Multiplication of Integers by a Fraction and a Mixed number - Examples, Exercises and Solutions

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

$5\times(3+\frac{1}{3})=$

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

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

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

$5\times3=15$

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

$5=\frac{5}{1}$

$\frac{5}{1}\times\frac{1}{3}=\frac{5\times1}{1\times3}=\frac{5}{3}$

And we get the exercise:

$15+\frac{5}{3}=15+1\frac{2}{3}=16\frac{2}{3}$

And now let's see the solution centered:

$5\times3\frac{1}{3}=5\times(3+\frac{1}{3})=(5\times3)+(5\times\frac{1}{3})=15+\frac{5}{3}=15+1\frac{2}{3}=16\frac{2}{3}$

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

$3\times(2+\frac{1}{4})=$

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

$(3\times2)-(3\times\frac{1}{4})=$

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

$3\times2=6$

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

$3=\frac{3}{1}$

$\frac{3}{1}\times\frac{1}{4}=\frac{3\times1}{1\times4}=\frac{3}{4}$

And we get the exercise:

$6+\frac{3}{4}=6\frac{3}{4}$

And now let's see the solution centered:

$3\times2\frac{1}{4}=3\times(2+\frac{1}{4})=(3\times2)+(3\times\frac{1}{4})=6+\frac{3}{4}=6\frac{3}{4}$

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

$\frac{1}{5}\times\frac{7}{8}\times\frac{3\times2+2}{3}=$

Let's solve the equation in the numerator:

$\frac{1}{5}\times\frac{7}{8}\times\frac{6+2}{3}=$

$\frac{1}{5}\times\frac{7}{8}\times\frac{8}{3}=$

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

$\frac{1\times7\times8}{5\times8\times3}=$

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

$\frac{1\times7}{5\times3}=$

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

$\frac{7}{15}$

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

$\frac{3\times2}{4\times3}\times2\frac{1}{4}x=$

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

$\frac{6}{12}\times2\frac{1}{4}x=$

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

$\frac{1}{2}\times2\frac{1}{4}x=1\frac{1}{8}x$

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

$\frac{7}{8}\times\frac{8\times2+7}{8}\times\frac{1}{4}=$

Let's solve the exercise in the numerator:

$\frac{7}{8}\times\frac{16+7}{8}\times\frac{1}{4}=$

$\frac{7}{8}\times\frac{23}{8}\times\frac{1}{4}=$

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

$\frac{7\times23\times1}{8\times8\times4}=$

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

$\frac{7\times23}{64\times4}=\frac{161}{256}$

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

$\frac{2}{3}\times\frac{7\times3+2}{3}\times\frac{3\times2+1}{2}=$

Let's solve the exercises in the fraction multiplier:

$\frac{2}{3}\times\frac{21+2}{3}\times\frac{6+1}{2}=$

$\frac{2}{3}\times\frac{23}{3}\times\frac{7}{2}=$

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

$\frac{2\times23\times7}{3\times3\times2}=\frac{46\times7}{9\times2}=\frac{322}{18}$

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

$\frac{3\times6+5}{6}\times\frac{5\times6+5}{6}\times\frac{1}{3}x=$

Let's solve the exercises with the eight fractions:

$\frac{18+5}{6}\times\frac{30+5}{6}\times\frac{1}{3}x=$

$\frac{23}{6}\times\frac{35}{6}\times\frac{1}{3}x=$

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

$\frac{23\times35}{6\times6\times3}x=\frac{805}{108}x$

We begin by converting the decimal numbers into mixed fractions:

$4\frac{1}{10}\times1\frac{6}{10}\times3\frac{2}{10}+4\frac{7}{10}=$

We then convert the mixed fractions into simple fractions:

$\frac{41}{10}\times\frac{16}{10}\times\frac{32}{10}+\frac{47}{10}=$

We solve the exercise from left to right:

$\frac{41\times16}{10\times10}=\frac{656}{100}$

This results in the following exercise:

$\frac{656}{100}\times\frac{32}{10}+\frac{47}{10}=$

We solve the multiplication exercise:

$\frac{656\times32}{100\times10}=\frac{20,992}{1,000}$

Now we get the exercise:

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

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

$\frac{47\times100}{10\times100}=\frac{4,700}{1,000}$

We obtain the exercise:

$\frac{20,992}{1,000}+\frac{4,700}{1,000}=\frac{20,992+4,700}{1,000}=\frac{25,692}{1,000}$

Lastly we convert the simple fraction into a decimal number:

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