Multiplying and Dividing Decimals
The decimal point is not decoration — it marks the ones place. Multiply and divide decimals by counting decimal places and shifting points, and the point will always land exactly where it belongs.
1 Understand
The core idea in plain language.
A decimal is a fraction wearing a disguise: 2.4 means 24/10 and 1.3 means 13/10. That disguise is the whole secret of decimal arithmetic — every rule below is just fraction arithmetic with the denominator hidden.
To multiply decimals, strip the points, multiply as whole numbers, then put the point back by counting decimal places. If the factors have 1 and 2 decimal places, the product needs 1 + 2 = 3 decimal places. No guessing, no feeling — just counting.
To divide by 10, 100, or 1000, slide every digit one, two, or three places to the right. 4.8 ÷ 10 = 0.48 because each digit becomes ten times smaller.
To divide by a decimal, first make the divisor a whole number by shifting its point right, then shift the dividend’s point the same distance. 12.6 ÷ 0.3 becomes 126 ÷ 3 — the same question, asked with whole numbers.
2 See it
Diagrams that make the idea visual.
Multiplying: count the places
Compute 2.4 × 1.3 as whole numbers: 24 × 13 = 312.
Now count: 2.4 has 1 decimal place, 1.3 has 1 decimal place, so the product needs 2. Count 2 digits back from the right of 312 and drop the point: 3.12.
Dividing: shift both points together
12.6 ÷ 0.3 looks hard because the divisor is a decimal. Shift the divisor’s point one place right to make it whole — and shift the dividend’s point the same distance so the quotient does not change.
Now it is 126 ÷ 3 = 42, an ordinary whole-number division.
3 The key rule
Multiply as whole numbers, then give the product exactly that many decimal places. If the product needs more digits than it has, pad with leading zeros: 0.2 × 0.3 = 0.06.
4 Worked examples
Follow each step. The pattern is always the same.
- Strip the points: 24 × 13 = 312.
- Count places: 2.4 has 1, 1.3 has 1 — total 2.
- Count 2 digits back from the right of 312: 3.12.
Check: Estimate: 2.4 × 1.3 is about 2 × 1 = 2, and 3.12 is close. A point in the wrong place would give 31.2 or 0.312.
- Strip the points: 25 × 4 = 100.
- Count places: 0.25 has 2, 0.4 has 1 — total 3.
- 100 needs 3 decimal places, so pad a zero: 0.100 = 0.1.
Check: Estimate: 0.25 × 0.4 is about a quarter of a half — about 0.1. Correct.
- Dividing by 100 shifts every digit two places right.
- 4.8 → 0.48 → 0.048.
Check: Multiply back: 0.048 × 100 = 4.8. Correct.
- The divisor 0.3 has 1 decimal place — shift both points 1 place right.
- 12.6 ÷ 0.3 becomes 126 ÷ 3.
- 126 ÷ 3 = 42.
Check: Multiply back: 42 × 0.3 = 12.6. Correct.
5 Common mistakes
These errors show up on almost every quiz. Spot them now and they will never cost you points.
6 Key vocabulary
Say these words like you mean them.
- Decimal places
- The number of digits to the right of the decimal point. 3.12 has two decimal places.
- Product
- The result of a multiplication.
- Dividend
- The number being divided (the first number in a ÷ b).
- Divisor
- The number you divide by (the second number in a ÷ b). Never zero.
- Quotient
- The result of a division.
- Power of 10
- 10, 100, 1000, … — dividing by one shifts digits right; multiplying shifts them left.
7 Quick check
Try each one on paper first, then reveal the answer.
Key points to remember
- Multiply decimals as whole numbers, then count total decimal places: places(a) + places(b).
- Pad with leading zeros when the product is short: 0.2 × 0.03 = 0.006.
- Dividing by 10, 100, 1000 shifts digits right; multiplying shifts them left.
- To divide by a decimal: make the divisor whole, shift the dividend the same amount.
- Dividing by a whole number: the point goes straight up into the quotient.
- Always sanity-check with an estimate — a misplaced point is off by a factor of 10.