 # Problem Solving with Percentages

Lesson

There are a number of different skills we have learnt with percentages. Now we are going to use these skills to solve word problems involving percentages.

## Finding a percentage of a quantity

We learnt how to convert percentages, decimals and fractions. So we know that if there is a class of $30$30 students and $50%$50% are girls, it means that half the class or $15$15 students are girls. This is an example of a word problem that involves finding a percentage of a quantity.

#### Example

##### QUESTION 1

When tickets to a football match went on sale, $71%$71% of the tickets were purchased in the first hour. If the stadium seats $58000$58000 people, what was the number of seats still available after the first hour?

Think: How can we work out how many tickets are left after the first hour?

Do

There are two ways we can approach this question.

1) We can find how many people bought tickets in the first hour, the subtract it from the total number of people.

 $58000\times71%$58000×71% $=$= $41180$41180 seat sold in the first hour $58000-41180$58000−41180 $=$= $16820$16820 seats still available

2) If $71%$71% of the tickets are sold in the first hour, $29%$29% of the tickets are still remaining. Let's see how it works.

 $100%-71%$100%−71% $=$= $29%$29% $58000\times29%$58000×29% $=$= $16820$16820 seats still available

We get the same answer either way!

## Percentage composition

We also learnt how to express amounts of things in percentages. In other words, we express the part of the whole as a percentage. This is known as percentage composition.

#### Example

##### QUESTION 2

In one day, the emergency line received $7520$7520 calls. They found that $1128$1128 of these calls were not an emergency.

a) What percentage of calls were not an emergency?

b) What percentage of calls were actual emergencies?

## Percentage change

Percentage change involves percentage increase and percentage decrease. This had a number of real world applications, particularly to do with buying and selling goods.

#### Example

##### QUESTION 3

Han bought a computer for $\$2580$$2580 and later sold it for \2540$$2540.

a) Express the loss as a percentage of the cost price to 1 d.p. Don't forget to include the percentage sign.

b) Express the loss as a percentage of the selling price to 1 d.p.

## The unitary method

The unitary method is a way of finding the total amount after being given a percentage of it by finding what $1%$1% is worth. We can also use other benchmarks fractions that we can easily multiplied to give $100%$100% such as $10%$10%, $25%$25% or $50%$50%.

#### Example

##### Question 4

When a rock sample was examined, it was found that $4.1$4.1 kilograms of it was copper.

If the sample represents $10%$10% of a rock bed, how many kilograms of copper are there in the rockbed?

Think: The total amount of copper is $100%$100%.

Do:

1) We can work out how much $1%$1% represents, then find $100%$100%.

 $10%$10% $=$= $4.1$4.1 kg $1%$1% $=$= $0.41$0.41 kg $100%$100% $=$= $41$41 kg

2) We can multiply the $10%$10% amount by $10$10 to work out $100%$100% straight away.

 $10%$10% $=$= $4.1$4.1 kg $100%$100% $=$= $41$41 kg

Again we get the same answer no matter which method we use!