Percentage for LIC AAO

Percentage is not really a chapter. It is the language the rest of Quant is written in. A DI question asks for growth or share, a profit question asks for profit per cent, an interest question is a percentage applied year after year. If you can take any percentage of any number in your head, and you always know which number is the base, you have made every other topic faster. This guide covers the handful of ideas that do the work, with the reason each one holds.

Fractions you should know as percentages

Division by 8 or 6 is slow. Multiplying by 12.5% or 16.67% is not, because you already know them as fractions.

So 37.5% of 640 is 3/8 × 640 = 240, and 16.67% of 1,020 is 1,020 ÷ 6 = 170.

Splitting. Any percentage can be built from 10%, 5% and 1%. For 35% of 640: 10% is 64, so 30% is 192; 5% is 32; total 224.

The core ideas and why they work

Change is always on the old value

Percentage change = (new − old) ÷ old × 100.

The base is the starting point because "change" means change from where you were. A rise from 800 to 1,000 is 25%, but a fall from 1,000 to 800 is 20%. The difference is the same 200, but the bases differ.

Successive changes

A change of a% followed by b% gives a net change of a + b + ab/100 per cent. Use a minus sign for a fall.

Why: multiplying by (1 + a/100) and then by (1 + b/100) gives 1 + (a + b)/100 + ab/10,000. Subtracting 1 and multiplying by 100 leaves a + b + ab/100.

"More than" and "less than" are not mirror images

If A is r% more than B, then B is r ÷ (100 + r) × 100 per cent less than A.

Why: take B = 100. Then A = 100 + r, and the gap r is now measured against A, which is 100 + r.

Keeping a total fixed

Expenditure = price × consumption. If the price rises by r%, consumption must fall by r ÷ (100 + r) × 100 per cent to keep expenditure unchanged. This is the same idea as the one above.

Percentage points versus per cent

If a claim settlement ratio moves from 92% to 96%, it has risen by 4 percentage points, but by 4/92 ≈ 4.35 per cent. DI questions use both, so read the words.

Worked examples

Example 1. A premium of ₹12,000 rises by 15%. What is the new premium?
10% = 1,200 and 5% = 600, so 15% = 1,800.
New premium = ₹13,800. Or directly: 12,000 × 1.15.

Example 2. A value rises by 20% and then falls by 10%. What is the net change?
20 − 10 + (20 × −10)/100 = 10 − 2 = +8%. Check: 100 → 120 → 108.

Example 3. A's premium is 25% more than B's. By what percentage is B's less than A's?
25 ÷ 125 × 100 = 20%. Check: B = 100, A = 125; 25/125 = 1/5.

Example 4. 36% of a number is 540. What is the number?
1% = 540 ÷ 36 = 15, so 100% = 1,500.

Example 5. The price of an item rises by 25%. By what percentage must a buyer cut consumption to keep spending the same?
25 ÷ 125 × 100 = 20%. Check: price 100 → 125; buying 80% as much costs 125 × 0.8 = 100.

Example 6. A settlement ratio rises from 92% to 96%. Describe the change.
4 percentage points, or 4 ÷ 92 × 100 ≈ 4.35% as a relative rise.

Example 7. A branch has 20,000 policyholders. The number grows by 10% a year. How many will there be after two years?
20,000 × 1.1 × 1.1 = 24,200. Growth compounds: the second year's 10% is on 22,000, not 20,000.

Example 8. A candidate who scores 30% fails by 30 marks. Another who scores 45% gets 30 marks more than the pass mark. Find the maximum marks and the pass mark.
The gap between the two scores is 30 + 30 = 60 marks, and it equals 45% − 30% = 15% of the maximum.
Maximum = 60 ÷ 0.15 = 400. Pass mark = 30% of 400 + 30 = 150. Check: 45% of 400 = 180, and 180 − 150 = 30.

Where percentages go wrong

• Using the new value as the base for a change.
• Treating "r% more" and "r% less" as reversible.
• Reading "150% of" as "150% more than". 150% of 100 is 150; 150% more than 100 is 250.
• Mixing percentage points with per cent in DI answers.

Practice set

1. What is 18% of 2,500? 450. 10% = 250, 8% = 200.
2. Two successive increases of 10%: net change? 21%. 10 + 10 + 1.
3. 60 is what percentage of 240? 25%. 60/240 = 1/4.
4. A value falls from 800 to 680. Percentage fall? 15%. 120 ÷ 800.
5. A value rises by 30% and then falls by 30%. Net change? −9%. 30 − 30 − 900/100.
6. A person spends 75% of their income. Income rises by 20% and spending by 10%. By what percentage do savings rise? 50%. Income 100 → 120, spending 75 → 82.5, savings 25 → 37.5.
7. 40% of A equals 60% of B. By what percentage is A more than B? 50%. 0.4A = 0.6B, so A : B = 3 : 2.
8. The price of an item falls by 20%. By what percentage can consumption rise for the same spending? 25%. 20 ÷ 80 × 100.

What to do next

• Learn the fraction table this week until each pair is instant.
• Do ten percentage-of calculations a day by splitting into 10%, 5% and 1%.
• Apply these ideas in profit and loss and table DI.
• Revise successive change again when you reach simple and compound interest.