The price of a bag was increased by 20%. It was later decreased by 20% of the new price. The bag now costs $96. What was the original price?
The bag cost $100 at first.
Ans: $100
How to solve it
Draw the original price as one bar. The whole bar is 100 percent.
Going up 20 percent makes 120 percent.
Taking 20 percent off the new price leaves 80 percent of 120 percent: 80 over 100 times 120 percent is 96 percent.
Now 96 percent of the original is 96 dollars, so 1 percent is 96 dollars divided by 96, which is 1 dollar.
The original price is 100 percent: 100 times 1 dollar is 100 dollars.
The first 20 percent is 20 percent of the original price. The second is 20 percent of the bigger price. A percentage of a larger amount is bigger, so more comes off than was added.
Practice questions with worked solutions
Question 1 is free to try. The other 9 open in the free trial, each with its full working.
Question 1 of 10 · Basic · free to tryA tree nursery had 5000 saplings. In the first year the number of saplings…A tree nursery had 5000…
A tree nursery had 5000 saplings. In the first year the number of saplings rose by 10%. In the second year it rose by 20% of the number at the end of the first year. How many saplings were there at the end of the second year?
How do you solve percentage of a percentage questions in P6?
When one percentage change follows another, the second is worked out on a different 100%.
What your child learns here
When one percentage change follows another, the second is worked out on a different 100%. This page teaches your child to redraw the whole after each change. It applies percentage increase and decrease to multi-step word problems.
Apply two percentage changes in turn, each on the amount it acts on.
Find a percentage of a percentage, such as 80% of 120%.
Work back to the original amount after two changes.
See why a rise and a fall of the same percentage do not cancel.
Does the answer make sense?
Two decreases in a row take off less than their percentages added, because the second acts on a smaller price, so your child's 20% off then 25% off must come to under 45%. A rise then a fall of the same percentage, like the bag's 20% up and 20% down, ends below the start.
How to practise these questions
The Basic questions train applying a second percentage change to the new amount, not the original. Intermediate questions add two falls combined into one overall percentage, a second discount, and working backwards through successive changes. Challenge questions take a percentage of a quantity that has itself grown, without that quantity ever being named. If a percentage of a percentage gets muddled, read the written step-by-step solution.
Questions parents ask
Why doesn't a 10% rise and a 10% fall cancel out? The fall is 10% of a bigger number. $200 rises to $220, and 10% of $220 is $22, leaving $198.
What does 'percentage of a percentage' mean? It means a percentage is taken of an amount that is itself a percentage of something else. For example, 50% of 40% of a class is 20% of the class.
Which syllabus outcome does percentage of a percentage belong to? It applies Primary 6 outcome 1.2, finding percentage increase/decrease, twice in a row, and outcome 1.1, finding the whole, when the question works backwards.
