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We have learned about whole numbers and fractions. Mixed numbers are a combination of whole numbers and fractional numbers. We will learn how to convert mixed numbers into improper fractions. We will also learn the steps involved in the multiplication operation of mixed numbers....Read MoreRead Less

Mixed numbers, as the name suggests, are a mixture of numbers: whole numbers and proper fractions. Mixed numbers are mostly used to express numbers between two whole numbers.

We can use a mixed fraction to represent \(1+\frac{2}{3}\), a number between the whole numbers 1 and 2, to make the number easily readable.

\(12\frac{3}{5},~ 4\frac{6}{7}, ~5\frac{1}{2},~ \text {and}~10\frac{1}{3}\) are some examples of mixed fractions.

A mixed number is always made up of three parts. It will have a whole number and a fraction with a numerator and a denominator. The important point to note here is that the fractional part of a mixed number will always be a proper fraction.

Mixed numbers and improper fractions are essentially the same things. All mixed numbers can be expressed as improper fractions, and all improper fractions can be expressed as mixed numbers. It is easier to perform mathematical operations on improper fractions. On the other hand, mixed numbers help us quickly find where the number lies on the number line.

Consider the mixed number \(a\frac{b}{c}\). To convert this mixed number into an improper fraction, we need to follow these steps.

\(a\frac{b}{c}~=~\frac{a~\times~c~+b}{c}\)

**For example,** \(2\frac{2}{3}~=~\frac{2~\times~ 3~+~2}{3}~=~\frac{6~+~2}{3}~=~\frac{8}{3}\)

It is easy to convert an improper fraction into a mixed number. Divide the numerator of the improper fraction by its denominator, and then write the result as follows:

\(Mixed~ fraction~=~Quotient\frac{Remainder}{Denomination}\)

**For example,** let’s find write \(\frac{13}{2}\) as a mixed number.

So, \(\frac{13}{2}~=~6\frac{1}{2}\).

At first glance, the multiplication of two mixed numbers might seem tricky. But, mixed numbers can be easily multiplied by either converting them into improper fractions or using an imaginary area model for multiplication.

**1. Multiplication of mixed numbers using improper fractions:**

To multiply two mixed numbers, we can convert the two mixed numbers into improper fractions.

**Example:** \(2\frac{3}{5}~\times~ 4\frac{2}{3}\)

\(2\frac{3}{5}\) can be written as

\(\frac{2~\times~ 5~+~3}{5}~=~\frac{13}{5}\)

\(4\frac{2}{3}\) can be written as

\(\frac{4~\times~ 3~+~2}{3}~=~\frac{14}{3}\)

Now, the two mixed numbers can be multiplied easily.

\(2\frac{3}{5}~\times ~4\frac{2}{3}~=~\frac{13}{5}~\times ~\frac{14}{3}~=~\frac{13~\times ~14}{5~\times~ 3}\)

\(=\frac{182}{15}\) or \(12\frac{2}{15}\)

2)**Multiplication using the area model:**

Two mixed numbers can be multiplied using an imaginary area model. This gives us a better understanding of the true meaning of multiplication of mixed numbers. Let’s use the same numbers in this case.

\(2\frac{3}{5}~\times ~4\frac{2}{3}\)

**Step 1:** Write each number as a sum.

\(2\frac{3}{5}~=2~+~\frac{3}{5}\) and \(4\frac{2}{3}~=~4~+~\frac{2}{3}\)

**Step 2:** Draw an area model that represents the product of the sums.

**Step 3:** Find the sum of the areas of the sections.

\(8~+~\left(\frac{12}{5}~+~\frac{4}{3}~+~\frac{2}{5}\right)\)

\(8~+~\left(\frac{12~\times~ 3}{5~\times ~3}~+~\frac{4~\times~ 5}{3~\times~ 5}~+~\frac{2~\times ~3}{5\times 3}\right)\) Take LCM of denominator

\(8~+~\frac{36~+~20+~6}{15}\) Simplify

\(8~+~\frac{62}{15}\) Add the fraction

Converting improper fractions into mixed numbers:

\(8~+~\frac{15~\times ~4~+~2}{15}~=~8~+~4~+~\frac{2}{15}~=~12\frac{2}{15}\)

Therefore, \(2\frac{3}{5}~\times ~4\frac{2}{3}~=~12\frac{2}{15}\)

**Example 1:** Evaluate \(5\frac{6}{7}~\times~ 8\frac{9}{10}\)

**Solution:** First, let’s convert these fractions into improper fractions.

\(5\frac{6}{7}\) can be written as

\(\frac{5~\times~ 7~+~6}{7}~=~\frac{35~+~6}{7}~=~\frac{41}{7}\)

\(8\frac{9}{10}\) can be written as

\(\frac{8~\times~ 10~+~9}{10}~=~\frac{80~+~9}{10}~=~\frac{89}{10}\)

\(5\frac{6}{7}~\times ~8\frac{9}{10}~=~\frac{41}{7}~\times ~\frac{89}{10}\) Multiply

\(=~\frac{3649}{70}\) 0r \(52\frac{9}{70}\)

**Example 2:** Evaluate \(3\frac{6}{9}\times \left(7\frac{1}{2}~-5~\frac{2}{3}\right)\)

**Solution:**

\(7\frac{1}{2}~-5~\frac{2}{3}~=~\frac{7~\times~ 2~+~1}{2}~-~\frac{5~\times~ 3~ +~2}{3}\) Solving the parenthesis

\(~=~\frac{15}{2}~-~\frac{17}{3}\) Conversion to improper fraction

\(~=~\frac{15\times 3-17\times 2}{3\times 2}\) Take LCM of denominator

\(~=~\frac{45~-~34}{6}\) Subtract the fractions

\(~=~\frac{11}{6}\) or \(1\frac{5}{6}\)

Therefore, \(3\frac{6}{9}~\times ~\left(7\frac{1}{2}~-5~\frac{2}{3}\right)~=~3\frac{6}{9}~\times ~\frac{11}{6}\)

\(~=~\frac{3~\times~ 9~+~6}{9}~\times~\frac{11}{6}\)

\(~=~\frac{33}{9}~\times ~\frac{11}{6}\) Conversion to improper fraction

\(~=~\frac{11~\times~ 11}{3~\times~ 6}\) Multiply

\(~=~\frac{121}{18}\) or \(6\frac{13}{18}\)

**Example 3:** The dimensions of a farmland is

\(1\frac{1}{3}~miles~\times~ 1\frac{3}{4}~miles\).

**Solution:**

Length of the farmland =\(1\frac{1}{3}~miles\)

Breadth of the farmland =\(1\frac{3}{4}~miles\)

Area of the farmland = length x breadth

\(=1\frac{1}{3}~miles~\times ~1\frac{3}{4}~miles\)

We can multiply these numbers using the area model of multiplication.

**Step 1:** Rewrite the mixed numbers as the sum of a whole number and a fraction.

\(1\frac{1}{3}~=~1~+~\frac{1}{3}\) and \(1\frac{3}{4}~ =~1~+~\frac{3}{4}\)

**Step 2:** Draw an area model that represents the product of the sums.

**Step 3:** Find the sum of the areas of the sections.

Sum of areas\(~=~1~+~\frac{1}{3}~+~\frac{1}{4}~+~\frac{1}{12}\)

\(~=~1~+~\frac{1~\times ~4}{3~\times ~4}~+~\frac{1~\times ~3}{4~\times ~3}~+~\frac{1}{12}\) Take LCM of denominator

\(~=~1~+~\frac{4~+~3~+~1}{12}\)

\(\therefore\) Sum of areas =\(1~+~\frac{8}{12}~=~1~+~\frac{2}{3}~=~1~\frac{2}{3}\) square miles

**Example 4:** If Michelle earns $18 \(\frac{2}{3}\) in an hour, how much money will she earn in \(5\frac{1}{2}\) hours?

**Solution:**

To find how much money Michelle earns in \(5\frac{1}{2}\) hours, we need to multiply the amount of money she earns in an hour by \(5\frac{1}{2}\).

First, let us convert the mixed numbers into improper fractions.

\(18\frac{2}{3}~=~\frac{18~\times ~3~+~2}{3}\) Conversion to improper fraction

\(~=\frac{54~+~2}{3}\) Addition of Fraction

\(=~\frac{56}{3}\)

\(5\frac{1}{2}~=~\frac{5~\times ~2~+~1}{2}\) Conversion to improper fraction

\(\frac{10~+~1}{2}\) Addition of Fraction

\(\frac{11}{2}\)

Multiplying $18

\(\frac{2}{3}\) by \(5\frac{1}{2}\) hours:

\(\frac{56}{3}~\times~ \frac{11}{2}~=~\frac{616}{6}\) or \(102\frac{2}{3}\)

\(\frac{56}{3}~\times ~\frac{11}{2}~=~\frac{616}{6}\) or \(102\frac{2}{3}\)

So, Michelle earns

$\(102\frac{2}{3}\) in \(5\frac{1}{2}\) hours.

Frequently Asked Questions on Multiplying Mixed Numbers

If the fractional part of a mixed number contains an improper fraction, that means the mixed number can be further simplified. The simplest form of a mixed number will have a proper fraction.

As mixed numbers are numbers that lie between two whole numbers, it is easy to find their approximate position on the number line. For the mixed number \(\)1\frac{1}{2}, the position will be between 1 whole and 2 whole. Now, we just need to locate the position of the fraction on the number line.