Asked to scale 4 : 6 so the first number becomes 6, many people answer 6 : 8. It is wrong.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
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.
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. Recognising 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.
Step 2: 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 3: Watch an Example
One step at a time.
Watch Ana Catch the Mistake
Ana is told that scaling 2 : 3 up to 4 gives 4 : 5.
- Step 1
She checks what happened to the first number: 2 became 4, so it doubled.
Step 4: Your Turn
Practice makes it stick.
The Right Answer
Problem 1 of 2
4 : 6 scaled so the first is 6. What is the second?
The Squash
Problem 2 of 2
1 : 4 squash to water. With 3 squash, how much water?
water
Multiply, Do Not Add
1 of 4
2 : 5 scaled so the first is 6. What is the second?
2 of 4
3 : 4 scaled so the first is 12. What is the second?
3 of 4
5 : 8 scaled so the first is 10. What is the second?
4 of 4
Is 3 : 5 the same comparison as 6 : 8?
Step 5: 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?
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.