MISSION · Number sense
Calculating with Decimals
Three different rules about where the point goes
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Each operation treats the decimal point differently, and knowing which rule applies is most of the work.
Adding and subtracting: line up the points
\[12.5 + 3.75\]Write them with the points vertically aligned, padding with zeros so the columns match:
12.50
+ 3.75
-------
16.25
Lining up the points is the same as lining up the place values — tenths under tenths, hundredths under hundredths. Lining up the last digits instead would add hundredths to tenths, which is adding pennies to pounds.
The padding zero costs nothing, since \(12.5 = 12.50\), and it removes the ragged edge that causes the mistake.
Multiplying: count the places
Here the points do not line up. Ignore them entirely, multiply as whole numbers, then count.
\[0.3 \times 0.4\]Multiply 3 by 4 to get 12. Now count decimal places in the question: one in \(0.3\), one in \(0.4\), two altogether. So the answer has two: \(0.12\).
The reason is place value. \(0.3\) is \(\frac{3}{10}\) and \(0.4\) is \(\frac{4}{10}\), so the product is \(\frac{12}{100}\) — the denominators multiplied, giving two zeros, giving two decimal places.
This is also where multiplying-makes-bigger finally dies. \(0.3 \times 0.4 = 0.12\), smaller than either. Multiplying by anything below 1 shrinks.
Dividing: move both points
\[4.8 \div 0.6\]Dividing by a decimal is awkward, so get rid of it. Multiply both numbers by 10 — which doesn’t change the answer, since you’ve scaled both parts of the fraction equally:
\[48 \div 6 = 8\]Move the point in the divisor until it’s a whole number, and move the point in the other number by exactly the same number of places. Then divide normally.
And note that \(4.8 \div 0.6 = 8\) is bigger than 4.8. Dividing by something less than 1 increases the result, because you’re asking how many small pieces fit into a larger amount.
Estimating catches everything
Decimal errors are almost always place-value errors, and they’re always off by a factor of ten or a hundred — which an estimate catches immediately.
\(0.3 \times 0.4\) is roughly a third of a half, so well under 1. An answer of 1.2 or 12 is wrong by inspection, no rechecking needed.
Common mix-ups
- Lining up the last digits instead of the points when adding.
- Lining up the points when multiplying, which is the addition rule in the wrong place.
- Moving only one point when dividing.
- Miscounting decimal places in a product. Count both numbers.
Check yourself
Work out \(1.2 \times 0.05\) and \(9 \div 0.3\).
\(12 \times 5 = 60\), and three decimal places altogether, so \(0.060 = 0.06\). For the division, move both points one place: \(90 \div 3 = 30\).
LEVEL 3 · REFLECT
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