Solving Percentage Problems

10 min

Teacher Prep
Setup
Display one problem at a time. 1 minute of quiet think time per problem, followed by a whole-class discussion.

Narrative

In this activity, students determine what percentage one number is of another number. They answer questions of the form “BB is what percentage of CC?” and match each question with an answer in a list of percentages.

Each pair of values is related by a factor that is a benchmark fraction, so students are likely to identify the matches quickly. They should spend some time discussing how they reason about each question.

Although students are likely to make the matches mentally, some may find it helpful to draw a tape diagram to visualize the relationship between the numbers.

Launch

Display this tape diagram for all to see.

Ask students to think about what question it could represent and to give a signal if they have a response. Select a couple of students to share their responses. Record questions along the lines of “7 is what percentage of 14?” and display the question(s) for all to see. Invite students to share the answer to the question and to explain how they know.

Tell students that they will now answer similar questions and think about what percentage one number is of another number.

Arrange students in groups of 2. Give students 2–3 minutes of quiet think time and another minute to share their responses and reasoning with a partner.

Action and Expression: Provide Access for Physical Action. Activate or supply background knowledge. Provide students with access to blank tape diagrams to support information processing.
Supports accessibility for: Visual-Spatial Processing, Organization

Student Task

Match each question in the left column with a percentage in the right column. One percentage will be left over. Be prepared to explain your reasoning.

  1. 5 is what percentage of 20?
  2. 3 is what percentage of 30?
  3. 6 is what percentage of 8?
  4. 20 is what percentage of 5?
  • 4%
  • 10%
  • 25%
  • 75%
  • 400%

Sample Response

  1. 25%. Sample reasoning: 5 times 4 is 20, so 5 is 14\frac{1}{4} (or 25%) of 20.
  2. 10%. Sample reasoning: 3 times 10 is 30, so 3 is 110\frac{1}{10} (or 10%) of 30.
  3. 75%. Sample reasoning: 2 is 14\frac{1}{4} of 8, so 6 is 34\frac{3}{4} (or 75%) of 8.
  4. 400%. Sample reasoning: 5 is 100% of 5, and 20 is 4 times that, so 20 is 400% of 5.
Activity Synthesis (Teacher Notes)

The goal of the discussion is to help students see that a number can be expressed as a percentage of another number by comparing the numbers multiplicatively and using fractions.

Invite students to briefly share their responses and reasoning. Highlight that when finding what percentage one number is of another number, it can be helpful to think about what fraction one number is of the other number, for instance:

  • 5 is what fraction of 20?
  • 3 is what fraction of 30?
  • 6 is what fraction of 8?

Once we know that 5 is 14\frac{1}{4} of 20, 3 is 110\frac{1}{10} of 30, and 6 is 68\frac{6}{8} (or 34\frac{3}{4}) of 8, we can relate the fractions to the percentages 25% of 20, 10% of 30, and 75% of 8, respectively.

Then discuss how students reasoned about the last question. If not mentioned by students, emphasize that unlike in the first question, the 5 in this question is the value that corresponds to 100%, so the percentage of 20 of 5 must be greater than 100. Because 20 is 4⋅54 \boldcdot 5, the percentages must also be related by a factor of 4. Consider using a table or a tape diagram to illustrate these relationships.

Anticipated Misconceptions

Because 5 goes into 20 four times, students might rush to say that 5 is 4% of 20. If this happens, encourage students to check their thinking by asking questions such as:

  • “10 is what percentage of 20?”
  • “If 10 is 50% of 20, what percentage of 20 is 5?”
  • “If 5 is 4% of 20, then does that mean 10 is 8% of 20?”
Standards
Building On
  • 5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
  • 5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Addressing
  • 6.RP.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
  • 6.RP.A.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

25 min