Section A Section A Checkpoint

Problem 1

The table represents mixtures of black and white paint that produce the same shade of gray.

Complete the table as you answer the questions.

  1. To make the same shade of gray, how many cups of white paint will they need to mix with 1 cup of black paint? Explain or show your reasoning.
  2. How many cups of black paint will they need to mix with 16 cups of white paint? Explain or show your reasoning.
  3. Make up a new pair of numbers that would make the same shade of gray.
  4. What is the constant of proportionality?
  5. What does the constant of proportionality mean in this situation?
black paint (cups) white paint (cups)
12\frac12 4
3 24
1
16
Show Solution
Solution
  1. 8 cups. Sample reasoning: From the row showing 12\frac12 and 4, I multiplied both by 2 to get 1 and 8. 
  2. 2 cups. Sample reasoning: I divided 16 by 8 to get 2.
  3. Sample response: 10 cups of black paint with 80 cups of white paint. (Any pair of numbers that makes a ratio equivalent to 1:81:8 is acceptable.)
  4. 8
  5. For every 1 cup of black paint, mix in 8 cups of white paint.
black paint (cups) white paint (cups)
12\frac12 4
3 24
1 8
2 16
10 80
Show Sample Response
Sample Response
  1. 8 cups. Sample reasoning: From the row showing 12\frac12 and 4, I multiplied both by 2 to get 1 and 8. 
  2. 2 cups. Sample reasoning: I divided 16 by 8 to get 2.
  3. Sample response: 10 cups of black paint with 80 cups of white paint. (Any pair of numbers that makes a ratio equivalent to 1:81:8 is acceptable.)
  4. 8
  5. For every 1 cup of black paint, mix in 8 cups of white paint.
black paint (cups) white paint (cups)
12\frac12 4
3 24
1 8
2 16
10 80