Skip to content

Twice as many, half as much

Ratio, rates and money · level 2, lesson 4 of 6

A question to start

Ben has 4 stickers. Ava has 12 stickers. How do they compare?

You might say: "Ava has 8 more than Ben." That is true. But you could also say: "Ava has 3 times as many as Ben." That is true too!

Now here is the puzzle. Next month, Ben has 10 stickers and Ava has 30. Is Ava still "8 more"? No: 30 is 20 more than 10. Is she still "3 times as many"? Yes! Three lots of 10 make 30.

So one way of comparing changed, and the other stayed the same. In this lesson you will learn the second way, comparing by times as many, and see why it is the way that matters for recipes, for prices and for sharing.

The idea

Two ways to compare

When you compare two amounts, you can ask two different questions.

  • How many more? This compares by taking away. Ava 12, Ben 4: 12 take away 4 is 8. Ava has 8 more.
  • How many times as many? This compares by groups. How many groups of Ben's 4 fit in Ava's 12? Count by fours: 4, 8, 12. Three groups. Ava has 3 times as many.

Both questions are good questions. They just tell you different things.

Seeing it with bars

A bar model is a picture where each amount is a bar. Equal bars mean equal amounts.

BenAva 4444 12 Ava's amount is 3 of Ben's bars

Ben's amount is one bar. Ava's amount is three of the same bar. "3 times as many" means "3 bars of the other amount".

You can see "how many more" in the picture too: it is the part of Ava's bars that sticks out past Ben's bar, 2 bars, which is 4 and 4, so 8.

Special words

Some "times as many" have their own names.

Words Means
twice as many, or double 2 times as many
three times as many, or triple 3 times as many
half as many the amount shared into 2 equal parts, and one part taken

Half as much is twice as many, turned around

If Ava has twice as many as Ben, then Ben has half as many as Ava. These are the same fact, said from the two sides.

LeoMia 555 105

Leo has 10 marbles. Mia has half as many. To find half, share 10 into 2 equal bars: 5 and 5. One bar is 5, so Mia has 5. And looking the other way: Leo has 2 of Mia's bars, so Leo has twice as many as Mia.

Why "times as many" stays the same when things grow

Back to the puzzle. Why did "3 times as many" stay true when Ben went from 4 to 10 and Ava from 12 to 30?

Look at the bars. Ava's amount is always 3 bars, and Ben's is always 1 bar. When Ben's bar gets longer (from 4 to 10), each of Ava's bars gets longer in the same way. So Ava still has 3 of Ben's bars. The number of bars did not change, only the size of each bar.

But "how many more" is the 2 extra bars. When the bars get longer, the 2 extra bars get longer too: from 8 to 20. So "how many more" changes.

This is why "times as many" is the right way to compare mixtures. In Doubling a recipe, there was twice as much flour as sugar (2 cups and 1 cup). In the double recipe there were 4 cups and 2 cups: still twice as much. The "how many more" changed (from 1 cup more to 2 cups more), but "twice as much" stayed. And it is "twice as much" that keeps the taste.

Going backwards: finding the smaller amount

Ava has 15 stickers. That is 3 times as many as Ben. How many does Ben have?

Draw Ava's 15 as 3 equal bars. Each bar is Ben's amount. Share 15 into 3 equal parts: 5, 5 and 5. So Ben has 5. Check: 3 times 5 is 5, 10, 15. Yes.

Worked examples

Example 1. Twice as many. Sam has 7 toy cars. Kim has twice as many. How many does Kim have?

  1. Kim has 2 bars of Sam's amount.
  2. Each bar is 7: 7 and 7 make 14.
  3. Kim has 14 cars. (She has 7 more than Sam, too.)

Example 2. Three times as many, in money. A comic costs 2 dollars. A book costs 3 times as much. What does the book cost?

  1. The book is 3 bars, each the price of a comic: 2 dollars.
  2. Count by twos: 2, 4, 6.
  3. The book costs 6 dollars.

Example 3. Half as much. A big bag of popcorn has 20 handfuls. A small bag has half as much. How many handfuls in the small bag?

  1. Half means sharing into 2 equal parts and taking one.
  2. 20 shared into 2 equal parts: 10 and 10.
  3. The small bag has 10 handfuls.

Example 4. Which comparison? Two plants: one is 3 cm tall, the other is 6 cm tall. Next week they are 6 cm and 12 cm. Compare them both ways, both weeks.

  1. This week: 6 take away 3 is 3, so the tall one is 3 cm taller. And 6 is 2 groups of 3, so it is twice as tall.
  2. Next week: 12 take away 6 is 6, so it is 6 cm taller. And 12 is 2 groups of 6, so it is still twice as tall.
  3. "How much taller" changed from 3 to 6. "Twice as tall" stayed the same.

Example 5. Working backwards. Lin read 24 pages. That is 4 times as many as her little brother. How many pages did he read?

  1. Lin's 24 is 4 equal bars. Each bar is her brother's amount.
  2. Share 24 into 4 equal parts. Count by a guess: is each part 5? Then 4 parts make 20, too few. Is it 6? Then 6, 12, 18, 24. Yes!
  3. Her brother read 6 pages.

Try it

  1. Jo has 5 pencils. Ana has twice as many. How many does Ana have?
  2. A small box holds 3 eggs. A big box holds 4 times as many. How many eggs in the big box?
  3. A toonie is worth how many times as much as a loonie? A loonie is worth how many times as much as a quarter?
  4. Dad is 36 years old. That is 4 times as old as Max. How old is Max?
  5. A blue ribbon is 16 cm long. A red ribbon is half as long. How long is the red ribbon?
  6. Ben has 6 stickers and Ava has 18. Compare them two ways: "how many more" and "how many times as many".
  7. Mia says: "My tower has 2 blocks and yours has 6. If we both add 10 blocks, yours will still be 3 times as tall." Is she right?
  8. A recipe has 3 cups of flour for every 1 cup of sugar. Raj uses 9 cups of flour. How much sugar should he use?
Answers
  1. 10 pencils. Two bars of 5: 5 and 5 make 10.
  2. 12 eggs. Four bars of 3: 3, 6, 9, 12.
  3. A toonie is worth 2 times as much as a loonie (2 dollars and 1 dollar). A loonie is worth 4 times as much as a quarter (4 quarters make 100 cents, which is 1 dollar).
  4. 9 years old. Dad's 36 is 4 equal bars. Try 9: 9, 18, 27, 36. Yes, 4 bars of 9 make 36.
  5. 8 cm. Share 16 into 2 equal parts: 8 and 8.
  6. Ava has 12 more (18 take away 6 is 12). Ava has 3 times as many (6, 12, 18 is three groups of 6).
  7. No. After adding 10, the towers have 12 and 16 blocks. 3 times 12 would be 36, not 16. Adding the same amount to both keeps "how many more" the same (still 4 more) but breaks "times as many". To stay 3 times as tall, Mia's tower should grow by 10 while the other grows by 30.
  8. 3 cups of sugar. Flour is always 3 times the sugar. 9 cups of flour is 3 bars, so each bar (the sugar) is 3 cups: 3, 6, 9.

Common mistake

Using "how many more" when it should be "times as many". "Ava has 3 times as many as Ben, Ben has 4" does not mean Ava has 4 and 3 more, which is 7. It means 3 lots of 4, which is 12. Draw the bars: three bars of 4.

Adding the same amount to keep a mixture the same. Adding 10 to both parts keeps the difference, but not the "times as many". To keep "times as many", each part must grow by its own number of bars.

Where it leads