4 .

In the context of the multiple-choice options provided, the correct answer is choice (4): 4 4 .

Answer

4 4

Exercise #7

3×67= 3\times\frac{6}{7}=

Video Solution

Step-by-Step Solution

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

  • Convert the whole number into a fraction.
  • Multiply the fractions.
  • Simplify the result.

Now, let's work through each step:

Step 1: Convert the whole number 3 into a fraction:
3 becomes 31 \frac{3}{1} .

Step 2: Multiply the fraction 31 \frac{3}{1} by 67 \frac{6}{7} :
The numerators are 3×6=18 3 \times 6 = 18 .
The denominators are 1×7=7 1 \times 7 = 7 .
The result is 187 \frac{18}{7} .

Step 3: Convert 187 \frac{18}{7} to a mixed number:
Divide the numerator by the denominator: 18 divided by 7 is 2 with a remainder of 4.
Thus, 187=247 \frac{18}{7} = 2\frac{4}{7} .

Therefore, the solution to the problem is 247 2\frac{4}{7} .

Answer

247 2\frac{4}{7}

Exercise #8

7×25= 7\times\frac{2}{5}=

Video Solution

Step-by-Step Solution

To solve the problem 7×25 7 \times \frac{2}{5} , we will follow a structured approach:

  • Step 1: Multiply the whole number by the numerator of the fraction.
  • Step 2: Retain the denominator of the fraction.
  • Step 3: Simplify the resulting fraction, if possible.

Let's work through each step:

Step 1: Multiply the whole number by the numerator.
We have 7×2=14 7 \times 2 = 14 .

Step 2: Keep the denominator the same.
The resulting fraction is 145\frac{14}{5}.

Step 3: Convert the improper fraction to a mixed number if possible.
Divide the numerator by the denominator: 14÷5=2 14 \div 5 = 2 with a remainder of 4 4 .
This results in the mixed number 245 2\frac{4}{5} .

Therefore, the solution to the problem 7×25 7 \times \frac{2}{5} is 245 2\frac{4}{5} , which corresponds to choice 3 in the provided options.

Answer

245 2\frac{4}{5}

Exercise #9

8×59= 8\times\frac{5}{9}=

Video Solution

Step-by-Step Solution

To solve the problem of multiplying 88 by 59\frac{5}{9}, we can follow these steps:

  • Step 1: Convert the whole number 88 into a fraction by expressing it as 81\frac{8}{1}.
  • Step 2: Multiply the numerators together: 8×5=408 \times 5 = 40.
  • Step 3: Multiply the denominators together: 1×9=91 \times 9 = 9.
  • Step 4: Write the product as a fraction: 409\frac{40}{9}.
  • Step 5: Convert the improper fraction 409\frac{40}{9} into a mixed number:
    • Divide 40 by 9, which gives 4 (quotient) with a remainder of 4.
    • Write the mixed number as 4494\frac{4}{9}.

Therefore, the solution to the multiplication problem 8×598 \times \frac{5}{9} is 449 4\frac{4}{9} .

Answer

449 4\frac{4}{9}

Exercise #10

3×812= 3\times\frac{8}{12}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll pursue a step-by-step method:

  • Step 1: Simplify the fraction 812\frac{8}{12}. The greatest common divisor of 8 and 12 is 4, so dividing both the numerator and denominator by 4 gives 23\frac{2}{3}.
  • Step 2: Multiply the integer 3 by the simplified fraction 23\frac{2}{3}.

Let's proceed with these steps:
Step 1: Simplify 812\frac{8}{12}:
812=8÷412÷4=23\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}.

Step 2: Multiply the integer by the fraction:
3×23=3×23=63=23 \times \frac{2}{3} = \frac{3 \times 2}{3} = \frac{6}{3} = 2.

Thus, the result of the multiplication is 2\boxed{2}.

Answer

2 2

Exercise #11

10×79= 10\times\frac{7}{9}=

Video Solution

Step-by-Step Solution

To solve the problem 10×79 10 \times \frac{7}{9} , we follow these steps:

  • Step 1: Multiply the whole number by the numerator. Calculate 10×7=70 10 \times 7 = 70 .
  • Step 2: Divide this product by the denominator. Calculate 709 \frac{70}{9} .
  • Step 3: If the result is an improper fraction, convert it to a mixed number. Divide 70 by 9, which goes 7 times with a remainder of 7. This can be expressed as the mixed number 779 7\frac{7}{9} .

Let's work through each step:
Step 1: Multiply 10 by 7 to get 70.
Step 2: Divide 70 by 9 to get 7 remainder 7.
Step 3: The proper whole number from the division is 7, with the remainder over the original fraction denominator giving us the final fraction 79 \frac{7}{9} .

Thus, the product 10×79 10 \times \frac{7}{9} is 779 7\frac{7}{9} .

Answer

779 7\frac{7}{9}

Exercise #12

7×68= 7\times\frac{6}{8}=

Video Solution

Step-by-Step Solution

To solve the multiplication of an integer with a fraction, we need to follow these steps:

  • Step 1: Multiply the integer 7 by the numerator of the fraction, which is 6.
  • Step 2: Keep 8 as the denominator.
  • Step 3: Simplify the resulting fraction.
  • Step 4: Convert to a mixed number if needed.

Now, let's work through each step:

Step 1: Multiply 7 by 6, which gives us 42 42 as the numerator.

Step 2: The denominator remains 8, so we have the fraction 428\frac{42}{8}.

Step 3: Simplify 428\frac{42}{8} by finding the greatest common divisor (GCD) of 42 and 8. The GCD is 2.

We divide the numerator and the denominator by 2: 42÷28÷2=214\frac{42 \div 2}{8 \div 2} = \frac{21}{4}.

Step 4: Convert 214\frac{21}{4} into a mixed number:

Divide 21 by 4, which equals 5 with a remainder of 1. Thus, 214\frac{21}{4} is equivalent to the mixed number 5145\frac{1}{4}.

Therefore, the solution to the problem is 5145\frac{1}{4}.

Answer

514 5\frac{1}{4}