3.7 + 2.6, adding
3.7
+ 2.6
6.3
Before this page: be able to multiply whole numbers with long multiplication, know what each digit is worth in integers and decimals, and know what happens to a number when it is multiplied or divided by ten.
The same digits, a different value
37 × 26 = 962, and 3.7 × 2.6 = 9.62 — the same three digits, with a point dropped into them.
The whole-number product supplies the digits. Place value supplies their size: 962, 96.2, 9.62 and 0.962 use the same digits at four different scales.
Multiplying by 10 moves the digits one place left past the point: 3.7 × 10 = 37 and 2.6 × 10 = 26.
3.7 × 2.6
3.7 × 2.6
The digits are those of 37 × 26, an ordinary multiplication; the points are the only difference.
The two ×10s on the factors and the ÷100 on the product cancel.
Every decimal multiplication goes the same way, both numbers made whole and the answer put back; only the power of ten changes.
Placing the decimal point
Each decimal place across the two factors contributes one factor of 10 when the numbers are made whole.
3.71 place
×
2.61 place
=
9.622 places
Count the places, multiply the digits, place the point
- 1
Count the decimal places in the question.
- 2
Ignore the points and multiply the digits.
- 3
Place the point to give that many places.
Setting out the multiplication
In an addition, tenths are added to tenths, so the points have to line up in one column. In a multiplication every digit on top meets every digit underneath, so the points are not lined up: 37 and 26 are written as any two whole numbers are, ones under ones.
3.7 × 2.6, multiplying
37
× 26
962
No points at all: 37 × 26 = 962, then two places counted back give 9.62.
Products smaller than one
0.3 × 0.4 = 0.12, and 0.12 is smaller than both factors. Multiplying by 0.4 takes four tenths of 0.3, so the product is part of the original amount.
0.3 across, 0.4 down
0.3 × 0.4
One whole square
The square is 1 along the top and 1 down the side, so its area is 1 × 1 = 1.
For a positive first number, a positive multiplier above 1 makes the product larger; a multiplier between 0 and 1 makes it smaller.
Removing trailing zeros
0.20 and 0.2 are the same number, so 0.5 × 0.4 is written 0.2 — but only after both places have been counted back:
0.5 × 0.4
0.5 × 0.4
Both numbers are below 1, so the answer comes out smaller than either; the digits still multiply as whole numbers, 5 × 4.
Two zeros turn up in that calculation doing opposite jobs: the zero before the point holds the ones column, so standard notation writes 0.2 rather than .2, and the zero on the end holds nothing.
Costs and areas
A price multiplied by a count gives a cost; two lengths multiplied together give an area. The calculation fixes the number, while the unit fixes what that number means.
Money
Six tins at £4.35 each. That is 435 × 6 = 2,610 with two places to come back, so £26.10.
The last zero stays: prices are written to the penny, so £26.10, not £26.1.
Area
A rug 2.4 m by 1.5 m. That is 24 × 15 = 360 with two places to come back, so 3.6 m².
Both measurements must be in the same unit: both are metres, so the area is in m².
The decimal-place calculation is the same in both examples. Write the cost as £26.10 to show the pence, and the area as 3.6 m².
Explore decimal multiplication
72 × 34 = 2,448 gives both 7.2 × 3.4 = 24.48 and 0.72 × 0.34 = 0.2448: one multiplication, then two places counted back and then four.
6.3 × 0.45
The digits are those of 63 × 45, an ordinary multiplication; the points are the only difference.
Key points
- Multiply as whole numbers, then give the product as many decimal places as the two factors have altogether.
- Place the decimal point before removing any trailing zeros.
- Multiplying by a positive number below one gives that fraction of the starting amount.