🧮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

How to use

53 + 79 = ?

✓

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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

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0 / 3 correct0 / 3 correct

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