Percent increase and decrease of a number

 Percent increase or decrease of a number

What is 84 increased by 20% and then decreased by 5%?
Convert 20% increase and 5% decrease to decimal and multiply all the numbers.
20% increase  =  1+0.2  =  1.2
5% decrease  =  1 − 0.05  =  0.95
1.2 × 0.95 × 84  =  95.76

It is important to remember that subsequent increases and decreases of a number cannot be added. In the above question, 20%  5%  =  15% increase is incorrect.

Explanation on percent increase and percent decrease:
Percent is about parts of a number per 100. 
20% increase means 20 more parts of a number per 100. 
100 + 20  = 120% 
5% decrease means 5 less parts of a number per 100. 
100 − 5  =  95% 

To make it simpler, start with decimals as shown in the above solution. 
As a decimal, 100% is 1.0. 
As a decimal, 20% is 0.2. Hence, 20% increase is 1 + 0.2  =  1.2. 
As a decimal, 5% is 0.05. Hence, 5% decrease is 1   0.05  =  0.95.

If you are not familiar with converting a percent to a decimal or vice versa, follow THIS LINK to read the post.

Try the following:
1.  What is 164 decreased by 12% and then increased by 30%?
2.  If a number 𝑥 is decreased by 60% and then decreased by 50%, what is the final value of 𝑥? The initial value of 𝑥 is 80.
3.  If a number is increased by 5% and then increased by 40%, what is the final percent increase of the number?

Show/Hide Answers
1.  0.88 × 1.3 × 164  =  187.616
2.  0.4 × 0.5 
× 80  =  16
3.  
1.05 × 1.4
  =  1.47. Since 1.47 is 147%, the final increase is 147 − 100 = 47%

Word problems on percent increase or decrease of a number

Things to remember on word problems:
- Percent discount is percent decrease.
- Sales tax added to an item is percent increase.

A toy store has discounted all the toys in the store by 20%. Every Sunday, an additional 5% discount is applied over the already discounted price on all the toys. If the original price of a toy Natanya bought last Sunday was $40 and no sales tax was collected, how much did Natanya pay for the toy? 
Identify the percent increases or decreases from the word problem. Convert them to decimal and multiply all the numbers.
20% discount  =  20% decrease  =  1  0.2  =  0.8
5% discount  =  5% decrease  =  1 − 0.05  =  0.95
0.8 × 0.95 × 40  =  30.40

Again remember not to add percent increases and decreases. The discount on Sunday is offered on the price that is already discounted by 20%. Adding the discounts is incorrect.

If the last sentence of the word problem is modified as 'If the original price of a toy Natanya bought last Sunday was $40 and 8% sales tax was collected, how much did Natanya pay for the toy, inclusive of sales tax?', then the solution will be as follows.
20% discount  =  20% decrease  =  1  0.2  =  0.8
5% discount  =  5% decrease  =  1 − 0.05  =  0.95
8% sales tax  =  8% increase  =  1  0.08  =  1.08
0.8 × 0.95 × 1.08 × 40  =  32.83

Answer page 376 in the sample screen images of my SAT Math Manual and Workbook book shows how every percent increase/decrease question can be solved using the above method. 

Try the following:
1.  The original price of the computer Roma wants to buy from an online store is $430. This week the online store is offering a 10% discount on that computer. Roma has a coupon that will give her 4% discount over the already discounted price. If Roma buys the computer this week and uses the coupon, how much will she pay in total, assuming no sales tax is collected?
2.  The current population of a certain town is 34,000. If the population of the town is expected to increase by 5% each year in the next two years, what will be the population of the town in 2 years?
3.  Jamie bought a scarf that was discounted by 20%. The original price of the scarf was $50. At the checkout, Jamie received an additional 5% discount on the discounted price and 6% sales tax was added. What was the final price Jamie paid for the scarf?

Show/Hide Answers
1.  0.9 × 0.96 × 430  =  371.52
2.  1.05 × 1.05 
× 34,000  =  37,485 (Population increase in second year will be on the population that was increased by 5% in the first year. Hence, adding them to 10% is incorrect.)
3.  
0.8 × 0.95
 × 1.06 × 50  =  40.28