The decimal point rule

Look back at the examples and count digits after the decimal point. In 9.5 × 5 = 47.5, the multiplier has 1, the multiplicand 0, and the product 1. In 12.5 × 7.5 = 93.75, it is 1 and 1, and the product has 2.

In 1.64 × 6 = 9.84, it is 2 and 0, and the product has 2. In 5.7 × 13.35 = 76.095, it is 1 and 2, and the product has 3. The pattern is plain: The number of digits after the point in the product is the sum of the numbers in the two factors.

Why must it be so? Multiplying decimals is multiplying their fractions. The NUMERATORS are just the numbers with the decimal points removed, so their product is the product of those whole numbers.

And the DENOMINATORS are each 1 followed by zeroes, so their product is 1 followed by the SUM of the zeroes. Dividing by that moves the point left by that many places — which is exactly the rule.

So here is a shortcut worth having. To multiply two decimals, multiply the two numbers you get by removing the points, then place the point so that the answer has as many decimal places as the two factors had between them.

Suppose you are told 596 × 248 = 147808. What is 5.96 × 24.8? The digits are 147808. The factors have 2 and 1 decimal places, so the product has 3: the answer is 147.808 — with no multiplication done at all.

One more: find 5.8 × 1.24. First 58 × 124 = 7192. The factors have 1 and 2 decimal places, so the product has 3: 7.192. Verify by converting to fractions and you will get the same.