Decimal points that have to match up

Liri wanted to add two stretches of the route: 4.25 light-years and 1.7 light-years. “Can I just write them one under the other and add as usual?”

Erklärus: “Almost! With decimals there is an important rule: the decimal points must line up exactly. Otherwise you’re adding apples and pears.”

➕➖ Adding and subtracting

Write the numbers one under the other so that the decimal points line up exactly. You fill missing places with zeros.

📐 4.25 + 1.7
4.25
1.70
────────
5.95

The missing place in 1.7 is filled with a 0: 1.70.

Work out: 3.4 + 2.15 =

✖️ Multiplying

Erklärus: “When multiplying, you first ignore the decimal points completely and calculate as with whole numbers.”
Liri: “And then?”
Erklärus: “Then you count how many decimal places the two starting numbers had altogether – and put the decimal point into the answer exactly that many places from the right.”
📐 2.5 × 0.4

Without decimal points: 25 × 4 = 100
Count the decimal places: 2.5 (1) + 0.4 (1) = 2 places
Put the decimal point 2 places from the right: 1.00 = 1

Work out: 1.5 × 0.2 =

➗ Dividing

Erklärus: “When dividing, you first move the decimal point in the divisor until it becomes a whole number.”
Liri: “And in the dividend?”
Erklärus: “Move it along by exactly the same number of places – then divide in writing as usual.”
📐 6.3 ÷ 0.3

Move the decimal point in 0.3 one place to the right → 3 (a whole number)
Move the decimal point in 6.3 one place to the right too → 63
Now in writing: 63 ÷ 3 = 21

Work out: 8.4 ÷ 0.2 =

🔄 Turning fractions into decimals

Some fractions are easy to turn into decimals – simply scale them up to the denominator 10, 100 or 1000.

Liri: “12 – how does that become a decimal?”
Erklärus: “Scale up by 5: 12 = 510. And 510 means exactly 5 tenths – so 0.5.”
Examples

12 = 510 = 0.5

14 = 25100 = 0.25

34 = 75100 = 0.75

Convert: 15 = 0.?

🎯 Rounding

With amounts of money you usually round to two decimal places (cents). The rule: from the digit 5 upwards you round up, below that you round down.

Rounding rule

€3.764 → round to the nearest cent → €3.76 (4 → round down)
€3.768 → round to the nearest cent → €3.77 (8 → round up)

Round €5.236 to the nearest cent: €5.?
🌌 Calculating with decimals in the universe

Fuel consumption is often calculated in decimals with many decimal places – even tiny rounding errors can have big effects over millions of kilometres.
Space-probe navigation therefore rarely uses rounded numbers, but works as precisely as possible.

❌ Common mistake

When adding or subtracting without lining up the decimal points you quickly end up with the wrong answer! 4.25 + 1.7 is not 4.32 (added wrongly) but 5.95 – first write the decimal points exactly one above the other.

📝 Exercise 1
Work out: 7.6 − 2.45 =
📝 Exercise 2
Work out: 0.6 × 0.3 =
📝 Exercise 3
Round €2.145 to the nearest cent: €2.?

Calculated exactly, travelling safely

Liri now added up all the stretches of her journey with confidence – decimal point under decimal point, place by place.

Liri: “Now I can work out all my fuel consumption with decimals without making a mistake. Every decimal place brings me a little closer to the truth about my way home.”
Erklärus: “That is exactly how real space probes navigate – with decimal precision, not with rough estimates. Anyone who calculates cleanly with decimal points doesn’t get lost.”
“Adding/subtracting: decimal points exactly one under the other. Multiplying: calculate without the decimal point, count the decimal places and put it back in. Dividing: move the decimal point in the divisor, move it along in the dividend. Rounding: round up from 5.”
What you learned today
Decimal point under decimal point
Adding/subtracting: line up · Multiplying: count places · Dividing: move the point
🌟

Chapter 34 mastered!

Liri can now calculate and round confidently with decimals.

Erklärus’s notebook – page 34:
“With decimals the position of the decimal point decides everything. Always line up first, then calculate.”
← Back to overview 34 – Calculating with Decimals