Adding, subtracting and multiplying decimals

Line up the points to add 12.5 and 3.75; count the places to multiply 2.4 by 0.35 and get 0.84.

A 12.50 sandwich and a 3.75 drink

You buy a sandwich for 12.50 dollars and a drink for 3.75 dollars. Adding 12.5 and 3.75 goes wrong if you line the numbers up on the right, because then you would be adding tenths to hundredths. The OpenStax steps are short: write the numbers in a column so the decimal points sit one above the other, fill empty places with zeros, add as if they were whole numbers, and put the point in the answer straight below the others. 12.50 + 3.75 = 16.25.

Subtraction works the same way, and the placeholder zeros matter even more. To find the change from 20 dollars on a bill of 7.36 dollars, write 20 as 20.00. Then 20.00 − 7.36 = 12.64. Without the zeros it is easy to subtract 6 from nothing and lose track of the borrowing.

A quick estimate catches most slips. 12.5 + 3.75 should be a little over 16, and 20 − 7.36 should be a little under 13. Wrote 4.0? Or 160? Then a point is in the wrong place.

2.4 × 0.35 starts as 24 × 35

For multiplication you do not line up the points at all. Multiply the digits as whole numbers, then place the point so that the answer has as many decimal places as the two factors together. For 2.4 × 0.35, multiply 24 × 35 = 840. The factors have 1 and 2 decimal places, 3 in total, so the product is 0.840, which is 0.84. A butcher's price works the same way. Meat at 24 a kilogram, 0.35 kilograms of it: 8.40.

Why count places? 2.4 is 24/10 and 0.35 is 35/100. Multiplying fractions multiplies the denominators too: 10 × 100 = 1,000. So the product is 840/1,000, and dividing by 1,000 means three decimal places. Counting places is a shortcut for multiplying the denominators.

Multiplying by a number between 0 and 1 makes the answer smaller. 0.35 is about a third, so 2.4 × 0.35 should be around 0.8. That check helps when the product has to be padded with zeros: 0.3 × 0.02 starts as 3 × 2 = 6 and needs 1 + 2 = 3 places, so it is 0.006.

Sliding the point: 4.07 × 1,000 and 36.5 ÷ 100

Multiplying by 10 moves the decimal point one place to the right; by 100, two places; by 1,000, three places. Write zeros at the end if you run out of digits: 4.07 × 1,000 = 4,070. Dividing by the same numbers moves the point left: 36.5 ÷ 100 = 0.365. The number of zeros tells you how many places to move.

Turning 7.2 ÷ 0.3 into 72 ÷ 3

To divide by a decimal, make the divisor a whole number first. Move its point to the right, and move the point in the number being divided the same number of places. 7.2 ÷ 0.3 becomes 72 ÷ 3 = 24. Both numbers were multiplied by 10, and multiplying both parts of a division by the same number does not change the answer, for the same reason that 2/4 and 20/40 are equal.