Mathematics · Classes 9-10 · Chapter 1

Multiplication and Division of Repeated Decimals.

0.3̇ and 0.6̇ both run on forever. Yet 0.3̇ ÷ 0.6̇ = 0.5: one digit after the point, and it stops. How does the quotient of two endless decimals stop? Because we never do the sum in decimals. We do it in fractions.

1. Fractions first, decimals last

There is one rule for multiplying or dividing repeated decimals. Turn each number into a common fraction first, multiply or divide with the fractions, then write the result back as a decimal. That decimal is your product or quotient. The same rule holds when a finite decimal meets a repeated one. Multiplying digit by digit is not safe with endless digits, and the fraction route is.

2. Multiplication, worked

Example 14. 4.3̇ = (43 − 4)/9 = 13/3 and 5.7̇ = (57 − 5)/9 = 52/9.
4.3̇ × 5.7̇ = 13/3 × 52/9 = 676/27 = 25.0̇37̇.

Example 15. 0.28̇ = (28 − 2)/90 = 26/90 = 13/45 and 42.1̇8̇ = (4218 − 42)/99 = 4176/99 = 464/11.
0.28̇ × 42.1̇8̇ = 13/45 × 464/11 = 6032/495 = 12.18̇5̇.

Example 16. Three numbers: 2.5 = 5/2, 4.35̇ = 392/90, 1.23̇4̇ = 611/495.
2.5 × 4.35̇ × 1.23̇4̇ = 119756/8910 = 13.440628... (approx.).

3. Division, worked

Example 17. 7.3̇2̇ = (732 − 7)/99 = 725/99 and 0.27̇ = (27 − 2)/90 = 25/90 = 5/18. Dividing means multiplying by the flipped divisor.
7.3̇2̇ ÷ 0.27̇ = 725/99 × 18/5 = 290/11 = 26.3̇6̇.

Example 18. 2.2̇718̇ = (22718 − 2)/9999 = 22716/9999 and 1.91̇2̇ = (1912 − 19)/990 = 1893/990.
Quotient = 22716/9999 × 990/1893 = 120/101 = 1.1̇881̇.

Example 19. 9.45 = 945/100 and 2.86̇3̇ = (2863 − 28)/990 = 2835/990.
9.45 ÷ 2.86̇3̇ = 945/100 × 990/2835 = 33/10 = 3.3.

Now run your own numbers. Type the whole decimal in the box and say how many of its last digits repeat. For 0.28̇, for instance, type 0.28 and pick the last 1 digit. The examples above can also be picked from the list.

4. The answer may not repeat

Whether an answer stops or repeats is decided by the denominator in lowest terms. Break it into primes: with only 2s and 5s, the decimal stops.

The product or quotient of repeated decimals may not be a repeated decimal. The answer 3.3 in Example 19 stops, and some products come out as a plain 2. Guess the answer to each of five sums first, then tap its chip to check.

Here are the sums in full.
0.3̇ × 0.6̇ = 1/3 × 2/3 = 2/9 = 0.2̇.
0.3̇ × 0.3̇ = 1/9 = 0.1̇.
2.4̇ × 0.8̇1̇ = 22/9 × 81/99 = 2.

0.2̇ × 0.16̇ = 2/9 × 15/90 = 2/9 × 1/6 = 1/27 = 0.0̇37̇.

Three divisions:
0.3̇ ÷ 0.6̇ = 1/3 ÷ 2/3 = 1/2 = 0.5.
0.35̇ ÷ 1.7̇ = 32/90 ÷ 16/9 = 0.2.
1.1̇85̇ ÷ 0.2̇4̇ = 4.8̇.

5. Check yourself

Three questions. Start with the first.

Do the second one in fractions too.

The last one asks whether the answer repeats.

The whole note on seven cards.

Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Multiplication and Division of Repeated Decimals” part.