Successive Percentage Change — The Fraction Chain Method

When a value changes by one percentage and then by another, you cannot simply add the percentages. The base changes after each step. This lesson teaches the fraction chain method — the fastest way to handle successive changes.

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Why You Can't Simply Add Percentages

Consider: A value increases by 10% in 2012, then by 20% in 2013.

Wrong: 10% + 20% = 30% total increase ✗

Right:

• Start: 1000
• After +10%: 1000 × 11/10 = 1100
• After +20%: 1100 × 6/5 = 1320
• Actual change = (1320 − 1000)/1000 × 100 = 32% (not 30%)

Why the difference? The 20% increase is applied to 1100 (not 1000), so you get an extra 20% of the first increase.

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The Fraction Chain Method

Convert each percentage change to a fraction multiplier and multiply them all together in a chain.

Steps:

1. Convert each percentage to a fraction (use the table from Lesson 2)
2. For increase: multiplier = 1 + fraction = (d+n)/d
3. For decrease: multiplier = 1 − fraction = (d−n)/d
4. Multiply all multipliers together
5. Apply to the initial value

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Two Successive Changes

Example 1: Price increases by 9.09% then by 25%. Overall change?

Change	Fraction	Multiplier
+9.09%	1/11	12/11
+25%	1/4	5/4

Chain: I × 12/11 × 5/4 = I × 15/11

Overall ratio I : F = 11 : 15. Change = 4/11 × 100 = 36.36% increase

Example 2: Mobile price Rs. 8,640. Increased by 12.50% then by 20%.

Change	Fraction	Multiplier
+12.5%	1/8	9/8
+20%	1/5	6/5

F = 8640 × 9/8 × 6/5 = 8640 × 54/40 = 11,664

Example 3: Share value Rs. 6,300. Reduced by 16.67% then by 14.29%.

Change	Fraction	Multiplier
−16.67%	1/6	5/6
−14.29%	1/7	6/7

F = 6300 × 5/6 × 6/7 = 6300 × 5/7 = 4,500

Notice how 6 cancels out — this simplification is the power of the fraction chain!

Example 4: 2,700 coins increase by 33.33%, then decrease by 11.11%.

Change	Fraction	Multiplier
+33.33%	1/3	4/3
−11.11%	1/9	8/9

F = 2700 × 4/3 × 8/9 = 2700 × 32/27 = 3,200

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Three Successive Changes

Example 5: 23,125 increased by 14.29%, 16.67%, and 20%.

Change	Fraction	Multiplier
+14.29%	1/7	8/7
+16.67%	1/6	7/6
+20%	1/5	6/5

F = 23,125 × 8/7 × 7/6 × 6/5

Look at the cancellations: 7 cancels, 6 cancels!

F = 23,125 × 8/5 = 23,125 × 8/5 = 37,000

Example 6: Salary Rs. 17,920. Increased by 12.50%, 11.11%, and 25%.

Change	Multiplier
+12.5%	9/8
+11.11%	10/9
+25%	5/4

F = 17,920 × 9/8 × 10/9 × 5/4

Cancellations: 9 cancels!

F = 17,920 × 10/8 × 5/4 = 17,920 × 50/32 = 28,000

Example 7: 37,800 decreased by 33.33%, 8.33%, and 9.09%.

Change	Multiplier
−33.33%	2/3
−8.33%	11/12
−9.09%	10/11

F = 37,800 × 2/3 × 11/12 × 10/11

Cancellations: 11 cancels!

F = 37,800 × 2/3 × 10/12 = 37,800 × 20/36 = 21,000

Example 8: Factory output 46,800. Decreased by 37.50%, 7.69%, and 5.56%.

Change	Multiplier
−37.5%	5/8
−7.69%	12/13
−5.56%	17/18

F = 46,800 × 5/8 × 12/13 × 17/18 = 25,500

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Finding Original from Final After Successive Changes

Reverse the chain — multiply by the inverse of each multiplier.

Example 9: After −22.22% and +10%, value becomes Rs. 16,170. Find original.

Change	Multiplier	Inverse
−22.22%	7/9	9/7
+10%	11/10	10/11

Original = 16,170 × 10/11 × 9/7 = 18,900

Example 10: After −55.56% and +18.18%, value becomes Rs. 13,000. Find original.

Change	Multiplier	Inverse
−55.56% (= 5/9)	4/9	9/4
+18.18% (= 2/11)	13/11	11/13

Original = 13,000 × 11/13 × 9/4 = 24,750

Example 11: After +87.50% and −28%, final = Rs. 12,960. Find original.

Change	Multiplier	Inverse
+87.5% (= 7/8)	15/8	8/15
−28% (= 7/25)	18/25	25/18

Original = 12,960 × 25/18 × 8/15 = 9,600

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The Key Insight: Cancellations

The fraction chain method is powerful because fractions cancel beautifully. When one change has a numerator that matches another's denominator, they simplify instantly.

This is especially true for percentages like:

• 14.28% (1/7) and 16.67% (1/6) → multipliers 8/7 and 7/6 → 7s cancel
• 11.11% (1/9) and 12.5% (1/8) → multipliers 10/9 and 9/8 → 9s cancel
• 33.33% (1/3) and 25% (1/4) → multipliers 4/3 and 5/4 → could simplify

Exam strategy: Always convert to fractions first. Look for cancellations before multiplying. This can turn a complex 3-step calculation into a single simple multiplication.

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Exam Speed Tricks for Successive Changes

Instant Formulas (no calculation needed)

Question Pattern	Instant Answer
+x% then −x% (same %)	Net loss = x²/100 % (always a loss). E.g., +20% then −20% = −4%
+10% then +10%	Net = +21% (not 20%)
+10% three times	Net = +33.1%
−10% then −10%	Net = −19% (not −20%)
Price ↑ x%, consumption ↓ to keep expenditure same	Consumption must decrease by x/(100+x) × 100 %
Price ↑ 25%, keep expenditure same	Consumption ↓ by 25/125 × 100 = 20% (use fraction: 1/4 up → 1/5 down)

The "Fraction Shortcut" for Expenditure-Constant Problems

If price rises by 1/n, reduce consumption by 1/(n+1) to keep spending same.

Price ↑ by	= fraction	Consumption ↓ by	= fraction
10%	1/10	9.09%	1/11
20%	1/5	16.67%	1/6
25%	1/4	20%	1/5
33.33%	1/3	25%	1/4
50%	1/2	33.33%	1/3

This table alone can solve 80% of price-consumption questions in under 10 seconds.