Percentage for CDS

Percentage is the most connected topic in CDS mathematics. Profit and loss, simple and compound interest, population growth, data questions and even some algebra depend on it. A candidate who is quick and accurate with percentages gains time across the whole paper, not just on the handful of questions labelled "percentage".

The habit that makes the difference is to think of percentages as fractions and multipliers. A 20% increase means multiply by 1.2. A 25% decrease means multiply by 0.75. Once this is automatic, most questions become one or two lines, and the traps (such as adding successive changes) stop working on you.

Percentages as fractions

With these, 37.5% of 640 is simply 3/8 × 640 = 240, and 16⅔% of 546 is 546 ÷ 6 = 91.

The core ideas

Percentage change

Change % = (new − old) ÷ old × 100. The base is always the old value, the one you are comparing from.

Multipliers

An increase of r% multiplies by (1 + r/100). A decrease of r% multiplies by (1 − r/100). Several changes in a row multiply together, which is why they cannot simply be added.

Successive changes

For changes of a% and then b%, the net change is a + b + ab/100 per cent, using negative values for decreases.

Why: (1 + a/100)(1 + b/100) = 1 + a/100 + b/100 + ab/10,000. Converting back to a percentage gives a + b + ab/100. For three or more changes, apply the formula twice or simply multiply the multipliers.

"More than" and "less than"

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

Why: the base changes. If B = 100 and A = 125, the difference is 25 either way, but it is 25% of B and only 20% of A.

Price and consumption

Spending = price × quantity. If the price rises by r%, consumption must fall by r ÷ (100 + r) × 100 per cent to keep spending the same. If the price falls by r%, consumption can rise by r ÷ (100 − r) × 100 per cent. This is the same "base changes" idea in another story.

Growth and depreciation

A population growing at r% a year for n years is multiplied by (1 + r/100)ⁿ. A machine depreciating at r% a year is multiplied by (1 − r/100)ⁿ. To go backwards in time, divide by the multiplier instead.

Worked questions

Question 1: A town's population grows by 10% a year and is now 12,100. What was it two years ago?

• Two years of growth multiply by 1.1 × 1.1 = 1.21.
• Going back: 12,100 ÷ 1.21 = 10,000.

Question 2: A's salary is 25% more than B's. By what per cent is B's salary less than A's?

• Take B = 100, so A = 125. The difference is 25.
• As a share of A: 25 ÷ 125 × 100 = 20%.

Question 3: The price of sugar rises by 25%. By what per cent must a household cut its consumption to keep its spending unchanged?

• Spending = price × quantity. The new price multiplier is 1.25, so quantity must be multiplied by 1/1.25 = 0.8.
• The cut is 20%. (The formula gives the same: 25 ÷ 125 × 100.)

Question 4: The price of an item rises by 20% and a family reduces its consumption by 10%. What happens to its spending?

• Spending multiplier = 1.2 × 0.9 = 1.08.
• Spending rises by 8%. The formula agrees: 20 − 10 − 200/100 = 8.

Question 5: In an election between two candidates, 10% of the voters on the list did not vote, and 10% of the votes cast were invalid. The winner got 60% of the valid votes and won by 1,620 votes. How many voters were on the list?

• Let the list have N voters. Votes cast = 0.9N. Valid votes = 0.9 × 0.9N = 0.81N.
• The winner has 60% and the loser 40% of the valid votes, so the margin is 20% of 0.81N = 0.162N.
• 0.162N = 1,620, so N = 10,000.

Question 6: One candidate scores 30% and fails by 15 marks. Another scores 45% and gets 30 marks more than the pass mark. Find the maximum marks and the pass percentage.

• Let the maximum be M. The pass mark is 0.30M + 15, and also 0.45M − 30.
• So 0.15M = 45, giving M = 300.
• Pass mark = 90 + 15 = 105, which is 105 ÷ 300 × 100 = 35%.

Percentage and percentage points

If a pass rate rises from 40% to 50%, it has risen by 10 percentage points but by 25 per cent (10 is a quarter of 40). CDS options sometimes include both numbers. Read whether the question asks about the rate or the change in the rate.

Practice set

1. Find 15% of 480.
2. A value is increased by 30% and then decreased by 30%. Find the net change.
3. A price falls by 20%. By what per cent can consumption rise for the same spending?
4. 25% of what number is 60?
5. A's income is 20% less than B's. By what per cent is B's income more than A's?
6. A number is increased by 10% and then by 20%. Find the total increase.
7. A town of 20,000 people grows by 5% in one year and falls by 5% the next. Find the population after two years.
8. A student needs 36% to pass. The student scores 123 marks and fails by 30 marks. Find the maximum marks.

Answers

1. 72. 10% is 48 and 5% is 24, so 15% is 72.
2. A 9% decrease. 30 − 30 − 900/100 = −9. Or 1.3 × 0.7 = 0.91.
3. 25%. 20 ÷ 80 × 100 = 25.
4. 240. 25% is a quarter, so the number is 4 × 60.
5. 25%. Take B = 100 and A = 80. The difference 20 as a share of A is 20 ÷ 80 × 100 = 25.
6. 32%. 10 + 20 + 200/100 = 32. Or 1.1 × 1.2 = 1.32.
7. 19,950. 20,000 × 1.05 × 0.95 = 20,000 × 0.9975 = 19,950.
8. 425. The pass mark is 123 + 30 = 153, which is 36% of the maximum. 153 ÷ 0.36 = 425.

What to do next

• Learn the percentage–fraction table until each pair comes to mind instantly
• Rewrite every percentage in your next ten questions as a multiplier before solving
• Solve the percentage questions from the last five CDS papers, timed at a minute each
• Apply the same multiplier method in profit and loss and simple and compound interest