Finding the Whole Given a Part and the Percentage
Learn to find 100% when you know a part and its percentage
1. Recall: Percentage Basics
The Big Idea
Percent means "per 100" or "out of 100". When we say 43%, we mean 43 out of 100.
Example 1: Visualising 43%
When a whole is divided into 100 equal parts, each part is 1% or 1/100 of the whole.
43 out of 100 = 43/100 = 43%
Example 2: Converting Fractions to Percentages
What percentage of the whole is 11 out of 50 equal parts?
Method 1: Equivalent Fractions
11/50 = 22/100 = 22%
(Multiply both by 2 to get denominator 100)
Method 2: Multiply by 100%
11/50 × 100% = 22%
Example 3: Converting Percentages
Percentage to Fraction
16% = 16/100 = 4/25 (simplest form)
Percentage to Decimal
45% = 45/100 = 0.45
2. Expressing One Quantity as a Percentage of Another
The Formula
Percentage = (Part ÷ Whole) × 100%
Example: Express 200 g as a Percentage of 800 g
200/800 × 100% = 25%
200 g is 25% of 800 g.
3. Finding the Whole - The Unitary Method
The Key Strategy
Step 1: Find 1% (divide by the percentage number). An easy percentage such as 5% or 10% works too.
Step 2: Find 100% (from 1%, multiply by 100; from 10%, multiply by 10)
Example 1: 10% of a Number is 5. What is the Number?
Finding the Whole
| Percentage | Value of number | |
|---|---|---|
| 10% | 5 | |
× 10 | × 10 | |
| 100% | 50 |
Example 2: Hassan's Money (finding 1% first)
Hassan spent $240 and had 70% of his money left. How much money did he have at first?
Key Insight: If 70% is left, then 100% - 70% = 30% was spent.
So 30% of Hassan's money = $240
Finding Hassan's Total Money
| Percentage | Value | |
|---|---|---|
| 30% | $240 | |
÷ 30 | ÷ 30 | |
| 1% | $8 | |
× 100 | × 100 | |
| 100% | $800 |
Step 1: 1% of Hassan's money = $240 ÷ 30 = $8
Step 2: 100% of Hassan's money = $8 × 100 = $800
Example 3: Concert Audience
At a concert, 2/5 of the audience were adults and 25% were girls. The rest were boys. There were 70 boys at the concert. How many people were there at the concert?
Finding the Total Audience
| Percentage | Value | |
|---|---|---|
| 35% | 70 | |
÷ 7 | ÷ 7 | |
| 5% | 10 | |
× 20 | × 20 | |
| 100% | 200 |
4. Finding the Whole - Direct Method
The Formula
Whole = Part ÷ (percentage written as a fraction)
For 30%, divide by 30/100. This is the same as multiplying by 100/30.
Example: Using the Direct Method
30% of a sum of money is $240. What is the sum of money?
Direct Method:
Whole = $240 ÷ (30/100) = $240 × (100/30) = $800
When to use which method:
- Unitary Method: Good for showing your working step-by-step
- Direct Method: Faster when you are comfortable dividing by a fraction
5. Real-World Applications
Example 1: Finding Price Before GST
Mr Wong paid $218 for a watch, including 9% GST. What was the price of the watch before GST?
Solution: Unitary Method
| Percentage | Value | |
|---|---|---|
| 109% | $218 | |
÷ 109 | ÷ 109 | |
| 1% | $2 | |
× 100 | × 100 | |
| 100% | $200 |
Example 2: Finding Original Price from Sale Price
During a sale, a dress was sold at a 25% discount. The sale price was $60. What was the original price of the dress?
Solution
| Percentage | Value | |
|---|---|---|
| 75% | $60 | |
÷ 75 | ÷ 75 | |
| 1% | $0.80 | |
× 100 | × 100 | |
| 100% | $80 |
Example 3: Finding Original Amount
After evaporation, a tank has 360 litres of water, which is 90% of its original amount. How much water was in the tank originally?
90% = 360 litres
1% = 360 ÷ 90 = 4 litres
100% = 4 × 100 = 400 litres
Key Points to Remember
- ✓ Percent means "per 100" - 43% means 43 out of 100
- ✓ To convert fraction to percentage: multiply by 100% or make denominator 100
- ✓ Unitary Method: Find 1% (or another easy percentage) first, then find 100%
- ✓ If "70% left", then "30% spent" (100% - 70% = 30%)
- ✓ GST is added to price, Discount is subtracted from price
- ✓ Always identify what percentage the given value represents!