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2.01 Finding the whole

Lesson

Introduction

Percentages are useful measures because they tell us how much of the whole we currently have. If we have 50\% then we have half the amount. If we have 100\%, what we have is equal to the amount. And if we have 200\% then we have double that amount.

How much is 'the whole'?

When looking at percentages, the 'whole' is our reference number that our percentage amount refers to.

For example: 14 is equal to half of 28, so 14 is 50\% of 28.

But also: 14 is equal to double of 7, so 14 is 200\% of 7.

In both cases our amount is 14. The numbers that change are the percentage value and our reference number - the whole.

Examples

Example 1

20\% of a quantity is the same as:

A
One twentieth of that quantity
B
One fifth of that quantity
C
One tenth of that quantity
D
One quarter of that quantity
Worked Solution
Create a strategy

Convert the percentage to a fraction by dividing by 100.

Apply the idea
\displaystyle 20\%\displaystyle =\displaystyle \frac{20}{100}Divide by 100
\displaystyle =\displaystyle \frac{1}{5}Simplify

So, 20\% of a quantity is the same as one fifth of that quantity.

The correct option is B.

Idea summary

When looking at percentages, the 'whole' is our reference number that our percentage amount refers to.

Use fractions to find the whole

Let's consider the case where we don't know the whole, or reference number:

14 is 50\% of some number. What is this number?

To work out the whole, we can use the percentage value to figure out how much of the whole we

currently have. Since 50\% is equal to \dfrac{1}{2}\,when written as a fraction, we know that 14 is half of our

missing number. If we let x represent our missing number, we can express this information as the equation:14=x\times \frac{1}{2}

We can solve for x by multiplying both sides of the equation by 2, and find that x=28, so we know the 'whole' we are looking for is 28.

We can apply this same technique to any amount and percentage.

Examples

Example 2

20\% is of number is equal to 7. What is this number?

Worked Solution
Create a strategy

Let x be the number and form an equation.

Apply the idea
\displaystyle 20\% \times x\displaystyle =\displaystyle 7Write the equation
\displaystyle \frac{1}{5} \times x\displaystyle =\displaystyle 7Write the percentage as a fraction
\displaystyle 5\times \frac{1}{5}\times x\displaystyle =\displaystyle 7\times 5Multiply both sides by 5
\displaystyle x\displaystyle =\displaystyle 35Evaluate

The number is 35.

The unitary method

If 39 is 30\% of some number, what is this number?

An approach to solving this would be to find 1\% of the number first. From there we can find the whole, or indeed any other percentage amount.

We know that 30\% of the number is 39, so if we divide 39 by 30, we now know 1\% of the number. Multiplying this amount by 100 will then tell us 100\% or the whole of the amount.x=\frac{39}{30}\times 100=1.3\times 100 = 130

This is known as the unitary method.

We can find the whole by:

  • Dividing the given amount by the percentage number. This finds 1\%of the whole.

  • Multiplying the result by 100. This will give us 100\% of the whole which is the number we are looking for.

Since the order of operations allows us to perform either multiplication and division in any order, we can do whichever operation is easier first.

Examples

Example 3

9\% of a number is 72.

a

Complete the statement:

Because 9\% of the number is ⬚, we know that 1\% of the number is ⬚.

Worked Solution
Create a strategy

Use the first step of the unitary method.

Apply the idea

The question tells us what 9\% of the unknown number is 72, so this is the number for the first box.

Tp find 1\% we divide 72 by 9 to get 72 \div 9 =8.

So we get the statement:

Because 9\% of the number is 72, we know that 1\% of the number is 8.

b

What is the number?

Worked Solution
Create a strategy

Use the second step of the unitary method and multiply our result from part (a) by 100.

Apply the idea
\displaystyle \text{Number}\displaystyle =\displaystyle 8 \times 100Multiply by 100
\displaystyle =\displaystyle 800Evaluate
Idea summary

We can find the whole by using the unitary method:

  • Dividing the given amount by the percentage number. This finds 1\%of the whole.

  • Multiplying the result by 100. This will give us 100\% of the whole which is the number we are looking for.

Outcomes

VCMNA276

Solve problems involving the use of percentages, including percentage increases and decreases and percentage error, with and without digital technologies

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