Dividing decimal numbers · Lập trình với Martin

Dividing by ten and a hundred by shifting the point, and dividing by a whole number.

The point moves to the left

Last time the point moved to the right when you multiplied. Division is the opposite operation — so the point goes left. 25 : 10 = 2.5. And 25 : 100 = 0.25.

That makes sense: when you split 25 korunas among 10 people, each gets less than three korunas. The result of dividing by ten is always smaller — by one place. That is the whole trick.

The second skill today: dividing a decimal number by an ordinary number, for example 7.5 : 3. You count almost the same as the division you know — just at the moment you cross the point in the division, you write a point in the result too.

And a bonus that will please you: thanks to decimal numbers, no division “comes out with a remainder” for ever. 7 : 2 was not 3 remainder 1 — it was 3.5, you just could not write it yet. Today you can.

Filip, Ema and Jonáš bought a pizza for 187.50 Kč and want to split it equally. “You cannot divide that, there will be a remainder,” Jonáš waves it off. Filip pulls out paper: “It comes out, just not in whole korunas.” He writes the division: 187.50 : 3. Wholes: 187 : 3 = 62, remainder 1. “And now I cross the point — so I write it in the result.” He goes on: 15 tenths : 3 = 5. It comes out 62.50. Ema puts 62.50 on the table, the others too, and the bill sits exactly. No remainder.

💡

Check the direction: with division the number gets smaller, so the point must go left. If after dividing by ten you get a bigger number, it went the wrong way.

Left by as many places as there are zeros

Dividing by 10, 100, 1000: shift the point to the left.

: 10 → the point 1 place to the left: 37.5 → 3.75
: 100 → 2 places: 37.5 → 0.375
: 1000 → 3 places: 37.5 → 0.0375

When the digits run out, you add zeros from the left: 4 : 100 = 0.04. And before a lonely point you always write a zero: not “.04”, but “0.04”.

Dividing by a whole number: divide as normal, write the point when you cross it.

For example 7.5 : 3:

Divide the wholes: 7 : 3 = 2, remainder 1. In the result I write 2.
Now I cross the point in the number 7.5 → in the result I write a point: “2.”.
Remainder 1 whole = 10 tenths, plus 5 tenths from the number = 15 tenths. 15 : 3 = 5. Result: 2.5.

Check by multiplying: 2.5 × 3 = 7.5 ✓.

And when division of whole numbers “does not come out”? Add zeros after the point and keep dividing: 7 : 2 → 7.0 : 2 = 3.5. Remainders are the past.

Work out 62.5 : 10 and 62.5 : 100

: 10 — the point one place to the left: 62.5 → 6.25.

: 100 — two places to the left: 62.5 → 0.625. The digits ran out, so a zero was written before the point.

Check the direction: I am dividing, so the result must be smaller than 62.5. Both fit (6.25 and 0.625 are smaller). If 625 came out, the point went right — wrong direction.

Split 9.6 metres of rope into 4 parts

I count 9.6 : 4.

Wholes: 9 : 4 = 2, remainder 1. I write 2.

I cross the point → in the result I write a point: “2.”.

Tenths: remainder 1 whole = 10 tenths, + 6 tenths = 16 tenths. 16 : 4 = 4.

Result: 2.4 metres for each part. Check: 2.4 × 4 = 9.6 ✓. Each gets a piece of rope 2 metres and 40 centimetres long.

Division that used not to come out: 9 : 4

In year 3: 9 : 4 = 2, remainder 1. Today: no remainder, we keep going.

I write 9 as 9.00 and divide: 9 : 4 = 2, remainder 1. I cross the point → “2.”. Then 10 : 4 = 2, remainder 2. Then 20 : 4 = 5.

Result: 2.25. Check: 2.25 × 4 = 9 ✓.

This is how you split 9 korunas among 4 people: each gets 2.25 Kč. With a remainder a koruna would sit on the table — with decimal numbers each gets exactly their share.

⚠️

Do not forget to write the point in the result. The most common mistake: 7.5 : 3 = 25. The digits are right, but without the point the result is ten times bigger than it should be. The moment you cross the point in the division, write it in the result — at once, not “I will remember later”.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

Train moving the point both ways: times 10, times 100 and divided by 10. Write the answer with a decimal point, for example 4.7.

Practise the questions

On paper

Division with a check — check every result at once by multiplying.

Work out by shifting the point: 84.2 : 10, 84.2 : 100, 7 : 1000. Before a lonely point do not forget a zero.
Divide in writing: 8.4 : 4 and 13.5 : 5. On each one mark the moment you wrote the point in the result, and do a check by multiplying.
Split 250 Kč among 4 friends with no remainder. (Hint: 250.00.) Write how much each gets.
Convert: 3200 m into kilometres and 45 cm into metres. Write which number you divided by and which way the point went.

Now you 💪

I can divide 62.5 by ten and by a hundred by shifting the point to the left.
In written division I know exactly when to write the point in the result.
I can finish the division 9 : 4 with no remainder as 2.25.

✅

Done when: You divide decimal numbers by ten and a hundred by shifting the point to the left, you divide by a whole number with the point at the right moment, and you check every result.

What to take from this lesson

Divided by 10 = the point 1 to the left, divided by 100 by 2 places. With division the number gets smaller.
When the digits run out, you add zeros from the left: 4 : 100 = 0.04.
In written division you write the point in the result exactly when you cross it.
Division “with a remainder” can be finished: add zeros after the point and keep dividing (7 : 2 = 3.5).
A check by multiplying finds every lost point: result × divisor = the original number.