Big Whole Numbers
Primary 5 · Mathematics
Big Whole Numbers
Lesson 3 of 3
15 min
Let's begin
Byiringiro's school is buying exercise books for the new term. The shop sells them in boxes, and each box holds 36 books. The head teacher orders 24 boxes.
How many books is that altogether? Nobody wants to tip out 24 boxes and count. So we multiply — and to multiply big numbers, you break them into pieces you already know.
We need 36 × 24.
Twenty-four is just 20 + 4. So 36 × 24 is the same as (36 × 20) + (36 × 4). Two easy multiplications instead of one scary one.
The school ordered 864 exercise books.
Set out in a column, the same work looks like this:
| 36 | |
| × 24 | |
| 144 | this is 36 × 4 |
| 720 | this is 36 × 20 — the 0 is what makes it 20 and not 2 |
| 864 | add the two lines |
That zero on the second line is the single most common slip in P5. Write 72 instead of 720 and you get 216 instead of 864 — a quarter of the right answer. The zero is not decoration. It is you saying "I multiplied by twenty, not by two."
Now try 247 × 132. Break 132 into 100 + 30 + 2.
Now add the three pieces:
494 + 7,410 = 7,904
7,904 + 24,700 = 32,604
So 247 × 132 = 32,604.
Never walk away from a big answer without asking "does that look about right?"
Round the numbers: 247 is about 250, and 132 is about 130.
250 × 130 = 25 × 13 × 100 = 325 × 100 = 32,500.
Your exact answer, 32,604, sits right next to 32,500. It makes sense. But if you had got 3,260 or 326,040, the estimate would shout at you at once: a digit has landed in the wrong place.
The school also receives 8,736 pencils, to be shared equally among 24 classrooms. How many pencils does each classroom get?
Before you start, write a small ladder of the 24 times table down the side of your page. It stops you guessing:
24, 48, 72, 96, 120, 144, 168, 192, 216, 240
Now divide 8,736 by 24, working from the left:
Reading the digits you wrote: 8,736 ÷ 24 = 364. Each classroom gets 364 pencils.
Check by multiplying back: 364 × 4 = 1,456 and 364 × 20 = 7,280. Add them: 1,456 + 7,280 = 8,736. That is exactly what we started with, so the answer is right.
Check with an estimate too: 8,736 is roughly 9,000 and 24 is roughly 30. 9,000 ÷ 30 = 300. Our answer 364 is in the hundreds, close to 300. Sensible.
Umutoni has 1,000 avocados to pack into crates. Each crate holds 24.
1,000 ÷ 24 = 41 remainder 16. Forty-one full crates, with 16 avocados still in the basket.
Here is the rule that tells you when to stop: the remainder must always be smaller than the number you are dividing by. If it is not, you can still make one more group.
Tip
Two habits will save you again and again
1. Before you divide, write the times table of the divisor down the edge of your page. Dividing then becomes reading, not guessing.
2. After you multiply or divide, do a quick rounded estimate in your head. If your exact answer is nowhere near the estimate, go hunting — you have almost certainly lost or gained a zero.
Work out 48 × 25.
Answer: 1,200. Split 25 into 20 + 5. 48 × 5 = 240 and 48 × 20 = 960. Then 240 + 960 = 1,200.
Work out 305 × 46, then check it with an estimate.
Answer: 14,030. Split 46 into 40 + 6. 305 × 6 = 1,830 and 305 × 40 = 12,200. Then 1,830 + 12,200 = 14,030. Estimate: 300 × 50 = 15,000, which is close, so the answer is believable.
Work out 4,275 ÷ 15.
Answer: 285. Ladder first: 15, 30, 45, 60, 75, 90, 105, 120, 135. Now: 15 goes into 42 twice (30), leaving 12. Bring down the 7 to make 127; 15 × 8 = 120, so write 8, leaving 7. Bring down the 5 to make 75; 15 × 5 = 75 exactly, so write 5 and nothing is left. Check: 285 × 15 = 2,850 + 1,425 = 4,275.
Uwase has 2,000 beads and threads 18 onto each necklace. How many complete necklaces, and how many beads are left?
Answer: 111 necklaces, 2 beads left. 18 × 100 = 1,800, leaving 200. 18 × 11 = 198, leaving 2. So 100 + 11 = 111 necklaces, remainder 2. The remainder 2 is smaller than 18, so no more necklaces are possible. Check: 18 × 111 = 1,998, and 1,998 + 2 = 2,000.
Estimate 612 × 39 first, then work it out exactly.
Answer: Estimate 600 × 40 = 24,000. Exactly: 612 × 9 = 5,508 and 612 × 30 = 18,360, so 5,508 + 18,360 = 23,868. That sits just under the estimate, exactly as it should — you rounded 612 up a little and 39 up a little too.
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