Asked to scale 4 : 6 so the first number becomes 6, many people answer 6 : 8. It is wrong.
Step 1: Picture It
Start with something you already know.
A mother and her son
- A mother is 36 years old and her son is 12.
- Right now she is three times as old as he is.
- In 12 years she will be 48 and he will be 24.
- She is still 24 years older, but now only twice as old.
Adding the same amount to both kept the difference but broke "times as much".
A ratio is a "times as much" comparison, so only multiplying both parts keeps it.
Where the picture stops: ages only grow by adding; a recipe can be multiplied.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A paint mix uses 2 cups of blue for every 5 cups of white. Dev wants the same shade and pours 6 cups of blue. How many cups of white does he need?
cups
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Where that comes from
4 became 6 by adding 2, so it feels right to add 2 to the other side. That is additive thinking.
Why it fails
A ratio compares by division, not by difference. 4 : 6 says the second is one and a half times the first.
The check that catches it
6 : 8 is not one and a half times. 6 : 9 is. Draw both and only one looks like the original.
Step 3: Try It Yourself
Both answers drawn against the same original. Only one matches.
| First | Second |
|---|---|
| 4 | 6 |
| 8 | 12 |
Multiplied: 8 to 12Same comparison.
Added 4 to each: 8 to 10A different comparison.
Adding 4 to both gives 8 to 10, which is not the same comparison. Scaling multiplies both.
Step 4: In Real Life
Lemonade
Lemonade is 1 cup of lemon juice to 4 cups of water. For a bigger jug, adding 1 to each gives 2 to 5, which is sourer. Multiplying both by 2 keeps the taste.
Step 5: Watch an Example
One step at a time.
Watch Ana Catch the Slip
A sticker is 4 inches wide and 6 inches tall. Sam says a bigger sticker 6 inches wide should be 8 inches tall.
- Step 1
Ana starts with what Sam did. The width went from 4 to 6 and the height went from 6 to 8, so Sam added 2 to each.
Watch Dev Scale a Juice Mix
A juice uses 3 cups of mango to 8 cups of water. Dev wants 12 cups of mango and the same taste. Dev writes the first line, and you write the rest.
- Step 1
The mix is 3 cups of mango to 8 cups of water, and Dev needs 12 cups of mango.
Step 6: Your Turn
Practice makes it stick.
The Poster
Problem 1 of 17
A bigger poster keeps the same comparison as the one drawn and is 18 inches wide. Printing costs $2 for every inch of its height. How many dollars does the bigger poster cost?
dollars
The Skate Ramp
Problem 2 of 17
A longer ramp keeps the same comparison as the one drawn and rises 12 feet. The rail along the ground costs $4 a foot. How many dollars does the longer ramp's ground rail cost?
dollars
Multiply, Do Not Add
Find the factor first, then use that same factor on the other part.
Problem 3 of 17
3 : 4 scaled so the first number is 12. What is the second?
Problem 4 of 17
5 : 8 scaled so the first number is 15. What is the second?
Problem 5 of 17
7 : 2 scaled so the second number is 8. What is the first?
Problem 6 of 17
9 : 5 scaled so the first number is 27. What is the second?
Problem 7 of 17
6 : 10 scaled down so the first number is 3. What is the second?
Whose Step?
One line in each page is a slip. Find it, then fix it.
Problem 8 of 17
Maya was asked to scale 2 : 7 so the first number is 6. One line is a slip. Which?
Maya's answer for: scale 2 : 7 so the first number is 6
- 12 : 7
- 22 × 3 = 6, so the factor is 3
- 3the second part: 7 + 3 = 10
- Answerso 2 : 7 becomes 6 : 10
Problem 9 of 17
What should the second number of Maya's scaled ratio be?
Problem 10 of 17
Dev was asked whether 3 : 5 and 8 : 10 are the same comparison. One line is a slip. Which?
Dev's answer for: are 3 : 5 and 8 : 10 the same comparison?
- 13 : 5 and 8 : 10
- 23 + 5 = 8 and 5 + 5 = 10
- 3both parts grew by 5, so the comparison is unchanged
- Answerso 3 : 5 and 8 : 10 are the same
Problem 11 of 17
Dev's page says 8 : 10 matches 3 : 5. Scale 3 : 5 properly so its second number is 10. What is its first number?
Problem 12 of 17
Noor was asked to scale 21 : 12 down so the first number is 7. One line is a slip. Which?
Noor's answer for: scale 21 : 12 down so the first number is 7
- 121 : 12
- 221 ÷ 3 = 7, so both parts are divided by 3
- 3the second part: 12 ÷ 4 = 3
- Answerso 21 : 12 becomes 7 : 3
Problem 13 of 17
What should Noor's second number be?
Pick the Move
Decide the first move before you make it. Nothing here needs working out in full.
Problem 14 of 17
A mix is 5 : 3. You need the same mix with 20 of the first. What is the first move?
Problem 15 of 17
Leo scaled 3 : 7 and got 9 : 13. You want to check it in one move. What is the best first move?
Same Comparison?
Decide what is true without working any of them out in full.
Problem 16 of 17
Select every ratio that is the same comparison as 2 : 3.
Problem 17 of 17
Select every statement that is true about scaling a ratio.
Step 7: Quick Check
Show what you know.
Question 1 of 2
6 : 9 scaled so the first is 12. What is the second?
Question 2 of 2
How does a ratio compare two numbers?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A drink is 2 cups of syrup to 9 cups of water. Sam pours in 2 more cups of syrup and 2 more cups of water. Is the new drink stronger, weaker, or exactly the same?
Stretch 2 of 2
Find the ratio that is the same comparison as 6 : 15 and whose first number is 10. What is its second number?
What You Learned
- Scaling a ratio multiplies both quantities.
- Adding the same amount to both changes the comparison.
- Draw it if you are unsure. Only the right answer keeps the shape.
For the grown-up
The misconception this lesson targets is additive scaling: the belief that adding the same amount to both parts of a ratio leaves the ratio alone. It is not carelessness. A child who reasons "4 became 6, so 6 becomes 8" is applying a rule that works perfectly for differences, and differences are what arithmetic has asked them to track for years. The repair is not "do not add" but "a ratio asks how many times, not how much more", and it has to be made out loud, on numbers where the two answers look visibly different. Asking "how many times bigger is the second part?" before and after a scaling catches it every time, because that number is exactly the one that must not move. The same error comes back later as a wrong scale factor on a drawing, and later still as a graph that misses the origin, so time spent on it here is repaid twice.
The mistake, stated plainly — Given a mix of 3 : 2, people often say that adding 1 to each gives 4 : 3 and keeps the mix the same. It does not. 3 : 2 means one and a half times as much red; 4 : 3 means one and a third. Adding changes the relationship. Only multiplying preserves it.
A test you can taste — Squash mixed 1 : 4 with water is a particular strength. Add one of each to get 2 : 5 and the drink is stronger — you added proportionally more squash. Doubling to 2 : 8 tastes identical. The tongue is a perfectly good detector of this error.
Ratios are multiplicative comparisons — A ratio answers "how many times as much", not "how many more". That is why it survives multiplication and not addition. Recognizing which kind of comparison a situation calls for is the central skill of this chapter and the next.
Spotting it in your own work — If you ever find yourself adding the same number to both parts of a ratio, stop. Check by reducing: 3 : 2 reduces to 3 : 2 and 4 : 3 reduces to 4 : 3. Different simplest forms means different ratios, which settles it immediately.
Watch adding go wrong — 3 : 2. Add 1 to each: 4 : 3. Test it with the recipe. 3 : 2 is one and a half cups of flour per cup of water. 4 : 3 is one and a third. The mixture changed. Multiplying keeps the relationship. Adding destroys it.