Count the decimal places first
A decimal place is a digit after the decimal point. To work out 0.3 × 0.4, start by counting the decimal places in both numbers. 0.3 has one decimal place, and so does 0.4, so together they have 2 decimal places.
Each number has one digit after its point: one decimal place each, 2 in total.
Multiply without the points
Next, multiply as if the points were not there: 3 × 4 = 12. This answer is too big, and it is easy to say by how much. 3 is 10 times 0.3, and 4 is 10 times 0.4, so 3 × 4 is 10 × 10 = 100 times as big as 0.3 × 0.4.
Put the point back
To undo the 100 times, divide 12 by 100. Dividing by 100 moves every digit two places to the right, and the decimal point stays where it is. The 1 moves from the tens to the tenths, the 2 moves from the ones to the hundredths, and a 0 fills the ones place. So 0.3 × 0.4 = 0.12.
The two places the digits moved are the 2 decimal places counted at the start. That is the whole rule: multiply without the points, then give the answer as many decimal places as the two numbers have together.
× 1 moves no digit, so the value is unchanged
Turn 12 into 0.12
Choose ÷ 100 and watch the digits of 12 move two places to the right while the point stays still: 12 becomes 0.12.
Tenths of tenths are hundredths
There is another way to see why the answer has two decimal places. Take a square that stands for 1 whole. Cut it into 10 strips one way and 10 strips the other way, and it is cut into 100 small squares, each one hundredth.
0.3 is 3 tenths of the width and 0.4 is 4 tenths of the height. Where they overlap there are 3 × 4 = 12 small squares: 12 hundredths, which is 0.12. A tenth of a tenth is a hundredth, so one decimal place times one decimal place gives two.
The answer 0.12 is smaller than both 0.3 and 0.4. Multiplying by a number less than 1 takes only a part of the other number, so the answer is less than the number you started with.
The square is 1 whole, cut into 100 hundredths. 3 tenths across and 4 tenths down overlap in 12 of the small squares, which is 0.12.
6/10 × 4/10 = 24/100 = 0.24: smaller than either factor, because it is a part of a part
Make 0.3 × 0.2 and count the hundredths
Drag the two edges to 3 tenths across and 2 tenths up. They overlap in 6 small squares, 6 hundredths: 0.3 × 0.2 = 0.06.
When the answer is short of digits
For 0.3 × 0.2, work out 3 × 2 = 6. The answer needs 2 decimal places, but 6 is only one digit. Write a 0 in front of it to fill the tenths place: 0.3 × 0.2 = 0.06.
The same happens with more places. 0.05 has 2 decimal places and 0.3 has 1, so 0.05 × 0.3 needs 3. 5 × 3 = 15, and 15 thousandths is written 0.015.
A decimal times a whole number
A whole number has no decimal places, so it adds none to the count. For 0.89 × 9, work out 89 × 9 = 801. Only 0.89 has decimal places, and it has 2, so the answer has 2: 0.89 × 9 = 8.01.
Estimate to check where the point goes. 0.89 is a little less than 1, so 0.89 × 9 is a little less than 9. 8.01 fits; 80.1 and 0.801 do not.
Count the places in both numbers
The usual mistake is to count the decimal places in only one of the numbers. 0.3 × 0.4 is not 1.2: 1.2 has one decimal place, and the two numbers have two between them. An estimate catches it too, because 0.3 × 0.4 must be less than 0.4, and 1.2 is more.
Worked example: Flour for a Bake Sale from a Recipe in Quarters
Question Ms Rao's muffin recipe uses 34 kg of flour for one batch. For a school bake sale she makes 5 batches. Flour is sold in bags of 2.5 kg, and she has no flour at home. (a) How many kilograms of flour does she need? Give your answer as a decimal. (b) How many bags of flour must she buy, and how much flour will be left over?
1.Three quarters is 0.75, because 34 = 75100. So one batch needs 0.75 kg of flour.
34 = 75100 = 0.75, so one batch needs 0.75 kg of flour. 2.Five batches need 5 × 0.75 = 3.75 kg of flour.
Five batches need 5 × 0.75 = 3.75 kg. 3.(a) She needs 3.75 kg of flour.
(a) She needs 3.75 kg of flour. 4.One bag holds only 2.5 kg, which is less than 3.75 kg. Two bags hold 2 × 2.5 = 5 kg, which is enough.
One bag of 2.5 kg is too little. Two bags hold 5 kg. 5.(b) She must buy 2 bags, and 5 − 3.75 = 1.25 kg of flour is left over. Check: 3.75 + 1.25 = 5.
(b) She buys 2 bags, and 5 − 3.75 = 1.25 kg is left over.
Answer: (a) 3.75 kg; (b) 2 bags, with 1.25 kg left over
Common mistakes
- Writing 34 as 0.34 or 0.3. The digits of a fraction are not the digits of its decimal: 34 = 75100 = 0.75.
- Answering 1.5 bags because 3.75 ÷ 2.5 = 1.5. Flour is sold only in whole bags, and one bag is too little, so she must buy 2.