How do you calculate successive percentage changes?

A shop raises the price of a Rs.100 phone cover by 20% to Rs.120. Then there is a 20% sale. 20% of 120 is 24, so the price becomes Rs.96. You end up 4% lower, not back at Rs.100, because the second percentage was taken on a bigger number. That is why you cannot just add percentages that happen one after another.

Formula for two successive changes a% and b%:

Net change = a + b + (a x b) / 100

• +20% then -20%: 20 - 20 + (20 x -20) / 100 = -4%.
• +10% then +10%: 10 + 10 + 100 / 100 = +21%.
• A salary rises by 10% and then by 20%: 10 + 20 + 200 / 100 = +32%.

The multiplier method, which works for any number of changes:

const multipliers = [1.10, 1.20];            // +10% then +20%
const net = multipliers.reduce((p, m) => p * m, 1);
console.log(Math.round((net - 1) * 100)); // Output: 32

A change of a% multiplies the value by (1 + a/100). Two changes multiply to (1 + a/100)(1 + b/100) = 1 + (a + b)/100 + ab/10000. Subtract 1 and multiply by 100, and you get a + b + ab/100. For three changes, apply the formula to the first two, then apply it again with the third, or just multiply all the multipliers.

Shopping discounts ("extra 10% off on already 40% off" is 46% off, not 50%), salary hikes over years, population growth, stock price moves and GST on a discounted price all use this. It appears in nearly every placement aptitude test and in bank and SSC exams.

• Adding the percentages directly: 40% off plus 10% off is not 50% off.
• Forgetting the sign: a decrease must be negative in the formula.
• Thinking an x% rise followed by an x% fall brings you back. It always gives a net loss of x squared / 100 percent.
• Rounding in the middle of a long calculation.

• A price rises 25%. By what percentage must you use less so your spending stays the same?
• Population grows 10% a year. What is the total growth in 3 years?
• A number is increased by 30% and then decreased by 30%. What is the net change?