Percentage Word Problems
Master complex percentage problems with multiple steps
1. Recall: GST, Discount and Interest
Quick Reference
- GST: ADD to price (increase)
- Discount: SUBTRACT from price (decrease)
- Interest: ADD to principal (increase)
Example: Does Same % Discount Mean Same $ Savings?
Shop A sells a bottle of shampoo for $50. Shop B sells the same shampoo for $40. Both shops offer a 20% discount. Does Amy (buying from Shop A) save the same dollar amount as Benny (buying from Shop B)?
Key Insight: Same percentage does NOT mean same dollar amount!
Amy saves: 20% of $50 = $10
Benny saves: 20% of $40 = $8
Answer: No, Amy saves $2 more, because 20% of a bigger price is a bigger amount.
2. Part-to-Whole Composition Problems
Key Concept
All parts of a whole add up to 100%. If you know some parts, you can find the rest.
Example: School Library Books
In a school library, 45% of the books are fiction, 30% are non-fiction, and the rest are reference books. There are 150 reference books. How many books are there in total? How many fiction books are there?
Finding Total Books
| Percentage | Value | |
|---|---|---|
| 25% | 150 | |
÷ 25 | ÷ 25 | |
| 1% | 6 | |
× 100 | × 100 | |
| 100% | 600 |
Part 2: Fiction books = 45% of 600 = 45/100 × 600 = 270 books
3. Multi-Step Problems
Example: Siti and Xinyi's Money
Siti spent 20% of her money. Xinyi spent 25% of her money. Each girl spent the same amount of money. They had $700 left altogether. How much money did they have in all at first?
Key Insight: Convert percentages to fractions for easier calculation!
20% = 1/5 (Siti spent 1 out of 5 parts)
25% = 1/4 (Xinyi spent 1 out of 4 parts)
Solution:
The girls spent the same amount, so 1 unit of Siti's money = 1 unit of Xinyi's money.
Amount left = 4 units (Siti) + 3 units (Xinyi) = 7 units
7 units = $700
1 unit = $700 ÷ 7 = $100
Total at first = 5 units + 4 units = 9 units = $900
Example: Bakery Muffins and Cupcakes
A bakery had 40 muffins and some cupcakes at first. It sold 6 muffins and 20% of the cupcakes. It then had 50 muffins and cupcakes left in total. How many cupcakes did the bakery have at first?
Finding Original Cupcakes
| Percentage | Value | |
|---|---|---|
| 80% | 16 | |
÷ 4 | ÷ 4 | |
| 20% | 4 | |
× 5 | × 5 | |
| 100% | 20 |
4. Remainder Problems
Key Concept
A fraction or a percentage "of the remainder" is taken from what is LEFT, not from the original!
Strategy
Convert all values to the same form (either all fractions or all percentages).
20% = 1/5
25% = 1/4
50% = 1/2
10% = 1/10
75% = 3/4
33⅓% = 1/3
Example: Fruit Seller's Apples
A fruit seller had a box of apples. He sold 25% of the apples in the morning and 1/6 of the remainder in the afternoon. He had 35 apples left. How many apples were sold altogether?
Step by step:
Morning: Sold 25%, Remainder = 75%
Afternoon: Sold 1/6 of 75% = 75% ÷ 6 = 12.5% of original
Left = 75% - 12.5% = 62.5% of original = 35 apples
62.5% = 35 apples
62.5% = 5 × 12.5%, so 12.5% = 35 ÷ 5 = 7 apples
100% = 8 × 12.5%, so 100% = 7 × 8 = 56 apples
Apples sold = 56 - 35 = 21 apples
5. Percentage of a Percentage
The Formula
X% of Y% = (X/100) × (Y/100) × 100%
70% of 80% = 0.70 × 0.80 × 100% = 56%
Example: Mrs Sim's Buttons
Mrs Sim has some buttons. 20% of the buttons are red and the rest are blue. 70% of the blue buttons are square-shaped and the remaining 60 blue buttons are round. How many buttons does she have in all?
Solution:
Blue buttons = 100% - 20% = 80%
Square blue = 70% of 80% = 56%
Round blue = 30% of 80% = 24%
24% = 60 buttons
4% = 60 ÷ 6 = 10 buttons
100% = 10 × 25 = 250 buttons
Finding Total Buttons
| Percentage | Value | |
|---|---|---|
| 24% | 60 | |
÷ 6 | ÷ 6 | |
| 4% | 10 | |
× 25 | × 25 | |
| 100% | 250 |
6. Percentages and Amounts Together
Example: John's Paper Clips
John has a box of red, green and yellow paper clips. He has 100 yellow paper clips. 30% of the paper clips are green. He has 20 more red paper clips than green paper clips. How many paper clips does John have in all?
Solution:
Green = 30% of the paper clips
Red = green + 20 paper clips = 30% + 20 paper clips
Green and red together = 30% + 30% + 20 paper clips = 60% + 20 paper clips
All the paper clips make 100%.
So the 20 extra red paper clips and the 100 yellow paper clips together make 100% - 60% = 40%.
40% = 20 + 100 = 120 paper clips
10% = 120 ÷ 4 = 30 paper clips
100% = 30 × 10 = 300 paper clips
John has 300 paper clips in all.
Check: green = 30% of 300 = 90, red = 90 + 20 = 110, yellow = 100, and 90 + 110 + 100 = 300.
Key Points to Remember
- ✓ All parts add to 100% in composition problems
- ✓ "Of the remainder" means a part of what is LEFT, not of the original
- ✓ Percentage of a percentage: 70% of 80% = 56%
- ✓ Convert fractions to percentages: 1/5 = 20%, 1/4 = 25%
- ✓ Use bar models to visualise complex relationships
- ✓ Break complex problems into smaller steps
- ✓ Always check: does my answer make sense?