Why You Multiply and Not Add

Step 1: Let's Learn

Read it, or press Listen and follow the words.

Watch the idea first — 59 seconds. Then read on, and try it yourself in the next step.

Asked to scale 4 : 6 so the first number becomes 6, many people answer 6 : 8. It is wrong.

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 2: Try It Yourself

Both answers drawn against the same original. Only one matches.

Compare multiplying both with adding the same amount to both.

A ratio table for 4 to 6
First	Second
4	6
8	12

Multiplied: 8 to 12

Same comparison.

Added 4 to each: 8 to 10

A different comparison.

Adding 4 to both gives 8 to 10, which is not the same comparison. Scaling multiplies both.

First quantity

Second quantity

Scale it up by

Step 3: 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 4: 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 5: 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 6: 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.

For the grown-up

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.