Finding the Original — Reverse Percentage Calculations

In exams, you're often given the final value after a percentage change and asked to find the original. This lesson teaches you three methods — and why the ratio method is the fastest.

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The Three Methods

For any "find the original" problem, you can use:

Method	Approach	Speed
1. Formula	F = I × multiplier → I = F ÷ multiplier	Medium
2. Ratio	Use I:F ratio, find the multiplier	Fastest
3. Direct	Calculate the percentage amount, add/subtract	Slowest

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Method Comparison: Increase 480 by 25%

Method 1 (Formula):

• +25% multiplier = 5/4
• F = 480 × 5/4 = 600

Method 2 (Ratio):

• I : F = 4 : 5
• 480 corresponds to 4 parts, so 1 part = 120
• F = 5 × 120 = 600

Method 3 (Direct):

• 25% of 480 = 480/4 = 120
• 480 + 120 = 600

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Finding Original from Final

Example 1: After 16.67% increase, marks become 105. Find original.

• 16.67% = 1/6
• +16.67% → I : F = 6 : 7
• F = 105 corresponds to 7 parts → 1 part = 15
• I = 6 × 15 = 90

Verify: 90 × 7/6 = 105 ✓

Example 2: After 25% increase, salary becomes Rs. 58,750. Find original.

• 25% = 1/4
• +25% → I : F = 4 : 5
• 58,750 corresponds to 5 parts → 1 part = 11,750
• I = 4 × 11,750 = Rs. 47,000

Example 3: After 16.67% decrease, weight becomes 2,345 kg. Find original.

• 16.67% = 1/6
• −16.67% → I : F = 6 : 5
• 2,345 corresponds to 5 parts → 1 part = 469
• I = 6 × 469 = 2,814 kg

Example 4: After 37.50% increase, money becomes Rs. 7,843. Find original.

• 37.5% = 3/8
• +37.5% → I : F = 8 : 11
• 7,843 corresponds to 11 parts → 1 part = 713
• I = 8 × 713 = Rs. 5,704

Example 5: After 87.50% increase, salary becomes Rs. 23,625. Find original.

• 87.5% = 7/8
• +87.5% → I : F = 8 : 15
• 23,625 corresponds to 15 parts → 1 part = 1,575
• I = 8 × 1,575 = Rs. 12,600

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Tricky Cases: Large Decrease Percentages

Example 6: After 93.33% decrease, toys become 372. Find original.

• 93.33% = 14/15 (since 100% − 93.33% = 6.67% = 1/15)
• −93.33% → remaining = 1/15 of original
• I : F = 15 : 1
• 372 corresponds to 1 part
• I = 15 × 372 = 5,580

Example 7: After 93.75% decrease, balls become 468. Find original.

• 100% − 93.75% = 6.25% = 1/16
• I : F = 16 : 1
• I = 16 × 468 = 7,488

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Finding the New Value After a Change

Example 8: Decrease 2,730 by 28.57%.

• 28.57% = 2/7
• −28.57% → I : F = 7 : 5
• 2,730 corresponds to 7 parts → 1 part = 390
• F = 5 × 390 = 1,950

Example 9: Increase 1,365 by 20%.

• 20% = 1/5
• +20% → I : F = 5 : 6
• 1,365 corresponds to 5 parts → 1 part = 273
• F = 6 × 273 = 1,638

Example 10: Decrease 4,158 by 16.67%.

• 16.67% = 1/6
• −16.67% → I : F = 6 : 5
• 4,158 corresponds to 6 parts → 1 part = 693
• F = 5 × 693 = 3,465

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Finding Original: Formula Method

When the ratio doesn't divide cleanly, use the formula:

I = F × (d / (d ± n)) for percentage = n/d

Example: After 15.38% increase, result = 3,225. Find original.

• 15.38% = 2/13
• +15.38% multiplier = 15/13
• I = 3,225 × 13/15 = 2,795

Example: After 28.57% decrease, result = 1,950. Find original.

• 28.57% = 2/7
• −28.57% multiplier = 5/7
• I = 1,950 × 7/5 = 2,730

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Practice Problems

#	Given	Change	Find	Answer
1	Final = 18,375	+25%	Original	14,700
2	Final = 3,225	+15.38%	Original	2,795
3	Final = 3,465	−16.67%	Original	4,158
4	Final = 1,950	−28.57%	Original	2,730
5	Initial = 1,365	+20%	Final	1,638
6	Final = 58,750	+25%	Original	47,000
7	Final = 23,625	+87.5%	Original	12,600
8	Final = 7,843	+37.5%	Original	5,704
9	Final = 2,345	−16.67%	Original	2,814
10	Final = 372	−93.33%	Original	5,580