8.3.4 Fact or Fiction?

Although percentages are used quite often in everyday life, they can be misleading and even a little tricky. In the next activity, you are asked to decide whether some statements are true or false, and to explain why.

Activity: Fact or Fiction?

For each of the following statements, say whether or not it is correct. Try working out some numerical examples to test your conjectures. Can you justify your results?

(a) If the price of a piece of jewelry is increased by 20% and then decreased by 20%, the item returns to its original price.

Hint icon

Comment

Pick any easy amount, like $100, to work with. Increase it by 20%, then take that new amount and decrease it by 20%.

Solutions icon

a. 

True


b. 

False


The correct answer is b.

Answer

(a) Let’s try it using $100. If we apply a 20% increase, we will have 120% of the starting amount. 120% of $100 is 1.20 postfix multiplication dollar 100 equals dollar 120. If we then apply a 20% decrease to this new amount, we will have 80% left. 80% of $120 is 0.8 postfix multiplication dollar 120 equals dollar 96. So, the item does not return to its original price. The statement is false.

This happens for any price. To apply an increase of 20%, you multiply the price by 1.2, and then to apply a decrease of 20%, you multiply by 0.8. This is the same as multiplying by 1.2 multiplication 0.8, which is 0.96. The new price will always be only 96% of the original price.

This also happens for any percentage. If, for instance, you apply an increase of 40% and then decrease it by 40%, the result is the same as multiplying by 1.4 and then by 0.6. With a 40% price increase and decrease, the end result will always be 84% of the original price.

(b) If the price of a pair of jeans is decreased by 20%, and two pairs are purchased, then the total paid is a 40% decrease from the original price.

Hint icon

Comment

Select an easy number to work with and see what happens!

Solutions icon

a. 

True


b. 

False


The correct answer is b.

Answer

(b) Increasing a number by a certain percent and then decreasing it by the same percent will not result in the original number.

Suppose the pair of jeans costs $100. If we apply a 20% reduction, the new price will only be 80% of the original cost. 80% of $100 is 0.8 postfix multiplication dollar 100 equals dollar 80. So, if you buy two pairs at $80 each, the total is $160 instead of $200, a reduction of $40.

The overall percent decrease is therefore 40 divided by 200 multiplication 100 percent equals 20 percent [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] . Thus, the statement is not correct.

To help convince you that this is the case, a diagram might help. You can see that the discount on the two items together is still only 20% of the total price.

8.3.5 Percentage Points