Mathematics · Classes 9-10 · Chapter 1

Converting Repeated Decimal Fractions into Common Fractions.

The 3s in 0.333... never stop, yet its value is exactly 1/3. The trick that ties an endless tail into a fraction is a single subtraction, and it has a shortcut rule too. Once the rule is in your hand, the next two topics get easy.

1. Multiply, then subtract

Start with the smallest case (Example 4). Multiply 0.3̇ by 10 and you get 3.333...; multiply it by 1 and you get 0.333.... The two tails are identical, so subtracting wipes them out.
0.3̇ × 10 = 3.333...
0.3̇ × 1 = 0.333...
Subtract: 0.3̇ × 9 = 3, so 0.3̇ = 3/9 = 1/3.

When two digits take turns, lining up the tails needs 100, not 10.
0.2̇4̇ × 100 = 24.2424...
0.2̇4̇ × 1 = 0.2424...
Subtract: 0.2̇4̇ × 99 = 24, so 0.2̇4̇ = 24/99 = 8/33.

Now one with a non-repeating part: 42.347̇8̇ = 42.34787878.... The 34 comes once; the 78 goes round and round.
42.347̇8̇ × 10000 = 423478.7878...
42.347̇8̇ × 100 = 4234.7878...
Subtract: 42.347̇8̇ × 9900 = 419244
so 42.347̇8̇ = 419244/9900 = 34937/825 = 42 287/825.

Not just any two multipliers will do. Choose them yourself and see which pair makes the tails match exactly and cancel when you subtract.

Why 10000 and 100? There are four digits written after the point (3478), so we multiply first by 1 followed by four 0s. There are two non-repeating digits (34), so we multiply again by 1 followed by two 0s. Subtracting the second product from the first cancels the two tails and leaves a whole number on the right. The steps below are shuffled, so put them in order.

2. The rule in one go

You do not need two multiplications every time; there is a rule. Numerator = the integer you get by deleting the decimal point and the repeating dots, minus the integer formed by the non-repeating portion, meaning the whole part together with the decimal digits before the loop.

Denominator = as many 9s as there are digits in the repeating part, followed by as many 0s as there are digits in the non-repeating part after the decimal point.

Before simplifying anything, practise just spotting the denominator: eight decimals, eight denominators. Count the repeating digits first, then the decimal digits before the loop.

Example 5. In 5.234̇57̇ the repeating part is 457 (three digits) and the non-repeating part is 5.23.
5.234̇57̇ = (523457 − 523)/99900 = 522934/99900 = 261467/49950 = 5 11717/49950.

Example 6. In 45.23̇46̇ the repeating part is 346 and the non-repeating part is 45.2.
45.23̇46̇ = (452346 − 452)/9990 = 451894/9990 = 225947/4995 = 45 1172/4995.

Example 7. Every decimal digit of 32.5̇67̇ repeats, so the denominator has no zeros.
32.5̇67̇ = (32567 − 32)/999 = 32535/999 = 3615/111 = 1205/37 = 32 21/37.

Now turn it over yourself. Enter the whole part, the non-repeating digits and the repeating digits, and the machine shows the numerator, the denominator and the fraction in lowest terms. You pick the digit counts separately so that 01 and 1 stay different. Nothing you type goes anywhere. The notebook is yours.

3. Every repeated decimal is rational

The rule never gets stuck: a repeated decimal fraction can always be converted into a common fraction. So every repeated decimal fraction is a rational number.

A pure repeated decimal has no non-repeating decimal digits, so the denominator is all 9s. 0.4̇ = 4/9, 0.2̇ = 2/9, 0.3̇5̇ = 35/99, and 0.9̇ = 9/9 = 1. Yes, 0.999... is exactly 1.

For a mixed repeated decimal with one repeating digit, the denominator is 90 (one 9, one 0).
0.13̇ = (13 − 1)/90 = 2/15
0.24̇ = (24 − 2)/90 = 11/45
0.69̇ = (69 − 6)/90 = 63/90 = 7/10
3.78̇ = (378 − 37)/90 = 3 71/90

The same rule holds for several repeating digits. A zero before the loop is also a non-repeating digit and earns a 0 in the denominator.
0.01̇2̇ = (12 − 0)/990 = 2/165
6.23̇09̇ = (62309 − 62)/9990 = 6 769/3330

4. Check yourself

In the first question, watch both the numerator and the denominator.

The second is about where the rule applies.

In the last one, count the denominator carefully.

The whole topic on six cards.

Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Converting Repeated Decimal Fractions into Common Fractions” part.