🧮Decimal Arithmetic

The moment the decimal point appears, many hands start to shake. Yet the four operations on decimals are hardly new — almost everything is whole-number arithmetic, and after each step you answer the same question: where does the point go back?

Add and subtract: line up the points first

Column addition and subtraction have exactly one iron rule: align the decimal points, which means lining up place values. To compute $3.75 + 2.6$, pad 2.6 into 2.60 first:

$3.75 + 2.60 = 6.35$

Hundredths over hundredths (5 over 0), tenths over tenths (7 over 6), add as if they were whole numbers, then copy the point down. If you lazily right-align the digits instead, the 6 slides under the 5 — that is really adding $3.75 + 0.26 = 4.01$. One slot off, and the answer is miles away.

Multiply: compute as integers, then count places

Multiplication needs no aligned points; it takes another road:

1. Erase the points and multiply as integers: $3.2 \times 0.5$ becomes $32 \times 5 = 160$;
2. Count decimal places: 3.2 has 1, 0.5 has 1 — 2 in total;
3. Count back from the right: 2 places into 160 gives $1.60$, that is $1.6$.

This is the "count decimal places" rule: the product carries as many decimal places as the factors have combined. Nothing mysterious — erasing the points enlarged each number 10-fold, and at the end that enlargement must be undone. This "compute as integers first" habit is the old skill from multiplication wearing a new coat.

Divide: whole divisors directly, decimal divisors after a shift

Dividing by a whole number is easy: divide straight away, and the quotient's point follows the dividend's. $7.2 \div 6$: first $72 \div 6 = 12$, the dividend has 1 decimal place, so the quotient is $1.2$.

Dividing by a decimal adds one move: turn the divisor into a whole — shift the divisor's point right by however many places, and shift the dividend's point just as far. The quotient never changes. In $7.2 \div 0.6$ both points move one place:

$7.2 \div 0.6 = 72 \div 6 = 12$

Sanity check: exactly twelve 0.6s fit into 7.2. Shifting points is really multiplying divisor and dividend by 10 together, and the quotient does not budge.

Estimate first, then compute precisely

67 − 39 = ?

✓

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The heart of decimal arithmetic is whole-number arithmetic: $3.75 + 2.6$ has the skeleton $375 + 260$. Drill the whole-number columns until they are fast and steady, then worry about where the point stands. Before computing, throw out a whole-number estimate: $3.75 + 2.6$ is about $4 + 3 = 7$, so 6.35 feels right — while an answer of 0.635 tells you instantly the point parked in the wrong slot.

Check yourself

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Quick quiz

↺ Try again

1. 1. 3.75 + 2.6 = ?

   4.016.356.45

2. 2. 3.2 × 0.5 = ?

   161.60.16

3. 3. 7.2 ÷ 0.6 = ?

   1.212120