Multiplying decimal numbers · البرمجة مع مارتن

The point moves to the right

How much is 2.5 × 10? Do not worry, you do not have to count anything in columns. Just shift the point one place to the right: 2.5 × 10 = 25. And 2.5 × 100? The point jumps two places: 250.

Why does it work? Multiplying by ten means each digit moves one place up: tenths become wholes, wholes become tens. The digits stay, they just have more weight — and on paper that looks exactly like shifting the point to the right.

The second thing you will handle today: a decimal number times an ordinary number, for example 3 × 2.5. That is just addition in disguise: 2.5 + 2.5 + 2.5 = 7.5. Or the money trick: 3 × 2 korunas and 3 × 50 hellers.

Shifting the point is one of the most useful things in all of maths. Unit conversions, percentages, shopping — everywhere the point moves back and forth. Today to the right, next lesson to the left.

Tereza is buying notebooks for the whole class: 10 of them at 12.50 Kč each. Filip is pulling out a phone with a calculator. “Wait,” Tereza stops him, “times ten I can do without a machine.” She writes 12.50 and shifts the point one place to the right: 125.0. “One hundred and twenty-five korunas.” Filip taps 12.5 × 10 into the calculator to be sure — 125. “How do you do that so fast?” “When you times ten, the point moves to the right. Times a hundred, two places. I do not count, I just move.”

💡

When the point runs out of digits, add zeros: 2.5 × 100 = 250 (the point jumps over the five and then over the extra zero). A missing place = a zero.

Two kinds of multiplying, you can do both

Multiplying by 10, 100, 1000: shift the point to the right.

× 10 → the point 1 place to the right: 3.75 → 37.5
× 100 → 2 places: 3.75 → 375
× 1000 → 3 places: 3.75 → 3750 (one place was missing, a zero was added)

The number of zeros in the number you multiply by = the number of jumps of the point. The result is always bigger — check with common sense: ten notebooks cost more than one.

Multiplying by a whole number: count without the point, put the point back at the end.

For example 3 × 2.5:

Hide the point and count 3 × 25 = 75.
In the number 2.5 there is one place after the point — so put it back into the result too: 7.5.
Estimate to check: 3 × 2.5 is about 3 × 3 = 9, and 7.5 is reasonably close. It fits.

It works because 2.5 is 25 tenths. Three times 25 tenths is 75 tenths — and 75 tenths you write as 7.5. The point was not lost, it just waited to the side while you finished the times table.

Work out 4.08 × 10 and 4.08 × 100

× 10: the point one place to the right: 4.08 → 40.8. The zero in the middle moved like every other digit — you must not skip it.

× 100: the point two places: 4.08 → 408. The point jumped all the way to the end, so you no longer write it.

Check: 4.08 is about 4. Times ten about 40 ✓, times a hundred about 400 ✓. If you got 4080, you jumped one place too many.

How much do 6 lollipops at 7.50 Kč cost?

I count 6 × 7.50.

Step 1: Without the point: 6 × 750 = 4500. (Or more smartly: 6 × 75 = 450 and I can drop the end zero at once — 7.50 = 7.5.)

Step 2: In 7.5 there is one place after the point, I put it back: 45.0 = 45 Kč.

Estimate: 6 × 7.5 is between 6 × 7 = 42 and 6 × 8 = 48. The result 45 is right in the middle — it fits.

Conversion: 3.2 km into metres

A kilometre has 1000 metres, so I count 3.2 × 1000.

A thousand has three zeros → the point jumps three places to the right: 3.2 → 32 → 320 → 3200. Two places were missing, two zeros were added.

3.2 km = 3200 m. That is exactly how unit conversions work — no magic, just multiplying by ten, a hundred or a thousand, so moving the point.

⚠️

When you multiply by ten you do not add a zero at the end. With whole numbers it happens to work (25 × 10 = 250), but 2.5 × 10 is not 2.50! A zero at the end after the point does not change the number. The right way is shifting the point: 2.5 × 10 = 25.

Practice: now you do it 🎲

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

Train shifting the point: times 10, times 100 and divided by 10. Write the answer with a decimal point when it does not come out as a whole number.

Practise the questions

On paper

Write the middle steps too — with a point shift, mistakes happen in the head, not on paper.

Work out by shifting the point: 6.04 × 10, 6.04 × 100, 0.7 × 1000. By each one write how many places the point jumped.
Work out with the “hide the point” trick: 4 × 3.25 and 7 × 1.5. By each one write an estimate first.
How much do 10 tickets at 24.50 Kč cost? And 100 tickets? Work out both by shifting the point.
Convert: 5.8 km into metres and 1.25 m into centimetres. Write which number you multiplied by.

Now you 💪

I can multiply 3.75 by ten and by a hundred just by shifting the point, with no counting.
I can work out 3 × 2.5 with the hide-the-point trick and put the point back correctly.
I can spot nonsense: I know why 2.5 × 10 cannot be 2.50.

✅

Done when: You multiply decimal numbers by ten, a hundred and a thousand by shifting the point to the right, and by a small whole number with the trick “hide the point, count, put the point back”.

What to take from this lesson

Times 10 = the point 1 to the right, times 100 by 2, times 1000 by 3. Number of zeros = number of jumps.
When the digits run out, you add zeros: 3.2 × 1000 = 3200.
Multiplying by a whole number: count without the point, then put back as many decimal places as there were.
The result of multiplying by ten and more is always bigger — an estimate saves you from jumping one place extra.
Unit conversions (km → m, m → cm) are just this multiplying in disguise.