##### 6.4 Multiplying improper fractions and mixed numbers

It’s pizza night, your friend is coming over to share it with you. She insists to split the bill based on the amount of slices that she would be eating. Your sister also wants to have at most two slices. You can eat up to 3 slices and your buddy can eat up to 4 slices. You quickly get a pad to write your shares down, 4/10, 3/10, 2/10. How much would each of you need to pay if the pizza is \$5?

Solving this problem would require you to know about fractions. The problem is quite easy to solve as long as you know the basic operations with fractions. This is why in this chapter, we will be talking all about fractions.

Fractions represent a part of a whole. Fraction is a word from Latin meaning “broken”, There two types of fractions, the proper fractions and improper fractions. Proper fractions are those that have a numerator that is smaller than the denominator; while the improper fraction, on the other hand, has a numerator that is larger than the denominator. Then, there’s also the mixed numbers which refers to the combination of both a whole number and a fraction like 5 ½, 39 4/5 and so on.

Examples of proper fractions: $\frac{2}{3}$ , $\frac{7}{{10}}$ , $\frac{{31}}{{40}}$

Examples of improper fractions: $\frac{6}{5}$, $\frac{{12}}{{11}}$, $\frac{{73}}{{45}}$

In the first part of the chapter, we will be looking at how to multiply whole numbers and fractions. In cases when we want to know how much ¼ of 20 or 5/6 of 75 is, we can be able to just multiply the two given numbers to solve the problem. Multiplying whole numbers and fractions would just involve multiplying the numerator to the whole number and simplifying the answer like this:

Example: $\frac{1}{4}\; \times 20$
Step 1: Multiply the whole number to the numerator (the number on top)
$\frac{{1 \times 20}}{4} = \;$ $\frac{{20}}{4}$
Step 2: Simplify the fraction
$\frac{{20}}{4} = \frac{5}{1} = 5$

In the second part, we will be dealing with the question of how to divide fractions with whole numbers, like 2/3 $\div$ 48 or 4/7 $\div$ 36. You simply need to turn the whole number which is your dividend into a fraction by putting 1 as its denominator. Then you get the reciprocal of the whole number turned into fraction, multiply it with the divisor which is the other fraction and then simplify the quotient if needed.

Example: $\frac{2}{3} \div 48$
Step 1: Turn the whole number into a fraction
$\frac{2}{3} \div \frac{{48}}{1}$
Step 2: Turn the second fraction upside down; and turn the division sign into multiplication sign
$\frac{2}{3} \times \frac{1}{{48}}$ $\leftarrow$ reciprocal
Step 3: Simplify the fractions

Step 4: Multiply the fractions
$\frac{1}{3} \times \frac{1}{{24}} = \;\frac{1}{{72}}$

For the third part, we will be looking at how to multiply proper fractions like ½ x ¾ or 7/10 x 5/6. The process is very much the same with how you multiply a whole number and a fraction without the part where you need to convert the whole number into a fraction.

Example: $\frac{1}{3} \times \frac{3}{5}$
Step 1: Multiply the fractions
$\frac{{1 \times 3}}{{3 \times 5}} = \;\frac{3}{{15}}$
Step 2: Simplify the fraction
$\frac{3}{{15}} = \frac{1}{5}$

The fourth part of this chapter will discuss about multiplying improper fractions and mixed numbers, a.k.a. mixed fractions. The process is similar to multiplying a proper fraction like the example given above except that the mixed numbers need to be converted into a fractional form.

Example: $\frac{4}{3} \times 2\frac{2}{5}$
Step 1: Turn the mixed numbers into improper fraction
$\frac{4}{3} \times \frac{{12}}{5}$
Step 2: Multiply the fractions
$\frac{{4\; \times \;12}}{{3\; \times \;5}} = \;\frac{{48}}{{15}}$
Step 3: Simplify the fraction

Step 4: Turn the improper fraction into mixed number (optional)
$\frac{{16}}{5} = 3\frac{1}{5}$

For the last two parts of this chapter, we will be learning how to divide fractions and mixed numbers and looking at how to use the things we learn about fractions in different math problems involving fraction operations.

Example: $1\frac{1}{5}\; \div \frac{2}{3}$
Step 1: Turn the mixed number to improper fraction
$\frac{6}{5} \div \frac{2}{3}$
Step 2: Turn the second fraction upside down; and turn the division sign into multiplication sign
$\frac{6}{5} \times \frac{3}{2}$
Step 3: Simplify the fractions

Step 4: Multiply the fractions
$\frac{{3\; \times 3}}{{5\; \times \;2}} = \frac{9}{{10}}$

We will be learning how to solve fractions operations by following the order of operations to get the correct answers. At the end of this chapter you can try to solve the pizza problem above and also try playing the free fraction games online.

### Multiplying improper fractions and mixed numbers

Improper fractions are fractions that their numerators are larger than or equal to their denominators. On the other hand, mixed numbers, also known as compound fractions, are numbers that have both a whole number and a fraction. In this section, a model will be used as a way to multiply mixed numbers and improper fractions. We will also practice questions of multiplying improper fractions and mixed numbers algebraically.