Doubling a recipe of 3 flour to 2 milk gives 6 flour to 4 milk. It tastes the same.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Sam packs 5 lunch boxes. Every box holds 3 apples and 4 crackers. How many crackers does he pack in all?
crackers
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Why the taste does not change
Both amounts grew by the same factor, so the comparison between them is untouched.
A ratio table
Writing the pairs in a table shows the pattern: 3:2, 6:4, 9:6, 12:8.
Both, or neither
Multiplying only one quantity changes the recipe. Both must be multiplied by the same number.
Step 2: Try It Yourself
Tap and try it out.
| Flour | Milk |
|---|---|
| 3 | 2 |
| 6 | 4 |
| 9 | 6 |
Every row is the same ratio. 3 to 2 is the same comparison as 9 to 6.
Step 3: In Real Life
Doubling a recipe
Pancakes: 2 cups of flour to 3 eggs. Double the guests, double both: 4 cups to 6 eggs. Same pancakes. A ratio table is a recipe scaled up.
Step 4: Watch an Example
One step at a time.
Watch Ana Scale a Mix Up
A juice mix uses 2 cups of mango to 5 cups of apple. Ana wants 10 cups of mango.
- Step 1
Ana has 2 cups of mango to 5 cups of apple, and she wants 10 cups of mango.
Watch Maya Scale a Ratio Down
A shelf holds 12 red mugs and 18 blue mugs.
- Step 1
Maya has 12 : 18 red to blue, and she wants that comparison in the smallest whole numbers.
Watch Dev Scale a Paint Mix
Green paint mixes 3 cups of blue to 4 cups of yellow. Dev needs 12 cups of yellow, and you fill in the steps.
- Step 1
Dev mixes green from 3 cups of blue to 4 cups of yellow, and he needs 12 cups of yellow.
Step 5: Your Turn
Practice makes it stick.
The Garden Path
Problem 1 of 17
A garden path repeats the same block of tiles over and over. The whole path uses 12 gray tiles. White tiles cost $2 each. What do the white tiles for the path cost, in dollars?
dollars
The Book Sale
Problem 2 of 17
A school book sale splits every $10 it raises the same way. So far it has spent $16 on printing. How much has gone on new books, in dollars?
dollars
Scale It
Find the factor first, then give both parts the same one.
Problem 3 of 17
5 : 2, scaled by 3. What is the second number?
Problem 4 of 17
3 : 7. With 9 of the first, how many of the second?
Problem 5 of 17
6 : 8. Written with the smallest whole numbers, what is the first number?
Problem 6 of 17
4 : 9, scaled by 5. What is the first number?
Problem 7 of 17
5 : 6. With 42 of the second, how many of the first?
Whose Step?
One line in each page is a slip. Find it, then fix it.
Problem 8 of 17
Priya was asked to scale 3 : 5 red to blue so that there are 12 red. One line is a slip. Which?
Priya's answer for: 3 : 5 red to blue, scaled to 12 red
- 13 : 5 red to blue, and 12 red
- 212 ÷ 3 = 4
- Answer3 : 5 becomes 12 : 9
Problem 9 of 17
Priya's line 3 should read 12 : ___ . What is the missing number?
Problem 10 of 17
Dev was asked to write 20 : 30 with the smallest whole numbers. One line is a slip. Which?
Dev's answer for: write 20 : 30 with the smallest whole numbers
- 120 : 30
- 210 goes into 20 and into 30
- Answer20 : 30 = 2 : 30
Problem 11 of 17
Dev's line 3 should read 2 : ___ . What is the missing number?
Problem 12 of 17
Sam was asked whether 4 : 6 and 6 : 9 make the same comparison. One line is a slip. Which?
Sam's answer for: do 4 : 6 and 6 : 9 make the same comparison?
- 14 : 6 and 6 : 9
- 24 : 6 divides by 2 to give 2 : 3
- 36 : 9 divides by 3 to give 2 : 3
- Answerso the two are different comparisons
Problem 13 of 17
What should Sam's line 4 read?
Pick the Move
Decide the first move before you make it. Nothing here has to be worked out.
Problem 14 of 17
A marinade mixes oil to vinegar in the ratio 3 : 8. You want a batch with 24 of the vinegar. What is the first move?
Problem 15 of 17
A ratio reads 18 : 24, and you want the same comparison in the smallest whole numbers. What is the first move?
Same Comparison?
Judge each one without working out a missing amount.
Problem 16 of 17
Select every ratio that makes the same comparison as 4 : 10.
Problem 17 of 17
A drink mixes 3 cups of syrup to 5 cups of water. Select every batch that tastes the same.
Step 6: Quick Check
Show what you know.
Question 1 of 2
4 : 5. With 16 of the first, how many of the second?
Question 2 of 2
To keep a ratio the same, what must you do?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A dye mix is 2 : 5 red to blue. One batch uses 35 cups of blue. How many cups of dye are in that batch in all?
cups
Stretch 2 of 2
Two numbers are in the ratio 2 : 3, and they add to 25. What is the larger number?
What You Learned
- Equivalent ratios are the same comparison written with different numbers.
- Multiply both quantities by the same number.
- A ratio table lays the pattern out in rows.
For the grown-up
The misconception this lesson targets is that a ratio is kept by adding. A child who turns 2 : 3 into 5 : 6 by adding 3 to each has kept the difference, not the comparison, and the drink tastes different. Two smaller slips ride alongside it. The first is scaling one part and forgetting the other, which is what "both, or neither" is there to catch. The second is giving each part its own factor — doubling one and tripling the other — which looks like multiplying and is not. One question catches all three: what was each number multiplied by? If the two answers are not the same number, the ratio has moved. Lesson 1.3 takes the adding error head on, so it is worth leaving a child who adds with the question rather than the correction.
Scaling both parts keeps the ratio — 3 : 2 is equivalent to 6 : 4, 9 : 6 and 30 : 20. Multiply both numbers by the same amount and the comparison is unchanged, exactly as multiplying the top and bottom of a fraction leaves its value alone.
A ratio table organizes the scaling — Write the two quantities in two columns and generate rows: 3 and 2, 6 and 4, 9 and 6. Each row is an equivalent ratio. Tables make it easy to jump to the row you actually need, and they make the multiplicative pattern visible.
Simplest form — 30 : 20 reduces to 3 : 2 by dividing both by 10. The simplest form uses the smallest whole numbers and is easiest to compare with other ratios. Finding it means dividing by the greatest common factor — the factor work you did earlier, still earning its keep.
Testing whether two ratios are equivalent — Are 12 : 8 and 15 : 10 equivalent? Reduce both: 3 : 2 and 3 : 2. Yes. Reducing both to simplest form is the most reliable test, and it works no matter how large or awkward the original numbers are.
Scale both parts together — 3 : 2. Multiply both by 2: 6 : 4. By 3: 9 : 6. By 10: 30 : 20. All the same ratio. Whatever you multiply one part by, you must multiply the other by too.