Doubling, halving and ×11
Reshape a multiplication until it becomes easy: halve one side, double the other, and watch 16 × 25 turn into 4 × 100. Then learn the ×11 trick, carry case included.
THE IDEA
Halve one side, double the other
A product is an area: 16 rows of 25. Cut the rectangle in half and lay the two halves side by side and you get 8 rows of 50. The shape changed, the area didn't. In symbols, (a ÷ 2) × (b × 2) = a × b, because the ÷2 and the ×2 cancel.
16 × 25 → 8 × 50 → 4 × 100 = 400. Each step is legal on its own, so you can keep going until one side is round.
16 × 25
step 1: 16 ÷ 2 = 8 25 × 2 = 50 → 8 × 50
step 2: 8 ÷ 2 = 4 50 × 2 = 100 → 4 × 100
read it off: 400400Stop as soon as one side is 10, 50, 100 or a number you know by heart. There is no prize for continuing, and halving an odd number drags you into fractions.
Check yourself
18 friends each put in 50 for a shared gift. How much is collected?
- 225
- 3,600
- 450
- 900
Show the answer
900
Right. Halve the 18 and double the 50: 9 × 100 = 900.
Which side do you halve?
Halve the even one
You need a side that survives halving, so pick the even number: 14 × 35 → 7 × 70 = 490. If both are even, halve whichever leaves you closest to a round partner.
Double towards a round number
Doubling is only worth it if the other side lands somewhere friendly. 25 → 50 → 100 is the classic. 35 → 70 is good. 37 → 74 is not, so don't start.
Check yourself
You worked out that 24 × 15 = 360. So 12 × 30 must also be 360.
Show the answer
True
True. Halving the 24 and doubling the 15 gives exactly the same product, so the two problems are the same problem wearing different clothes. This also makes 12 × 30 a free check on your first answer.
ELEVEN
×11 is ×10 plus one more copy
11 × 43 is ten 43s plus one more 43: 430 + 43 = 473. Look at what came out: 4, then 4+3, then 3. That's the famous trick, and it is not magic, it's just the 10-copy and the 1-copy sitting on top of each other with the middle column adding up.
11 × 43 = 473. 11 × 36: outside 3 and 6, middle 3 + 6 = 9, so 396.
Check yourself
A crate holds 11 jars and each jar costs 52. What's the crate worth?
- 512
- 562
- 572
- 582
Show the answer
572
Yes. 520 + 52 = 572, or read it straight off: 5, then 5+2, then 2.
When the middle is bigger than 9
- The problem: 11 × 87
Outside digits are 8 and 7. The middle would be 8 + 7 = 15, and 15 is not a single digit, so it cannot just sit there.
- Write the middle as it is: 8 | 15 | 7
Nothing is wrong yet. This is a real number, it just isn't written in digits yet.
- Carry the ten out of the 15
The 15 means 10 + 5. That 10 belongs one column to the left, so the 8 becomes 9 and 5 stays in the middle.
- Read the answer: 957
Check it the slow way if you like: 870 + 87 = 957. Same number.
Check yourself
Eleven times each of these. Match them up.
Show the answer
- 11 × 36 → 396
- 11 × 74 → 814
- 11 × 25 → 275
- 11 × 43 → 473
Make it a habit
- Before multiplying, ask: can I reshape this? One side even and the other near 25, 50 or 5 is the signal.
- Halve and double as many times as you like; stop when one side is round.
- ×11 = ×10 + one copy. The digit trick is that sum written out, nothing more.
- If the middle digit sum passes 9, carry the ten left, exactly as you would on paper.
Check yourself
The hard one. A hall is set out with 11 rows of 68 seats. How many seats?
- 648
- 748
- 6,148
- 758
Show the answer
748
Exactly. Outside 6 and 8, middle 6 + 8 = 14, so carry the ten: 6 becomes 7 and 4 stays. 748, which is 680 + 68.
Lesson recap
- Halving one side and doubling the other leaves the product unchanged, because ÷2 and ×2 cancel.
- Keep reshaping until one side is 10, 50 or 100: 16 × 25 → 8 × 50 → 4 × 100 = 400.
- Never do the same thing to both sides, and never halve an odd number.
- ×11 is ×10 plus one more copy, which is why the digits go outside, sum, outside, with a carry when the middle passes 9.