Lesson 3

Increasing and Decreasing

Let’s use percentages to describe increases and decreases.

3.1: Improving Their Game

Here are the scores from 3 different sports teams from their last 2 games.

sports team total points in game 1 total points in game 2
football team 22 30
basketball team 100 108
baseball team 4 12
  1. What do you notice about the teams’ scores? What do you wonder?
  2. Which team improved the most? Explain your reasoning.

3.2: More Cereal and a Discounted Shirt

  1. ​​

    A cereal box says that now it contains 20% more. Originally, it came with 18.5 ounces of cereal. How much cereal does the box come with now?

    Picture of a cereal box with the label "20% more free" on the box.
  2. ​​

    The price of a shirt is \$18.50, but you have a coupon that lowers the price by 20%. What is the price of the shirt after using the coupon?

    Light blue t-shirt with a 20 percent off tag attached.

3.3: Using Tape Diagrams

  1. Match each situation to a diagram. Be prepared to explain your reasoning.

    1. Compared with last year’s strawberry harvest, this year’s strawberry harvest is a 25% increase.
    2. This year’s blueberry harvest is 75% of last year’s.
    3. Compared with last year, this year’s peach harvest decreased 25%.
    4. This year’s plum harvest is 125% of last year’s plum harvest.
    Two tape diagrams. Last year, labeled 100 percent, one long section. This year, with a smaller blue section labeled 25 percent that extends past the 100 percent tape.
    Two tape diagrams. Last year, labeled 100 percent, one long section. This year, with a smaller blue section labeled 25 percent.

  2. Draw a diagram to represent these situations.

    1. The number of ducks living at the pond increased by 40%.

    2. The number of mosquitoes decreased by 80%.


What could it mean to say there is a 100% decrease in a quantity? Give an example of a quantity where this makes sense.

3.4: Agree or Disagree: Percentages

Do you agree or disagree with each statement? Explain your reasoning.

  1. Employee A gets a pay raise of 50%. Employee B gets a pay raise of 45%. So Employee A gets the bigger pay raise.

  2. Shirts are on sale for 20% off. You buy two of them. As you pay, the cashier says, “20% off of each shirt means 40% off of the total price.”

Summary

Imagine that it takes Andre \(\frac34\) more than the time it takes Jada to get to school. Then we know that Andre’s time is \(1\frac34\) or 1.75 times Jada’s time. We can also describe this in terms of percentages:

Two tape diagrams, Jada's time and Andre's time.

We say that Andre’s time is 75% more than Jada’s time. We can also see that Andre’s time is 175% of Jada’s time. In general, the terms percent increase and percent decrease describe an increase or decrease in a quantity as a percentage of the starting amount.

For example, if there were 500 grams of cereal in the original package, then “20% more” means that 20% of 500 grams has been added to the initial amount, \(500+(0.2)\boldcdot 500=600\), so there are 600 grams of cereal in the new package.

Picture of a cereal box with the label "20% more free" on the box.

We can see that the new amount is 120% of the initial amount because

\(\displaystyle 500+(0.2)\boldcdot 500 = (1 + 0.2)500\)

Tape diagram.

Glossary Entries

  • percentage decrease

    A percentage decrease tells how much a quantity went down, expressed as a percentage of the starting amount.

    For example, a store had 64 hats in stock on Friday. They had 48 hats left on Saturday. The amount went down by 16.

    This was a 25% decrease, because 16 is 25% of 64.

    a tape diagram
  • percentage increase

    A percentage increase tell how much a quantity went up, expressed as a percentage of the starting amount.

    For example, Elena had $50 in the bank on Monday. She had $56 on Tuesday. The amount went up by $6.

    This was a 12% increase, because 6 is 12% of 50. 

    a tape diagram