In general, people tend to think of percent increases and decreases as absolute measurements. They should be thought of as relative measurements. This post includes sample problems that cover global warming, subway fare hikes, the stock market, and the unreasonably high cost of coffee. New challenges will be sent out on a weekly basis.

“The price of a sandwich was raised from $6.25 to $6.75. What was the percent increase in the price?”

— 2021 SHSAT Sample Test A, #97

Why Percents are tricky, and should not be thought of as absolute measurements

Consider this…

On Monday, the price of pair of gloves was $10
On Tuesday, the price increased by 20% – bringing the price up to $12
On Wednesday, the price decreased by 20% – what was the final price?
- Most people answer: $10
- Correct answer: $9.60

Decomposing % problems into starting points and ending points – kind of like a journey

Let’s unpack the first example point by point:

On Monday
- Our starting point = $10
- Nothing happens on Monday
- Our ending point = $10
On Tuesday
- Our starting point = $10
- Something happens – the price increases by 20%
- 20% of what? 20% of our starting point… which is $10
- 20% of $10 = $2
- If the price increases by $2, the price of gloves is now $12
- Our ending point = $12
On Wednesday
- Our starting point = $12
- Something happens – the price decreases by 20%
- 20% of what? 20% of our starting point… which is $12
- 20% of $12 = $2.40
- If the price decreases by $2.40, the price of gloves is now $9.60
- Our ending point = $9.60

Typical SHSAT examples

Shares of XYZ stock were trading at $130 this morning, but later fell by 10%. The ending share price was $117.
The tree in our back yard was 60 inches tall. A year later, it grew 5%. The new height was 63 inches.
The price of a blouse is $50 without tax. With tax, it’s $54.50. The 9% sales tax added $4.50 to the cost.
A subway token used to be $1.25. It went up 60%, making the new cost $2.00.

A car’s fuel efficiency used to be 20mpg. Major engineering changes raised this to 30mpg – a 50% improvement.

Don’t forget! `% OF` some number is just multiplication by that number

Any percentage OF a number is the decimal equivalent of the percentage multiplied by that number. In math speak, when you see/hear the word OF… it’s an indication that you have to multiply.

20% of 80 = .2 x 80 = 16
120% of 100 = 1.2 x 100 = 120
0% of 1,000,000 = 0 x 1000000 = 0

The other area you see “OF” expressions implying multiplication is fractions. For example…

1/5 of 80 = 1/5 x 80 = 16
6/5 of 100 = 6/5 x 100 = 120
0/2 of 1,000,000 = 0/2 x 1000000 = 0

Real-world example:

97% of scientists agree that humans are the primary cause of global warming
There are roughly 7 million scientists employed in the United States
If we applied this percentage (97%) to the roughly 7 million scientists employed in the United States, we would get 6,790,000 scientists
This exceeds the population of Indiana

Weekly Challenge: 5/16/2021 (Percents)

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This week’s theme: 🇵🇷 The Bronx

A shout-out to my Puerto Rican cousins – the Vegas.