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2.08 Solve contextual problems with fractions

Lesson

Contextual problems with fractions

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.

Outcomes

MA4-5NA

operates with fractions, decimals and percentages

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