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Ratios and Data 6-8 / Week 01 / Tuesday
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Week 01 · Ratios

Tuesday

Equivalent ratios
// Two quantities, one comparison
⏱ about 20 min

Tuesday: Equivalent Ratios

Raven tapes a sheet of paper to the Numbers Desk and draws two columns: Blue and Yellow.

Rocket fills it in fast. "2 and 3. Then 3 and 4. Then 4 and 5. I just add one each time."

He mixes the 4 and 5 batch. It comes out a dull blue-green, nothing like the sign color.

"What do you notice?" Raven asks. Rocket frowns. "It is wrong, but the numbers went up together."

Nova dips lower, her light pulsing slowly. "Yesterday you made a batch twice as big. What did you do to both numbers?"

"Multiplied," Rocket says. "Both of them. By the same number."

Raven hands him the marker. "Then fix the table."

Multiply both parts by the same number

Equivalent ratios show the same relationship. To make one, multiply both parts by the same number.

2:3 becomes 4:6 when both parts are multiplied by 2. It becomes 6:9 when both are multiplied by 3.

Adding the same number to both parts does not work. 3:4 is not the same green as 2:3.

  1. Write the ratio you know: 2 blue to 3 yellow.
  2. Pick a number to multiply by, like 4.
  3. Multiply both parts: 2 × 4 = 8 and 3 × 4 = 12.
  4. Write the new row in the table: 8 blue, 12 yellow.
A table with columns Blue and Yellow: 2 and 3; 4 and 6; 6 and 9; 8 and 12

Find a missing value

A ratio table may have a blank. Find what the known part was multiplied by, then multiply the other part the same way.

In the table below, blue went from 2 to 10. That is multiplying by 5, so yellow is 3 × 5.

A table with columns Blue and Yellow: 2 and 3; 10 and a blank
MISSING VALUES
  • Read the question.
  • Tap your answer.
The table shows 2 blue and 3 yellow in one row. Blue is 10 in the next row. What is yellow?
A table with columns Blue and Yellow: 2 and 3; 8 and a blankBlue 2 and yellow 3 in the first row. Blue 8 in the next row. What is yellow?
A table with columns Blue and Yellow: 2 and 3; a blank and 18Blue 2 and yellow 3 in the first row. Yellow 18 in the next row. What is blue?
Lemonade uses 1 cup of lemon juice for every 4 cups of water. With 28 cups of water, how many cups of lemon juice?
BUILD THE TABLE
  • Read the question.
  • Tap your answer.
2 spoons of blue for every 3 spoons of yellow. With 12 spoons of yellow, how many spoons of blue?
3 spoons of yellow for every 2 spoons of blue. With 12 spoons of blue, how many spoons of yellow?
Which ratio is equivalent to 2:3?
Which ratio is equivalent to 1:4?
BATCHES OF LEMONADE FROM SMALLEST TO LARGEST
  • ?3 cup lemon juice and 12 cups water
  • ?4 cup lemon juice and 16 cups water
  • ?1 cup lemon juice and 4 cups water
  • ?2 cup lemon juice and 8 cups water
WHY THIS EXERCISEEach row of a ratio table is the same recipe at a different size.
StatementTrue or false?
2:3 is the same ratio as 4:6.?
2:3 is the same ratio as 3:4.?
1:4 is the same ratio as 3:12.?
5:2 is the same ratio as 10:6.?
WHY THIS EXERCISERocket's add-one mistake looks fair but changes the color. Equivalent ratios come from multiplying.
Try it
Tear a paper strip into 5 equal pieces, 2 for blue and 3 for yellow.
Make a second strip with 10 pieces, 4 blue and 6 yellow. Hold them side by side.
What stays the same about the two strips? What changes?
Draw a ratio table for lemonade at 1 cup of juice to 4 cups of water. Fill in four rows.

Well fixed. Tomorrow the Numbers Lab opens and you mix real batches to test the table.

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