Successive discounts compound

The idea

Successive discounts written as a sum of percentages are applied one after another, each on the already reduced price, so they compound and give less than a single discount of their total.

"30% + 20%" or "50%"?

Cakely offers a 30% + 20% discount on all cakes. Cakify offers 50% off. A cake costs ₹200.

Which bakery gives the cheaper cake?

• Cakely, with 30% + 20%
• Cakify, with 50%
• Both give the same price

One discount after another

Set the first and the second discount. The second one is taken off the price that is left after the first. Compare with one single discount of the two added together.

The second discount is on the reduced price

Successive discounts written as a sum of percentages, like 30% + 20%, are applied one after another, each on the already reduced price, so they compound and give less than a single discount of their total.

Cakely, 30% + 20% on a ₹200 cake: First discount: 200 − 60 = ₹140 Second discount, 20% of ₹140: 140 − 28 = ₹112

Cakify, 50% on ₹200: 200 − 100 = ₹100 Cakify is cheaper by ₹12.

In one step: after 30% off you pay 70%, and after 20% off you pay 80% of that: 0.7 × 0.8 = 0.56. You pay 56%, so the two discounts together are only 44% off, not 50%. Taking 20% first and 30% second gives the same ₹112.

Notes

Successive discounts written as a sum of percentages, like 30% + 20%, are applied one after another, each on the already reduced price, so they compound and give less than a single discount of their total.

Check yourself

A jacket is marked ₹500. The shop offers 20% + 10% off, one after the other. What is the final price (in ₹)?

Answer: 360

After 20% off: 500 − 100 = ₹400 After 10% more: 400 − 40 = ₹360

A shirt is marked ₹1000. It gets two discounts of 10% one after the other. What is the final price (in ₹)?

Answer: 810

After the first 10%: 1000 − 100 = ₹900 After the second 10%: 900 − 90 = ₹810

A shop offers 40% + 10% off. How does this compare with 50% off?

A sofa is marked ₹10,000. The shop gives 50% + 20% off. What is the final price (in ₹)?

Answer: 4000

After 50% off: 10,000 − 5,000 = ₹5,000 After 20% more: 5,000 − 1,000 = ₹4,000

• It takes off less than 50% — correct. Yes! 40% off leaves 60%, then 10% off leaves 90% of that: 0.6 × 0.9 = 0.54. You pay 54%, so the discount is only 46%.
• It is exactly the same as 50% off. The 10% is taken off the already reduced price, so the two do not simply add up. The discount is 46%, not 50%.
• It takes off more than 50%. The second discount is on a smaller price, so it takes off less than the 10% of the original. Together they are only 46%.