Another Peek Beyond the Point: Multiplying and Dividing Decimals

The number 34.267 is really a sum of place values: 3 Tens, 4 Ones, 2 Tenths, 6 Hundredths and 7 Thousandths. Each place to the right of the decimal point is ten times smaller than the one before it — tenths, then hundredths, then thousandths. This means any decimal can be written as a fraction: 34.267 = 34 + 2/10 + 6/100 + 7/1000, which combines into 34267/1000.

This connection between decimals and fractions-with-denominator-10-or-100-or-1000 is exactly why decimal multiplication and division work the way they do — every rule in this chapter is really a fraction rule in disguise, just written without the fraction bar.

A useful shortcut follows directly from place value: to divide a number by 10, 100 or 1000, simply move the decimal point left by as many places as there are zeroes in the divisor. For example, 48.5 ÷ 10 = 4.85 (one place left), and 48.5 ÷ 100 = 0.485 (two places left).

Suppose a pen costs 9.5 rupees and you buy 5 of them. Writing 9.5 as the fraction 95/10, the total cost is 5 × 95/10 = 475/10 = 47.5 rupees. Notice what happened: we multiplied the whole numbers 5 and 95 to get 475, then divided by 10 because the original fraction had a 10 in its denominator — which is the same as placing the decimal point one place from the right, since 9.5 has exactly one decimal digit.

This gives the general rule: multiply the two numbers completely ignoring the decimal points, then count the total number of decimal digits across both numbers being multiplied, and place the decimal point that many digits from the right in the answer. For example, to find 12.5 × 7.5: first multiply 125 × 75 = 9375. Since 12.5 has one decimal digit and 7.5 has one decimal digit, the answer needs 1 + 1 = 2 decimal digits total, giving 93.75.

A slightly longer example: 5.8 × 1.24. Multiplying 58 × 124 gives 7192. Since 5.8 has one decimal digit and 1.24 has two, the product needs 1 + 2 = 3 decimal digits, giving 7.192.

With counting numbers, multiplying always gives something bigger (or equal): 5 × 3 = 15 is bigger than both 5 and 3. Decimals break this pattern completely, because multiplying by a number smaller than 1 actually shrinks a quantity rather than growing it.

There are three situations worth noticing. When both numbers are greater than 1 — like 2.5 × 4 = 10 — the product is bigger than both numbers, just like with whole numbers. When both numbers are between 0 and 1 — like 0.25 × 0.4 = 0.1 — the product is smaller than both numbers, since you're really taking “a fraction of a fraction.” When one number is between 0 and 1 and the other is greater than 1 — like 0.25 × 8 = 2 — the product lands in between: smaller than the bigger number (8) but bigger than the smaller number (0.25).

This is worth checking before you even calculate: if you see 0.6 × 0.7, you can predict straight away that the answer will be less than 1 (both factors are below 1), without working it out. It comes out to exactly 0.42, confirming the prediction.

Dividing a decimal by a whole number uses the same long division you already know, just continued past the Ones place. To divide 9.5 by 4: 9 Ones divided by 4 gives 2 Ones with 1 One remaining. That leftover 1 One is regrouped as 10 Tenths; adding the 5 Tenths already there gives 15 Tenths, and 15 ÷ 4 gives 3 Tenths with 3 Tenths remaining. Regroup those as 30 Hundredths, and 30 ÷ 4 gives 7 Hundredths with 2 Hundredths remaining. Regroup as 20 Thousandths, and 20 ÷ 4 gives exactly 5 Thousandths with nothing left over. Reading off the digits: 9.5 ÷ 4 = 2.375.

When the divisor itself is a decimal, convert it into a whole number first by multiplying both the divisor and the dividend by the same power of 10 — this doesn't change the value of the division, since you're really multiplying the whole fraction by 10/10 or 100/100. For example, to find 4.68 ÷ 1.3, multiply both numbers by 10 to get 46.8 ÷ 13, which divides out to 3.6. To find 4.68 ÷ 0.13, multiply both numbers by 100 to get 468 ÷ 13, which divides out to exactly 36.

Notice how the same dividend (4.68) gave two completely different quotients depending on where the divisor's decimal point sat — 3.6 when dividing by 1.3, but 36 when dividing by the ten-times-smaller 0.13. Shrinking the divisor by a factor of 10 grew the quotient by the same factor of 10.

Ajay's school is 0.827 km from his home, and he walks there and back, 6 days a week. In one day, he walks 0.827 × 2 = 1.654 km. Over 6 days, the week's total is 1.654 × 6 = 9.924 km. Both steps are plain decimal multiplication, just applied to a distance context.

Average speed problems hide a division: if a car covers 126 km in 2.5 hours, its average speed is distance ÷ time = 126 ÷ 2.5 = 50.4 km per hour. Converting the decimal divisor 2.5 into the whole number 25 (by multiplying dividend and divisor by 10) turns this into the familiar 1260 ÷ 25 = 50.4.

Shopping bills often combine both operations. Meenu buys 4 notebooks at 15.50 rupees each and 3 erasers at 2.75 rupees each. The notebooks cost 4 × 15.50 = 62 rupees, and the erasers cost 3 × 2.75 = 8.25 rupees, so the total bill is 62 + 8.25 = 70.25 rupees. A wholesaler buying notebooks at 23.6 rupees and selling them at 30 rupees each makes a profit of (30 − 23.6) = 6.4 rupees per notebook; selling 50 notebooks in a week gives a total profit of 50 × 6.4 = 320 rupees.