We use fractions to solve many everyday problems. For example, in recipes, ingredients are often measured in fractions of a cup. If we wanted to know the total volume of the ingredients, we could use fraction addition.

Here are some tips for applying fractions to real world problems:

When we describe equal parts out of a whole, we can write the situation as a fraction. For example, if a prize was split 6 ways, each recipient would get \dfrac{1}{6} of the total.
The denominator is the total number of parts. In some cases, we can find it by adding together all of the parts. For example, if Mick picked 4 strawberries and Rachel picked 5 then Mick picked \dfrac{4}{4+5} = \dfrac{4}{9} of the strawberries.
If we want to find a fraction of a quantity, we can multiply the fraction by the quantity. This works if the quantity is a fraction as well. For example, if we want to find \dfrac{2}{3} of \dfrac{1}{10} of a minute in seconds, we would find \dfrac{2}{3}\times \dfrac{1}{10} \times 60.
Fractions are also a way to write division. If a piece of timber was divided into 5 parts, each part would be \dfrac{1}{5} of the original piece.
Improper fractions and mixed numbers can be used to represent more than one whole. For example, if Francisco ran one lap around a track and then ran another third of the track, he has run \dfrac{4}{3} or 1 \, \dfrac{1}{3} laps.

Examples

Example 1

At a party, Bill makes a drink by combining 5 \, \dfrac{1}{3} \text{ L} of water with 1 \, \dfrac{1}{2} \text{ L} cordial.

What is the total amount of the drink?

Worked Solution

Create a strategy

Combine the amount of drinks.

Apply the idea

\displaystyle \text{Total}	\displaystyle =	\displaystyle 5 \, \dfrac{1}{3} + 1 \, \dfrac{1}{2}	Add the values
	\displaystyle =	\displaystyle 5 + \dfrac{1}{3} + 1 + \dfrac{1}{2}	Split the mixed numbers into whole and fraction parts
	\displaystyle =	\displaystyle 6 + \dfrac{1}{3} + \dfrac{1}{2}	Evaluate
	\displaystyle =	\displaystyle 6 + \dfrac{1 \times 2}{3 \times 2} + \dfrac{1 \times 3}{2 \times 3}	Multiply to have the same denominator
	\displaystyle =	\displaystyle 6 + \dfrac{2}{6} + \dfrac{3}{6}	Evaluate
	\displaystyle =	\displaystyle 6 + \dfrac{2 + 3}{6}	Add the numerators over the common denominator
	\displaystyle =	\displaystyle 6 + \dfrac{5}{6}	Evaluate
	\displaystyle =	\displaystyle 6 \, \dfrac{5}{6} \text{ L}	Simplify

Example 2

Jack is making bags for his friends. He has 3 \, \dfrac{1}{2} \text{ m} of fabric.

If each bag requires \dfrac{2}{5} \text{ m} of fabric, how many bags can he make?

Express your answer as an improper fraction.

Worked Solution

Create a strategy

Divide the length of fabric by the amount needed for each bag.

Apply the idea

\displaystyle \text{Number}	\displaystyle =	\displaystyle \dfrac{7}{2} \div \dfrac{2}{5}	Divide the values
	\displaystyle =	\displaystyle \dfrac{7}{2} \times \dfrac{2}{5}	Multiply by the reciprocal
	\displaystyle =	\displaystyle \dfrac{7 \times 5}{2 \times 2}	Multiply the numerators and denominators
	\displaystyle =	\displaystyle \dfrac{35}{4}	Simplify

Idea summary

Tips for applying fractions to real world problems:

When we describe equal parts out of a whole, we can write the situation as a fraction.
The denominator is the total number of parts.
If we want to find a fraction of a quantity, we can multiply the fraction by the quantity.
Fractions are also a way to write division.
Improper fractions and mixed numbers can be used to represent more than one whole.

2.08 Solve contextual problems with fractions

Ideas

Contextual problems with fractions

Examples

Example 1

Example 2

Outcomes

MA4-5NA

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