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Doubling, halving and quick products

A train has 18 cars, and each car carries 25 crates of eggs. Is there a way to find 18 \times 25 in your head, without writing out any steps?

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Doubling, halving and quick products
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When you reach for it

Harsha watches a freight train pull into the egg farm. It has 18 cars, and each car carries 25 crates of eggs. How many crates are on the train?

The answer is 18 \times 25. You could break it into partial products and add them up. But there is a quicker way. Change the problem into an easier one that has the same answer. Here,

18 \times 25 = 9 \times 50,

and 9 \times 50 is 9 times 5 tens, which is 45 tens, or 450. The train carries 450 crates.

The move is called doubling and halving: halve one factor, double the other, and the product does not change. This lesson is about that move, and about a few other quick products that work the same way. You reach for them when a problem has a factor that is close to a friendly number such as 10, 50 or 100, and when you want an answer in your head.

Part 1 is doubling and halving itself, and why it works. Part 2 uses the same ideas for quick products: times 5, times 9, and halving to divide by 2 or 4.

The procedure

Part 1: double one factor, halve the other

The picture. Think of 18 \times 25 as a rectangle that is 18 squares tall and 25 squares wide. Its area is the product: the number of unit squares inside.

Now cut the rectangle in half across the middle. Each half is 9 squares tall and still 25 wide. Slide the bottom half up and set it beside the top half. The new rectangle is 9 squares tall and 25 + 25 = 50 squares wide.

25 18 9 × 25 9 × 25 50 9 9 × 25 9 × 25
Figure 1Cut the 18 by 25 rectangle in half and set the halves side by side. The new rectangle is 9 by 50, and it covers exactly the same squares, so 18 \times 25 = 9 \times 50.

No square was added and none was lost. They were only moved. So the two rectangles have the same area:

18 \times 25 = 9 \times 50 = 450.

The height was halved, from 18 to 9. The width was doubled, from 25 to 50. Halving one side and doubling the other keeps the area.

Why it is true. The picture works for these numbers. The properties of multiplication show it works for any numbers. Since 18 = 9 \times 2,

\begin{aligned} 18 \times 25 &= (9 \times 2) \times 25 \\ &= 9 \times (2 \times 25) \\ &= 9 \times 50. \end{aligned}

The second line moves the parentheses. That is allowed by the associative property of multiplication: when you multiply three numbers, you may group them either way and the product is the same. The 2 that came out of the 18 joined the 25 and doubled it.

Either factor may be halved. The order of the factors does not matter, by the commutative property of multiplication: 25 \times 18 is the same as 18 \times 25. So you may halve the first factor or the second, whichever is even and whichever makes the problem easier.

Doing it more than once. Each step keeps the product, so you may double and halve again. For 4 \times 75, halve the 4 and double the 75: 2 \times 150. Do it again: 1 \times 300, which is 300. Stop when one factor is friendly.

What to aim for. The move never changes the answer, but it does not always make the problem easier. It helps when doubling turns a factor into a round number: 5 becomes 10, 25 becomes 50, 50 becomes 100, 35 becomes 70, 15 becomes 30. So look for a factor ending in 5, and halve the other one.

Part 2: quick products and quick quotients

Times 5: times 10, then halve. Five is half of ten. So 5 copies of a number is half of 10 copies. For 5 \times 68:

\begin{aligned} 5 \times 68 &= \text{half of } (10 \times 68) \\ &= \text{half of } 680 = 340. \end{aligned}

Half of 680 is found in pieces: half of 600 is 300 and half of 80 is 40, so half of 680 is 340. This is doubling and halving again: double the 5 to get 10, and halve the 68 to get 34, so 5 \times 68 = 10 \times 34 = 340. Either way works. Multiplying by 10 writes each digit one place to the left, as in multiplying by ten.

Times 9: times 10, then take one copy away. Nine copies of a number is ten copies, minus one copy. Tonatiuh needs 9 \times 47. Ten rows of 47 make 470. Nine rows is one row fewer:

\begin{aligned} 9 \times 47 &= 10 \times 47 - 1 \times 47 \\ &= 470 - 47 = 423. \end{aligned}
47 10 10 × 47 = 470 1 × 47 = 47 9 × 47 = 423
Figure 2Ten rows of 47 make 470. Take away one row of 47 and nine rows are left, so 9 \times 47 = 470 - 47 = 423.

The step that splits 9 \times 47 into two products is the distributive property: since 9 = 10 - 1,

(10 - 1) \times 47 = 10 \times 47 - 1 \times 47.

Multiplying a difference is the same as multiplying each part and then subtracting. In the picture, that is the whole rectangle minus its top row.

Halving to divide by 2 or 4. Dividing by 2 means splitting into two equal parts, so it is halving. Dividing by 4 means splitting into four equal parts. You can do that in two steps: split in half, then split each half in half. So dividing by 4 is halving twice.

Tama has 312 eggs to share equally among 4 crates. Half of 312 is 156. Half of 156 is 78. Each crate gets 78 eggs.

312 312 156 156 312 ÷ 2 78 78 78 78 312 ÷ 4
Figure 3Halving 312 gives 156. Halving each half again cuts the bar into four equal parts of 78, so 312 \div 4 = 78.
312 \div 4 = (312 \div 2) \div 2 = 156 \div 2 = 78.

Halving twice works because 4 = 2 \times 2. Check with multiplication, as with any division fact (a quotient times the divisor gives back the number divided): 78 \times 4 is 78 doubled twice, 156 and then 312. In the same way, dividing by 8 is halving three times, since 8 = 2 \times 2 \times 2.

When the move does not help. Halving needs an even number. In 13 \times 17, both factors are odd, and half of 13 is not a whole number. Doubling and halving cannot start, so use another way, such as partial products. And if halving a number for division gives an odd number too early, the second halving will not come out even (see Boundaries).

Worked examples

Example 1

Part 1: one double and one halve

A train has 14 cars. Each car carries 35 crates of eggs. How many crates does the train carry?

Solution

The total is 14 \times 35. The factor 35 ends in 5, so doubling it gives a round number, 70. The factor 14 is even, so it can be halved to 7.

14 \times 35 = 7 \times 70.

This is allowed because 14 = 7 \times 2, and by the associative property (7 \times 2) \times 35 is the same as 7 \times (2 \times 35), which is 7 \times 70.

Now 7 \times 70 is 7 times 7 tens, which is 49 tens, or 490. The train carries 490 crates.

Example 2

Part 1: doubling and halving twice

Harsha stacks egg cartons in 28 rows on a loading dock, with 25 cartons in each row. How many cartons are on the dock?

Solution

The total is 28 \times 25. Halve the 28 and double the 25:

28 \times 25 = 14 \times 50.

That is easier, but 14 \times 50 can be made easier still. The 14 is even, so halve it and double the 50:

14 \times 50 = 7 \times 100 = 700.

There are 700 cartons. Each step kept the product the same, so

28 \times 25 = 14 \times 50 = 7 \times 100 = 700.

Two halvings of 28 divide it by 4, and two doublings of 25 multiply it by 4. Since 25 \times 4 = 100, any number times 25 is the same as a quarter of that number, times 100.

Example 3

Part 1: choosing which factor to halve

Tonatiuh needs 15 \times 32. He starts by halving the 15. What goes wrong, and what works instead?

Solution

Half of 15 is 7 and a half, which is not a whole number. So the 15 cannot be halved here.

The 32 is even. By the commutative property, it does not matter that 32 is the second factor. Halve the 32 and double the 15:

15 \times 32 = 30 \times 16.

Now 30 \times 16 is 3 tens times 16, which is 48 tens, or 480. You could also go one step further: 30 \times 16 = 60 \times 8 = 480. Both ways give 480.

So 15 \times 32 = 480. The odd factor was the one to double, not the one to halve.

Example 4

Part 2: times five as times ten, then halve

Each egg crate on Tama’s truck holds 5 trays. There are 86 crates. How many trays are there?

Solution

The total is 5 \times 86. Since 5 is half of 10, find 10 \times 86 first, then halve it.

10 \times 86 = 860.

Half of 860: half of 800 is 400, and half of 60 is 30, so half of 860 is 430.

5 \times 86 = \text{half of } 860 = 430.

There are 430 trays. As doubling and halving, this is 5 \times 86 = 10 \times 43 = 430, the same answer.

Example 5

Part 2: times nine as times ten, minus one copy

A train has 64 cars, and each car holds 9 crates of eggs. How many crates are on the train?

Solution

The total is 64 \times 9, which is the same as 9 \times 64 by the commutative property. Nine copies of 64 is ten copies minus one copy. By the distributive property,

9 \times 64 = 10 \times 64 - 1 \times 64 = 640 - 64.

To find 640 - 64, take away 60 to get 580, then take away 4 more to get 576.

There are 576 crates. Check another way: 9 \times 64 is close to 9 \times 60 = 540, and 9 \times 4 = 36 more makes 576.

Example 6

Part 2: halving twice to divide by four

The farm packs 348 eggs into 4 crates, with the same number in each crate. How many eggs go in each crate?

Solution

Each crate gets 348 \div 4 eggs. Dividing by 4 is halving twice, because 4 = 2 \times 2.

Half of 348: half of 300 is 150, half of 40 is 20, and half of 8 is 4. So half of 348 is 174.

Half of 174: half of 160 is 80, and half of 14 is 7. So half of 174 is 87.

348 \div 4 = 174 \div 2 = 87.

Each crate gets 87 eggs. Check: double 87 to get 174, and double again to get 348. So 87 \times 4 = 348.

Example 7

Part 2: two quick moves in one problem

A train has 45 cars, each with 18 crates. Find 45 \times 18 in your head.

Solution

The 45 ends in 5, so it is the one to double. The 18 is even, so halve it:

45 \times 18 = 90 \times 9.

Halving again will not help, since 9 is odd. But 9 copies of 90 is 10 copies minus one copy, by the distributive property:

9 \times 90 = 900 - 90 = 810.

So 45 \times 18 = 810. The train carries 810 crates. One problem used doubling and halving first, then the times-nine move.

Boundaries

  • An even number to halve. Halving keeps whole numbers only when the number is even. If both factors are odd, as in 13 \times 17, doubling and halving cannot start with whole numbers. Use partial products instead.
  • Double one, halve the other. The two changes must go in opposite directions, on the two factors of one product. Doubling both factors makes the product four times as big: 6 \times 7 = 42, but 12 \times 14 = 168.
  • Products only. The move belongs to multiplication. It does not keep a sum: 18 + 25 = 43, but 9 + 50 = 59.
  • Take away one copy, not one. Times 9 is ten copies minus one copy of the number: 9 \times 47 = 470 - 47, not 470 - 1.
  • Halving twice needs an even half. To divide by 4 with no remainder, the first half must itself be even. For 342 \div 4, half of 342 is 171, which is odd, so it cannot be halved into whole numbers. In fact 342 = 4 \times 85 + 2, a remainder of 2.
  • The answer stays, the ease may not. Every step keeps the product, but not every step makes it easier. Aim for a factor such as 10, 50 or 100, and stop once you have one.

Where this goes next

Doubling and halving is the first time you change a problem into an easier one with the same answer, and the reason it is allowed is a property of multiplication, not a trick. The same idea works with other numbers: tripling one factor and taking a third of the other also keeps the product, as in 6 \times 35 = 2 \times 105. It works when you multiply by multiples of ten, where 40 \times 25 can become 20 \times 50 or 10 \times 100. Later, halving a factor that is odd gives a fraction, and the same move works with fractions and decimals: 4.5 \times 8 = 9 \times 4.

Exercises

Part 1: doubling and halving

  1. Find 16 \times 15 by halving one factor and doubling the other.
  2. Find 22 \times 35.
  3. Find 25 \times 24. Double and halve more than once.
  4. Find 14 \times 45.
  5. Find 50 \times 36.
  6. Find 12 \times 15.
  7. Find 26 \times 50.
  8. Find 8 \times 125. Keep halving the 8 until it is 1.
  9. A train has 24 cars, and each car carries 35 egg crates. How many crates are on the train?
  10. Fill in the boxes: 34 \times 15 = 17 \times \square = \square.

Part 2: quick products and halving to divide

  1. Find 5 \times 46 by finding 10 \times 46 and halving.
  2. Find 5 \times 124.
  3. Find 9 \times 38 by finding 10 \times 38 and taking one copy away.
  4. Find 9 \times 76.
  5. Find 9 \times 125.
  6. Find 456 \div 2 by halving.
  7. Find 920 \div 4 by halving twice.
  8. Harsha loads 138 crates, and each crate holds 5 trays of eggs. How many trays does she load?
  9. A train has 9 cars, and each car carries 87 egg crates. How many crates are on the train?
  10. Tama shares 372 eggs equally among 4 crates. How many eggs go in each crate?
  11. Find 888 \div 8 by halving three times.

Reasoning

  1. Tonatiuh says that since 14 \times 30 = 7 \times 60, it must also be true that 14 + 30 = 7 + 60. Is he right? Explain.
  2. Harsha tries to find 6 \times 15 by doubling both factors: 12 \times 30. Find both products. How do they compare, and why?
  3. Tama wants to find 15 \times 21 by doubling and halving. Explain why the move does not help here, and find the product another way.
  4. Explain, using a rectangle, why 12 \times 35 and 6 \times 70 have the same product. Then explain it again with the associative property, and find the product.
Answers
  1. 240. Halve 16 and double 15: 16 \times 15 = 8 \times 30, which is 24 tens, or 240.
  2. 770. 22 \times 35 = 11 \times 70, which is 77 tens, or 770.
  3. 600. 25 \times 24 = 50 \times 12, and then 100 \times 6 = 600.
  4. 630. 14 \times 45 = 7 \times 90, which is 63 tens, or 630.
  5. 1{,}800. Halve 36 and double 50: 50 \times 36 = 100 \times 18 = 1{,}800.
  6. 180. 12 \times 15 = 6 \times 30, which is 18 tens, or 180.
  7. 1{,}300. 26 \times 50 = 13 \times 100 = 1{,}300.
  8. 1{,}000. 8 \times 125 = 4 \times 250, then 2 \times 500, then 1 \times 1{,}000, which is 1{,}000.
  9. 840 crates. 24 \times 35 = 12 \times 70, which is 84 tens, or 840.
  10. 34 \times 15 = 17 \times 30 = 510. Halving 34 gives 17 and doubling 15 gives 30, and 17 \times 30 is 51 tens.
  11. 230. 10 \times 46 = 460, and half of 460 is 230.
  12. 620. 10 \times 124 = 1{,}240, and half of 1{,}240 is 620 (half of 1{,}200 is 600, half of 40 is 20).
  13. 342. 10 \times 38 = 380, and 380 - 38 = 342.
  14. 684. 760 - 76 = 684.
  15. 1{,}125. 1{,}250 - 125 = 1{,}125.
  16. 228. Half of 400 is 200, half of 50 is 25, half of 6 is 3, and 200 + 25 + 3 = 228.
  17. 230. Half of 920 is 460, and half of 460 is 230. Check: 230 \times 4 = 920.
  18. 690 trays. 5 \times 138 is half of 1{,}380, which is 690.
  19. 783 crates. 9 \times 87 = 870 - 87 = 783.
  20. 93 eggs. Half of 372 is 186, and half of 186 is 93. Check: 93 \times 4 = 372.
  21. 111. Half of 888 is 444, half of 444 is 222, and half of 222 is 111. Halving three times divides by 8, since 8 = 2 \times 2 \times 2.
  22. No. 14 + 30 = 44 but 7 + 60 = 67. Doubling and halving keeps a product because of the associative property of multiplication: (7 \times 2) \times 30 = 7 \times (2 \times 30). Addition has no such step. Halving 14 takes away 7, and doubling 30 adds 30, and those do not cancel.
  23. 6 \times 15 = 90 and 12 \times 30 = 360. The second is four times the first. Doubling one factor doubles the product, and doubling the other doubles it again, so 12 \times 30 = 4 \times 90. To keep the product, one factor must be halved while the other is doubled.
  24. Both 15 and 21 are odd, so neither can be halved into a whole number. Another way: by the distributive property, 15 \times 21 = 15 \times 20 + 15 \times 1 = 300 + 15 = 315.
  25. 420. Draw a rectangle 12 squares tall and 35 wide. Cut it across the middle into two rectangles, each 6 tall and 35 wide, and set them side by side. The new rectangle is 6 tall and 70 wide, and it holds the same squares, so the areas match. With the associative property: 12 \times 35 = (6 \times 2) \times 35, which equals 6 \times (2 \times 35), and that is 6 \times 70, or 42 tens, or 420.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0