Multiplying a whole number by a fraction and a mixed number is solved in the following steps:

The first step:

Convert each whole number and mixed number into a similar fraction and rewrite the problem.

The second stage:

Multiply the numerators and the denominators separately.

The multiplication of numerators will be written in the new numerator.

The multiplication of denominators will be written in the new denominator.

Suggested Topics to Practice in Advance

  1. Mixed Numbers and Fractions Greater Than 1
  2. Addition and Subtraction of Mixed Numbers

Practice Multiplication of Integers by a Fraction and a Mixed number

Examples with solutions for Multiplication of Integers by a Fraction and a Mixed number

Exercise #1

9×389= 9\times3\frac{8}{9}=

Video Solution

Step-by-Step Solution

We will use the distributive property of multiplication and break down the fraction into a subtraction exercise between a whole number and a fraction. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication allows us to break down the larger term in a multiplication problem into a sum or difference of smaller numbers, which makes multiplication easier and gives us the ability to solve the problem even without a calculator

9×(419)= 9\times(4-\frac{1}{9})=

We will use the distributive property formula a(b+c)=ab+ac a(b+c)=ab+ac

(9×4)(9×19)= (9\times4)-(9\times\frac{1}{9})=

Let's solve what's in the left parentheses:

9×4=36 9\times4=36

Note that in the right parentheses we can reduce 9 by 9 as follows:

9=91 9=\frac{9}{1}

91×19=9×11×9=99=11=1 \frac{9}{1}\times\frac{1}{9}=\frac{9\times1}{1\times9}=\frac{9}{9}=\frac{1}{1}=1

And we get the exercise:

361=35 36-1=35

And now let's see the solution centered:

9×389=9×(419)=(9×4)(9×19)=361=35 9\times3\frac{8}{9}=9\times(4-\frac{1}{9})=(9\times4)-(9\times\frac{1}{9})=36-1=35

Answer

35 35

Exercise #2

3×214= 3\times2\frac{1}{4}=

Video Solution

Step-by-Step Solution

We will use the distributive property of multiplication and separate the fraction into an addition exercise between fractions. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication allows us to break down the larger term in the multiplication exercise into a sum or difference of smaller numbers, which makes the multiplication operation easier and gives us the ability to solve the exercise without a calculator

3×(2+14)= 3\times(2+\frac{1}{4})=

We will use the distributive property formula a(b+c)=ab+ac a(b+c)=ab+ac

(3×2)(3×14)= (3\times2)-(3\times\frac{1}{4})=

Let's solve what's in the left parentheses:

3×2=6 3\times2=6

Let's solve what's in the right parentheses:

3=31 3=\frac{3}{1}

31×14=3×11×4=34 \frac{3}{1}\times\frac{1}{4}=\frac{3\times1}{1\times4}=\frac{3}{4}

And we get the exercise:

6+34=634 6+\frac{3}{4}=6\frac{3}{4}

And now let's see the solution centered:

3×214=3×(2+14)=(3×2)+(3×14)=6+34=634 3\times2\frac{1}{4}=3\times(2+\frac{1}{4})=(3\times2)+(3\times\frac{1}{4})=6+\frac{3}{4}=6\frac{3}{4}

Answer

634 6\frac{3}{4}

Exercise #3

5×313= 5\times3\frac{1}{3}=

Video Solution

Step-by-Step Solution

We will use the distributive property of multiplication and separate the fraction into an addition exercise between fractions. This allows us to work with smaller numbers and simplify the operation

Reminder - The distributive property of multiplication actually allows us to separate the larger term in the multiplication exercise into a sum or difference of smaller numbers, which makes the multiplication operation easier and gives us the ability to solve the exercise without a calculator

5×(3+13)= 5\times(3+\frac{1}{3})=

We will use the distributive property formula a(b+c)=ab+ac a(b+c)=ab+ac

(5×3)+(5×13)= (5\times3)+(5\times\frac{1}{3})=

Let's solve what's in the left parentheses:

5×3=15 5\times3=15

Let's solve what's in the right parentheses:

5=51 5=\frac{5}{1}

51×13=5×11×3=53 \frac{5}{1}\times\frac{1}{3}=\frac{5\times1}{1\times3}=\frac{5}{3}

And we get the exercise:

15+53=15+123=1623 15+\frac{5}{3}=15+1\frac{2}{3}=16\frac{2}{3}

And now let's see the solution centered:

5×313=5×(3+13)=(5×3)+(5×13)=15+53=15+123=1623 5\times3\frac{1}{3}=5\times(3+\frac{1}{3})=(5\times3)+(5\times\frac{1}{3})=15+\frac{5}{3}=15+1\frac{2}{3}=16\frac{2}{3}

Answer

1623 16\frac{2}{3}

Exercise #4

34×23×214x= \frac{3}{4}\times\frac{2}{3}\times2\frac{1}{4}x=

Video Solution

Step-by-Step Solution

Let's begin by combining the simple fractions into a single multiplication exercise:

3×24×3×214x= \frac{3\times2}{4\times3}\times2\frac{1}{4}x=

Let's now proceed to solve the exercise in the numerator and denominator:

612×214x= \frac{6}{12}\times2\frac{1}{4}x=

Finally we'll simplify the simple fraction in order to obtain the following:

12×214x=118x \frac{1}{2}\times2\frac{1}{4}x=1\frac{1}{8}x

Answer

118x 1\frac{1}{8}x

Exercise #5

15×78×223= \frac{1}{5}\times\frac{7}{8}\times2\frac{2}{3}=

Video Solution

Step-by-Step Solution

First, let's convert the mixed fraction to an improper fraction as follows:

15×78×3×2+23= \frac{1}{5}\times\frac{7}{8}\times\frac{3\times2+2}{3}=

Let's solve the equation in the numerator:

15×78×6+23= \frac{1}{5}\times\frac{7}{8}\times\frac{6+2}{3}=

15×78×83= \frac{1}{5}\times\frac{7}{8}\times\frac{8}{3}=

Since the only operation in the equation is multiplication, we'll combine everything into one equation:

1×7×85×8×3= \frac{1\times7\times8}{5\times8\times3}=

Let's simplify the 8 in the numerator and denominator of the fraction:

1×75×3= \frac{1\times7}{5\times3}=

Let's solve the equations in the numerator and denominator and we get:

715 \frac{7}{15}

Answer

715 \frac{7}{15}

Exercise #6

5(212+116+34)= 5\cdot\big(2\frac{1}{2}+1\frac{1}{6}+\frac{3}{4}\big)=

Video Solution

Step-by-Step Solution

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

We'll start by simplifying the expression inside the parentheses.

In this expression, there are addition operations between mixed fractions, so in the first step we'll convert all mixed fractions in this expression to improper fractions.

We'll do this by multiplying the whole number by the denominator of the fraction, and adding the result to the numerator.

In the fraction's denominator (which is the divisor) - nothing will change of course.

We'll do this in the following way:

212=(2×2)+12=4+12=52 2\frac{1}{2}=\frac{(2\times2)+1}{2}=\frac{4+1}{2}=\frac{5}{2}

116=(1×6)+16=6+16=76 1\frac{1}{6}=\frac{(1\times6)+1}{6}=\frac{6+1}{6}=\frac{7}{6}

Now we'll get the exercise:

5(52+76+34) 5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big)

We'll continue and perform the addition of fractions in the expression inside the parentheses.

First, we'll expand each fraction to the common denominator, which is 12 (since it is the least common multiple of all denominators in the expression), we'll do this by multiplying the numerator of the fraction by the number that answers the question: "By how much did we multiply the current denominator to get the common denominator?"

Then we'll perform the addition operations between the expanded numerators:

5(52+76+34)=556+72+3312=530+14+912=55312= 5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big) =\\ 5\cdot\frac{5\cdot6+7\cdot2+3\cdot3}{12} =\\ 5\cdot\frac{30+14+9}{12} =\\ 5\cdot\frac{53}{12} =\\ We performed the addition operation between the numerators above, after expanding the fractions mentioned.

Note that since multiplication comes before addition, we first performed the multiplications in the fraction's numerator and only then the addition operations,

We'll continue and simplify the expression we got in the last step, meaning - we'll perform the multiplication we got, while remembering that multiplying a fraction means multiplying the fraction's numerator.

In the next step, we'll write the result as a mixed fraction, we'll do this by finding the whole numbers (the answer to the question "How many complete times does the denominator go into the numerator?") and adding the remainder divided by the divisor:

55312=55312=26512=22112 5\cdot\frac{53}{12}=\\ \frac{5\cdot53}{12}=\\ \frac{265}{12}=\\ 22\frac{1}{12}

Let's summarize the steps of simplifying the given expression:

5(212+116+34)=5(52+76+34)=556+72+3312=55312=22112 5\cdot\big(2\frac{1}{2}+1\frac{1}{6}+\frac{3}{4}\big)= \\ 5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big)=\\ 5\cdot\frac{5\cdot6+7\cdot2+3\cdot3}{12} =\\ 5\cdot\frac{53}{12} =\\ 22\frac{1}{12}

Therefore the correct answer is answer B.

Answer

22112 22\frac{1}{12}

Exercise #7

356×556×13x= 3\frac{5}{6}\times5\frac{5}{6}\times\frac{1}{3}x=

Video Solution

Step-by-Step Solution

First, let's convert all mixed fractions to simple fractions:

3×6+56×5×6+56×13x= \frac{3\times6+5}{6}\times\frac{5\times6+5}{6}\times\frac{1}{3}x=

Let's solve the exercises with the eight fractions:

18+56×30+56×13x= \frac{18+5}{6}\times\frac{30+5}{6}\times\frac{1}{3}x=

236×356×13x= \frac{23}{6}\times\frac{35}{6}\times\frac{1}{3}x=

Since the exercise only involves multiplication, we'll combine all the numerators and denominators:

23×356×6×3x=805108x \frac{23\times35}{6\times6\times3}x=\frac{805}{108}x

Answer

805108x \frac{805}{108}x

Exercise #8

23×723×312= \frac{2}{3}\times7\frac{2}{3}\times3\frac{1}{2}=

Video Solution

Step-by-Step Solution

First, we'll convert the mixed fractions to simple fractions as follows:

23×7×3+23×3×2+12= \frac{2}{3}\times\frac{7\times3+2}{3}\times\frac{3\times2+1}{2}=

Let's solve the exercises in the fraction multiplier:

23×21+23×6+12= \frac{2}{3}\times\frac{21+2}{3}\times\frac{6+1}{2}=

23×233×72= \frac{2}{3}\times\frac{23}{3}\times\frac{7}{2}=

Since the only operation in the exercise is multiplication, we'll combine everything into one exercise and get:

2×23×73×3×2=46×79×2=32218 \frac{2\times23\times7}{3\times3\times2}=\frac{46\times7}{9\times2}=\frac{322}{18}

Answer

1789 17\frac{8}{9}

Exercise #9

78×278×14= \frac{7}{8}\times2\frac{7}{8}\times\frac{1}{4}=

Video Solution

Step-by-Step Solution

First, let's convert the mixed fraction to a simple fraction as follows:

78×8×2+78×14= \frac{7}{8}\times\frac{8\times2+7}{8}\times\frac{1}{4}=

Let's solve the exercise in the numerator:

78×16+78×14= \frac{7}{8}\times\frac{16+7}{8}\times\frac{1}{4}=

78×238×14= \frac{7}{8}\times\frac{23}{8}\times\frac{1}{4}=

Since the only operation in the exercise is multiplication, we'll combine everything into one exercise:

7×23×18×8×4= \frac{7\times23\times1}{8\times8\times4}=

Let's solve the exercises in the numerator and denominator:

7×2364×4=161256 \frac{7\times23}{64\times4}=\frac{161}{256}

Answer

161256 \frac{161}{256}

Exercise #10

4.11.63.2+4.7=? 4.1\cdot1.6\cdot3.2+4.7=\text{?}

Step-by-Step Solution

We begin by converting the decimal numbers into mixed fractions:

4110×1610×3210+4710= 4\frac{1}{10}\times1\frac{6}{10}\times3\frac{2}{10}+4\frac{7}{10}=

We then convert the mixed fractions into simple fractions:

4110×1610×3210+4710= \frac{41}{10}\times\frac{16}{10}\times\frac{32}{10}+\frac{47}{10}=

We solve the exercise from left to right:

41×1610×10=656100 \frac{41\times16}{10\times10}=\frac{656}{100}

This results in the following exercise:

656100×3210+4710= \frac{656}{100}\times\frac{32}{10}+\frac{47}{10}=

We solve the multiplication exercise:

656×32100×10=20,9921,000 \frac{656\times32}{100\times10}=\frac{20,992}{1,000}

Now we get the exercise:

20,9921,000+4710= \frac{20,992}{1,000}+\frac{47}{10}=

We then multiply the fraction on the right so that its denominator is also 1000:

47×10010×100=4,7001,000 \frac{47\times100}{10\times100}=\frac{4,700}{1,000}

We obtain the exercise:

20,9921,000+4,7001,000=20,992+4,7001,000=25,6921,000 \frac{20,992}{1,000}+\frac{4,700}{1,000}=\frac{20,992+4,700}{1,000}=\frac{25,692}{1,000}

Lastly we convert the simple fraction into a decimal number:

25,6921,000=25.692 \frac{25,692}{1,000}=25.692

Answer

25.692

Exercise #11

6×34= 6\times\frac{3}{4}=

Video Solution

Answer

412 4\frac{1}{2}

Exercise #12

2×57= 2\times\frac{5}{7}=

Video Solution

Answer

137 1\frac{3}{7}

Exercise #13

4×23= 4\times\frac{2}{3}=

Video Solution

Answer

223 2\frac{2}{3}

Exercise #14

3×12= 3\times\frac{1}{2}=

Video Solution

Answer

112 1\frac{1}{2}

Exercise #15

Solve:

7×38= 7\times\frac{3}{8}=

Video Solution

Answer

258 2\frac{5}{8}