2.4 Percentages

Why This Matters

Percentages power a third of the arithmetic paper indirectly. Profit-loss, simple and compound interest, population, election statistics, and data interpretation all rest on percentage fluency. RRB typically sets 3 to 4 direct percentage questions per cycle.

Percentage Basics

"Percent" means "per hundred". The symbol % is shorthand for /100, so 25% = 25/100 = 0.25 = 1/4.

Three forms to memorise for common values:

Converting any fraction to a percentage: multiply by 100 and add %. For 3/8: (3/8) * 100 = 37.5%.

The Two Master Formulas

• Percentage increase or decrease = (Change / Original) * 100
• New value = Original * (1 + p/100) for an increase, or Original * (1 - p/100) for a decrease, where p is the percentage.

Be careful which value is the "original" (the base). The base is always the number you are taking the percentage of. "X% of Y" means (X/100) * Y, with Y as the base.

Successive Percentage

When two changes happen one after another, the net percentage is not their simple sum. Use:

Net percentage change = a + b + (a * b)/100

where a and b are the percentage changes (with sign: increases positive, decreases negative).

Worked example. A salary rises 10% in year 1 and 20% in year 2. Net rise = 10 + 20 + (10 * 20)/100 = 32%. A common wrong answer is 30%.

This formula also handles decreases. A price falls 20% then rises 20%: net = -20 + 20 + (-20 * 20)/100 = -4%. The price ends up 4% lower, so symmetry does not cancel out.

Percentage Increase and Decrease Word Problems

Problem 1. A number is increased by 20% then the new number is decreased by 20%. Find the net change. Using the formula: 20 - 20 + (20 * -20)/100 = -4. Net change = -4%, so the number is 4% smaller than the original.

Problem 2, population. A town's population is 50,000. It grows at 10% per year. What is the population after 2 years? This is successive growth. After year 1: 50,000 * 1.10 = 55,000. After year 2: 55,000 * 1.10 = 60,500. Directly, 50,000 * (1.10)^2 = 50,000 * 1.21 = 60,500.

Problem 3, election. In an election with two candidates, a candidate got 40% of the votes and lost by 600 votes. Find the total number of voters. The winner's share = 60%, the loser's = 40%, so the difference is 20%. This 20% equals 600 votes, so 1% = 30 and 100% = 3,000 voters.

Successive Percentage Quick Table

Notice that equal up-then-down (or down-then-up) always produces a loss, a useful trap detector.

More Worked Examples

Example, fraction to percentage. Convert 7/25 to a percentage. Multiply by 100: (7/25) * 100 = 7 * 4 = 28%. Equivalently, 7/25 = 28/100 = 0.28, which is 28% by definition. For a quick mental path, find the equivalent fraction with denominator 100.

Example, percentage comparison with different bases. If A is 25% more than B, what percentage is B less than A? Let B = 100, so A = 125. The difference is 25, but the base for the reverse comparison is A (125), so B is (25/125) * 100 = 20% less than A. The two percentages differ because the bases differ: 25% of B equals 20% of A. RRB loves this asymmetry.

Example, price fall and consumption adjustment. The price of sugar falls by 20%. By what percentage must a family's consumption increase so that the monthly expenditure on sugar stays unchanged? Let the original price be P and the original consumption be C, so expenditure = P * C. The new price is 0.8 P. To keep expenditure the same: 0.8 P * new consumption = P * C, giving new consumption = C / 0.8 = 1.25 C. So consumption must rise by 25%. The shortcut formula is: if the price falls by x%, consumption must rise by (100x / (100 - x))% to keep expenditure fixed. For x = 20, that is (20 * 100) / 80 = 25%.

Example, price rise and consumption adjustment. If the price rises by 25% instead, the family must cut consumption by (100x / (100 + x))% = (25 * 100) / 125 = 20% to keep expenditure fixed. The two formulas are mirror images: falling price needs rising consumption, rising price needs falling consumption, and the percentage is never the same as the price change because the base shifts.

Example, voter turnout with non-voters. In an election, 20% of voters on the roll did not cast their votes. Of the votes cast, one candidate secured 40% and lost by 600 votes. Find the total roll. Let the total roll be N. Votes cast = 0.8 N. The losing candidate gets 40% of 0.8 N = 0.32 N; the winner gets 60% of 0.8 N = 0.48 N. The difference is 0.16 N = 600, so N = 3,750. Two traps sit inside this one problem: the 20% who stayed home shrink the base from N to 0.8 N, and the candidate with 40% of the votes cast is necessarily the loser in a two-way contest — if the question says that candidate won, the wording is inconsistent and you should re-read it.

Example, successive reduction of a quantity. A tank's water level drops by 10% on day 1 and by a further 20% on day 2 of what remains. What is the net drop? Use the successive formula with negative signs: -10 - 20 + ((-10) * (-20))/100 = -30 + 2 = -28%. The net drop is 28%, not 30%, because the second percentage applies to the smaller amount left after day 1.

Common Percentage Shortcut Results

The last row is the same result that appears in profit-loss: equal up-then-down always loses. Memorising these four shortcuts turns a multi-line word problem into a single-line substitution, saving 30 to 40 seconds per question under the 90-minute clock.

Exam Scenarios and Common Traps

• Wrong base: "A is 20% more than B" means A = 1.2 B, so B = A/1.2 = 0.833 A. The percentage that B is less than A is not 20%; it is (0.2/1.2) = 16.67%.
• Adding successive percentages: never simply add; use a + b + (a * b)/100.
• Election "lost by" wording: the gap in percentage points equals the difference of vote shares; convert that to vote count via the total.
• "Of" before "off": "20% of 50" gives 10 (a part); "20% off 50" gives 40 (a discounted result). The wording changes the operation entirely.