Percentage Calculator with Step-by-Step Working

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Understanding Percentage Changes

Percentage is one of the trickiest PSLE topics because questions come in many forms. Let’s break down each type.

The Golden Rule: What is 100%?

💡 Always Ask: What is 100%?

In every percentage problem, 100% represents the original value (the starting point).

After a 20% increase, the new value = 120% of original
After a 20% decrease, the new value = 80% of original

Type 1: Percentage Increase

Question type: “A shirt costs $40. After a 25% increase, what is the new price?”

Method 1: Two-Step Method

$\text{Step 1: Find increase} = 25\% \times \$40 = \$10$

$\text{Step 2: Add to original} = \$40 + \$10 = \$50$

Method 2: One-Step Method (Faster!)

$\text{New Value} = \text{Original} \times \frac{100 + \%}{100}$

$\text{New Price} = \$40 \times \frac{125}{100} = \$40 \times 1.25 = \$50$

💡 One-Step Shortcut

For increases: multiply by (100 + percentage) ÷ 100

10% increase → × 1.10
25% increase → × 1.25
50% increase → × 1.50

Type 2: Percentage Decrease

Question type: “A bag costs $80. During a 30% sale, what is the sale price?”

Method 1: Two-Step Method

$\text{Step 1: Find decrease} = 30\% \times \$80 = \$24$

$\text{Step 2: Subtract from original} = \$80 - \$24 = \$56$

Method 2: One-Step Method (Faster!)

$\text{New Value} = \text{Original} \times \frac{100 - \%}{100}$

$\text{Sale Price} = \$80 \times \frac{70}{100} = \$80 \times 0.70 = \$56$

💡 One-Step Shortcut

For decreases: multiply by (100 − percentage) ÷ 100

10% decrease → × 0.90
25% decrease → × 0.75
30% decrease → × 0.70

Type 3: Finding the Original (Reverse Percentage)

This is where most students struggle! The question gives you the final value and asks for the original.

After an Increase

Question type: “After a 20% increase, a TV costs $600. What was the original price?”

⚠️ Common Mistake

WRONG: $600 − 20% of$ 600 = $600 −$ 120 = $480 ❌

This is wrong because 20% should be of the original, not $600!

Correct Method:

After 20% increase: Final = 120% of Original

$120\% \rightarrow \$600$

$1\% \rightarrow \$600 \div 120 = \$5$

$100\% \rightarrow \$5 \times 100 = \$500$

Or using the formula:

$\text{Original} = \text{Final} \div \frac{100 + \%}{100} = \$600 \div 1.20 = \$500$

After a Decrease

Question type: “After a 25% discount, a dress costs $150. What was the original price?”

After 25% decrease: Final = 75% of Original

$75\% \rightarrow \$150$

$1\% \rightarrow \$150 \div 75 = \$2$

$100\% \rightarrow \$2 \times 100 = \$200$

Or using the formula:

$\text{Original} = \text{Final} \div \frac{100 - \%}{100} = \$150 \div 0.75 = \$200$

Type 4: Finding the Percentage Change

Question type: “The price increased from $50 to$ 60. What is the percentage increase?”

Formula

$\text{Percentage Change} = \frac{\text{Change}}{\text{Original}} \times 100\%$

Worked Example

$\text{Change} = \$60 - \$50 = \$10$

$\text{Percentage Increase} = \frac{\$10}{\$50} \times 100\% = 20\%$

❌ Critical: Divide by ORIGINAL

Always divide by the original value, not the new value!

If price goes from $50 to$ 60: divide by $50 (original)
If price goes from $60 to$ 50: divide by $60 (original)

Type 5: Finding Percentage Of

Question type: “Find 35% of 240.”

Method 1: Direct Calculation

$35\% \text{ of } 240 = \frac{35}{100} \times 240 = 84$

Method 2: Break Into Parts

For tricky percentages, break them into easier parts:

$35\% = 25\% + 10\%$

$25\% \text{ of } 240 = 240 \div 4 = 60$

$10\% \text{ of } 240 = 240 \div 10 = 24$

$35\% \text{ of } 240 = 60 + 24 = 84$

Percentage Shortcuts to Memorize

Percentage	Shortcut	Example: % of 80
10%	÷ 10	80 ÷ 10 = 8
20%	÷ 5	80 ÷ 5 = 16
25%	÷ 4	80 ÷ 4 = 20
50%	÷ 2	80 ÷ 2 = 40
5%	÷ 20 (or 10% ÷ 2)	8 ÷ 2 = 4
1%	÷ 100	80 ÷ 100 = 0.8
75%	× 3 ÷ 4	80 × 3 ÷ 4 = 60

PSLE Worked Examples

Example 1: Price After Discount

Problem:

A laptop costs $1,200. During a sale, the price is reduced by 15%. What is the sale price?

Method 1 (Two-Step):

Discount amount = 15% × $1,200 =$ 1,200 × 15 ÷ 100 = $180

Sale price = $1,200 −$ 180 = $1,020

Method 2 (One-Step):

Sale price = $1,200 × (100 − 15) ÷ 100 =$ 1,200 × 0.85 = $1,020

Example 2: Find Original After Increase

Problem:

After a 25% pay raise, John earns $5,000 per month. What was his salary before the raise?

After 25% increase, the new salary represents 125% of the original.

125% → $5,000

1% → $5,000 ÷ 125 =$ 40

100% → $40 × 100 = **$ 4,000**

Check: $4,000 + 25% of$ 4,000 = $4,000 +$ 1,000 = $5,000 ✓

Example 3: Two Successive Changes

Problem:

A shop increases prices by 20%, then offers a 20% discount. If an item originally cost $100, what is the final price?

Step 1: After 20% increase

New price = $100 × 1.20 =$ 120

Step 2: After 20% discount

Final price = $120 × 0.80 = **$ 96**

⚠️ Not $100!

Many students think +20% then −20% gives back $100. But the 20% discount is on$ 120, not $100!

Example 4: Combined GST and Discount

Problem:

A meal costs $80 before GST. After adding 9% GST, a 10% member discount is applied. What is the final bill?

Step 1: Add 9% GST

Price with GST = $80 × 1.09 =$ 87.20

Step 2: Apply 10% discount

Final bill = $87.20 × 0.90 = **$ 78.48**

Common Mistakes to Avoid

❌ Mistake 1: Wrong Base for Reverse Problems

When finding the original after an increase/decrease, don’t calculate % of the final value!

Wrong: Original = $600 − 20% of$ 600 ❌ Right: $600 represents 120%, find 100% ✓

❌ Mistake 2: Adding/Subtracting Percentages Directly

A 20% increase followed by 20% decrease does NOT equal 0% change!

The second percentage applies to the NEW value, not the original.

❌ Mistake 3: Dividing by New Instead of Original

When finding percentage change, always divide by the ORIGINAL value.

$50 →$ 60 is a 20% increase (divide by 50) $60 →$ 50 is a 16.67% decrease (divide by 60)

❌ Mistake 4: Forgetting to Convert % to Decimal

25% means 25 ÷ 100 = 0.25, not 25!

25% of 80 = 0.25 × 80 = 20 (not 2000!)

Quick Reference Card

Problem Type	Formula
Find X% of Y	Y × X ÷ 100
Increase Y by X%	Y × (100 + X) ÷ 100
Decrease Y by X%	Y × (100 − X) ÷ 100
Original (after X% increase)	Final ÷ (100 + X) × 100
Original (after X% decrease)	Final ÷ (100 − X) × 100
% change from A to B	(B − A) ÷ A × 100%

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