Percentage Algebra & Chain Percentages

Two powerful question types that appear frequently in exams: algebraic percentage equations (find the unknown number) and chain percentage comparisons (merge ratios to connect three or more quantities).

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Part 1: Percentage Algebra — Finding the Unknown Number

Type 1: Difference of Two Percentages = Given Value

Example 1: 40% of a number is 60 more than 25% of the same number. Find the number.

• 40% of N − 25% of N = 60
• 15% of N = 60
• N = 60 × 100/15 = 400

Example 2: 12% of N exceeds 18% of N by 96. Wait — 12% < 18%, so:

• 18% of N − 12% of N = 96
• 6% of N = 96
• N = 96 × 100/6 = 1,600

Type 2: Sum/Average of Percentages = Given Value

Example 3: The average of 30% and 50% of a number is 144. Find the number.

• Average = (30% + 50%) / 2 = 40% of N
• 40% of N = 144
• N = 144 × 100/40 = 360

Type 3: Percentage of a Percentage

Example 4: 35% of 40% of a number is 1,050. Find the number.

• 35/100 × 40/100 × N = 1,050
• 14/100 × N = 1,050
• N = 1,050 × 100/14 = 7,500

The General Strategy

1. Convert all percentages to fractions
2. Set up the equation
3. Solve for the unknown
4. Always verify your answer by plugging back in

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Part 2: Chain Percentages — Merging Ratios

When you're told "A is X% of B" and "B is Y% of C", you can find "A as % of C" by merging the ratios.

The Core Method

Convert each percentage to a ratio, then multiply the ratios.

Example 5: A is 75% of B, B is 80% of C. A is what % of C?

Statement	Ratio
A = 75% of B	A/B = 3/4 → A : B = 3 : 4
B = 80% of C	B/C = 4/5 → B : C = 4 : 5

Since B = 4 in both ratios (already equal), merge directly:

A : B : C = 3 : 4 : 5

A as % of C = (3/5) × 100 = 60%

Example 6: A is 60% of B, B is 40% of C. A is what % of C?

Statement	Ratio
A = 60% of B	A : B = 3 : 5
B = 40% of C	B : C = 2 : 5

B is 5 in the first ratio and 2 in the second. Make B equal:

• A : B = 3 : 5 → multiply by 2 → 6 : 10
• B : C = 2 : 5 → multiply by 5 → 10 : 25

Merged: A : B : C = 6 : 10 : 25

A as % of C = (6/25) × 100 = 24%

Example 7: A is 120% of B, B is 150% of C. A is what % of C?

Statement	Ratio
A = 120% of B	A : B = 6 : 5
B = 150% of C	B : C = 3 : 2

Make B equal: A : B = 6 : 5, multiply by 3 → 18 : 15. B : C = 3 : 2, multiply by 5 → 15 : 10.

Merged: A : B : C = 18 : 15 : 10

A as % of C = (18/10) × 100 = 180%

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4-Variable Chains

Example 8: A is 50% of B, B is 60% of C, C is 80% of D. Find A as % of D.

Statement	Ratio
A : B	1 : 2
B : C	3 : 5
C : D	4 : 5

Make B common in first two: A : B = 3 : 6, B : C = 6 : 10 → A : B : C = 3 : 6 : 10

Make C common: A : B : C = 3 : 6 : 10, multiply by 2 → 6 : 12 : 20. C : D = 4 : 5, multiply by 5 → 20 : 25.

Merged: A : B : C : D = 6 : 12 : 20 : 25

A as % of D = (6/25) × 100 = 24%

Shortcut: Just multiply the fractions directly:

A/D = (A/B) × (B/C) × (C/D) = 1/2 × 3/5 × 4/5 = 12/50 = 24%

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"Difference as % of New Value"

A tricky variant: "If salary increases by 16.67%, the difference is what % of the NEW salary?"

• 16.67% = 1/6. I : F = 6 : 7
• Difference = 1, but now base = new value (F) = 7
• (1/7) × 100 = 14.29%

Always check whether the question asks "% of original" or "% of new value" — the base changes the answer completely.

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Practice Problems

#	Problem	Answer
1	35% of N − 20% of N = 225. Find N.	1,500
2	Average of 25% and 45% of N is 280. Find N.	800
3	20% of 30% of N = 1,200. Find N.	20,000
4	A = 80% of B, B = 75% of C. A is what % of C?	60%
5	A = 66.67% of B, B = 60% of C, C = 50% of D. A as % of D?	20%
6	Salary increases by 25%. Increase is what % of new salary?	20%