Is 3 : 4 the same strength as 5 : 7? The numbers are all different, so looking is not enough.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A double number line pairs spoons of syrup with cups of water. How many cups of water line up with 12 spoons of syrup?
cups
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Make one quantity match
Scale both ratios until their first numbers agree, then compare what is left.
With those two
3 : 4 scaled by 5 is 15 : 20. 5 : 7 scaled by 3 is 15 : 21. Different, so not equivalent.
Which is stronger
At the same 15 of the first, one has 20 of the second and one has 21. Less of the second means a stronger mix.
Step 2: Try It Yourself
Tap and try it out.
| Syrup | Water |
|---|---|
| 3 | 4 |
| 6 | 8 |
| 9 | 12 |
| 12 | 16 |
Every row is the same ratio. 3 to 4 is the same comparison as 15 to 20.
Step 3: In Real Life
Two juice mixes
One juice is 2 parts concentrate to 5 water. Another is 3 to 8. Scale to the same water: 16 to 40 and 15 to 40. The first is stronger.
Step 4: Watch an Example
One step at a time.
Watch Ana Compare Two Jugs
Jug A is 2 : 3 syrup to water. Jug B is 3 : 5 syrup to water. Which one tastes stronger?
- Step 1
The two jugs hold different amounts of syrup. So their water counts cannot be set against each other as they stand.
Watch Dev Compare Two Paints
Mix P is 2 : 5 blue to white. Mix Q is 3 : 7 blue to white. Dev starts it off, and you finish it.
- Step 1
Dev wants the blue to match in both mixes. 2 and 3 both go into 6, so he aims for 6 of blue.
Watch Maya Match the Water Instead
Jug C is 5 : 2 syrup to water. Jug D is 7 : 3 syrup to water.
- Step 1
Either quantity can be the one that matches. Maya picks the water, because 2 and 3 both go into 6.
Step 5: Your Turn
Practice makes it stick.
Two Lemonade Stands
Problem 1 of 17
Maya’s stand takes $3 for every 2 cups sold. Dev’s stand takes $5 for every 4 cups sold. Scaled to 4 cups, how many dollars does Maya’s stand take?
dollars
Two Tile Paths
Problem 2 of 17
Each path repeats the pattern shown until it has 6 gray tiles. White tiles cost $2 each. What do Path B’s white tiles cost, in dollars?
dollars
Scale, Then Compare
Scale a ratio until the number you need matches. When two mixes already match on one quantity, the other one decides.
Problem 3 of 17
Scale 2 : 7 so the first number is 8. What is the second number?
Problem 4 of 17
Scale 5 : 3 so the first number is 20. What is the second number?
Problem 5 of 17
Scale 4 : 9 so the second number is 27. What is the first number?
Problem 6 of 17
Two mixes are already scaled to 15 spoons of syrup. Mix A is 15 : 8 and Mix B is 15 : 11, syrup to water. Which mix is stronger?
Problem 7 of 17
Is 4 : 6 equivalent to 10 : 15?
Whose Step?
One line on each page is a slip. Find it, then put it right.
Problem 8 of 17
Priya is scaling a paint mix of 3 : 4 blue to white, so that the blue reaches 12. One line is a slip. Which?
Priya’s answer for: scale 3 : 4 so the blue is 12
- 13 : 4
- 23 × 4 = 12
- Answer3 : 4 → 12 : 4
Problem 9 of 17
Priya’s mix has to keep its colour. When the blue reaches 12, what is the white number?
parts
Problem 10 of 17
Dev was asked which is stronger, a 4 : 5 syrup mix or a 5 : 7 one. One line is a slip. Which?
Dev’s answer for: which is stronger, 4 : 5 or 5 : 7 syrup to water?
- 1A = 4 : 5, B = 5 : 7
- 2B has more syrup than A: 5 is more than 4
- Answerso Mix B is the stronger one
Problem 11 of 17
What should Dev’s line 3 read?
Problem 12 of 17
Maya was asked to scale a 4 : 3 mix and a 6 : 5 mix so that both hold 12 spoons of syrup. One line is a slip. Which?
Maya’s answer for: scale 4 : 3 and 6 : 5 so both hold 12 of syrup
- 1A = 4 : 3, B = 6 : 5
- 2A × 3 → 12 : 9× 3
- AnswerB × 2 → 12 : 15× 2
Problem 13 of 17
Mix B is 6 : 5, doubled so its syrup reaches 12. What is its water number?
cups
Pick the Move
Decide the first move before making it.
Problem 14 of 17
You want to know whether 3 : 8 or 5 : 12 is the stronger syrup-to-water mix. What is the first move?
Problem 15 of 17
You want to know whether 5 : 8 or 5 : 6 is the stronger mix. What is the first move?
Same Strength?
Decide what is true without working every answer out in full.
Problem 16 of 17
A mix is 3 : 5 syrup to water. Select every mix below that is exactly the same strength.
Problem 17 of 17
Juice X is 2 : 5 and Juice Y is 4 : 9, juice to water. Select every statement that is true.
Step 6: Quick Check
Show what you know.
Question 1 of 1
Scale 4 : 7 so the first number is 12. What is the second?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A punch is 3 : 4 syrup to water. Anna wants a stronger punch that uses 12 cups of water. Which amount of syrup gives her one?
Stretch 2 of 2
Mix A is 6 : 15 syrup to water. Mix B holds 4 spoons of syrup and must be stronger than Mix A. What is the largest whole number of cups of water Mix B can hold?
cups
What You Learned
- Two ratios are equivalent when one scales to the other.
- To compare, scale until one quantity matches in both.
- Then whichever has less of the other quantity is the stronger mix.
For the grown-up
The misconception this lesson targets is comparing two ratios one number at a time. A child who sees 2 : 5 beside 3 : 8 says the second is stronger because 3 is more than 2, or weaker because 8 is more than 5, and never notices that both numbers moved. Nothing is comparable until one quantity is made to match in both, and the safest question to ask out loud is "the same what?" A second slip rides alongside it: scaling by adding rather than multiplying, so 3 : 4 becomes 12 : 13 instead of 12 : 16. Both numbers must take the same multiplier, and a quick check is whether the two numbers still look roughly the same distance apart in size, not in difference.
Make one part match, then compare — Which is stronger, 3 : 2 or 4 : 3? Scale both so the second part matches: 9 : 6 and 8 : 6. With the same amount of the second thing, 9 beats 8, so 3 : 2 is stronger. This is finding a common denominator, applied to ratios.
Or reduce both to a unit — 3 : 2 is 1.5 : 1 and 4 : 3 is about 1.33 : 1. Now they are directly comparable and 3 : 2 is clearly stronger. Reducing to "per one" is the most general method and it is what unit rates in the next chapter formalise.
Ratios as fractions — Writing 3 : 2 as 3/2 and 4 : 3 as 4/3 turns the comparison into an ordinary fraction comparison. Since 9/6 beats 8/6, the answer is the same. The three methods are the same idea in three costumes.
Check it makes sense — Before calculating, guess: 3 : 2 is one and a half parts of the first per part of the second, while 4 : 3 is only one and a third. So 3 : 2 should be stronger. When the calculation agrees with the estimate, you can trust it.
Make one part match — Which is stronger, 3 : 2 or 5 : 3? Scale both so the second part is 6: 9 : 6 and 10 : 6. Same water, more flour in the second. 5 : 3 is stronger.