Mathematics · Classes 9-10 · Chapter 1

Similar Repeated Decimals, Addition and Subtraction.

Try adding 3.8̇9̇ and 2.17̇8̇ and the trouble starts at once: their digits cycle out of step. One starts repeating straight after the point, the other one place later. So before you add, you cut both to the same pattern, and at the end you make one small correction. This note is about those two jobs.

1. Similar and dissimilar

Repeated decimals are similar when they have the same number of non-repeating digits and the same number of repeating digits. Otherwise they are dissimilar. 12.4̇5̇ and 6.3̇2̇ are similar: neither has a decimal digit before the repeat starts, and both repeat two digits. So are 9.453̇ and 125.897̇: two non-repeating digits each, one repeating digit each. How big the whole-number part is does not count.

Dissimilar decimals can be made similar thanks to one plain fact: writing the repeating digits out again does not change the value. Write 6.453̇7̇ as 6.4537̇3̇ or as 6.453̇737̇ and every one of them is 6.45373737..., which is 63892/9900. Once the non-repeating part is over, the repeating block may be taken to start at any digit.

2. Two steps to make them similar

Before the rule, push it around yourself. Slide the dots to the right or make the repeating block longer. The way it is written changes; the fraction stays put.

Now the rule, in two steps.

Step 1. Find the number with the most non-repeating digits. Every number gets that many non-repeating digits; the shorter ones stretch theirs by writing repeating digits out once more.

Step 2. Count the repeating digits in each and take the LCM of the counts. Every repeating block is written out to that many digits. The longest for the non-repeating part, the LCM for the repeating part: two different rules, so keep them apart.

Example 8. Make 5.6̇, 7.34̇5̇ and 10.784̇23̇ similar. They have 0, 1 and 2 non-repeating digits, so each gets 2. They repeat 1, 2 and 3 digits, and the LCM is 6, so each block becomes 6 digits long.
5.6̇ = 5.666̇66666̇
7.34̇5̇ = 7.345̇45454̇
10.784̇23̇ = 10.784̇23423̇

Example 9. A finite decimal gets 0s added at its end to make a repeating part. Of 1.7643, 3.2̇4̇ and 2.783̇46̇, the most non-repeating digits belong to 1.7643: four. The blocks are 2 and 3 digits long, so the LCM is 6.
1.7643 = 1.76430̇00000̇
3.2̇4̇ = 3.24242̇42424̇
2.783̇46̇ = 2.78346̇34634̇

Now run your own numbers. Type the whole decimal in the box and say how many of its last digits repeat. For 7.34̇5̇, type 7.345 and pick the last 2 digits. The list also has the book's examples and two pairs from the exercises. Your numbers go nowhere.

3. Addition: the carry comes round

Once they are similar, add them column by column as if they were finite decimals. One job is left. Whatever you carry when you add the leftmost digits of the repeating parts, add it to the last digit of the sum. Only then do you have the actual sum.

Why the last digit? Because the repeating part does not stop at the line: past it, the same digits start again. Pick an example, open the digits past the line one at a time, and watch where the carry comes from.

Example 10. Find 3.8̇9̇ + 2.17̇8̇ + 5.897̇98̇. Walk it through one step at a time, digits past the end of the repeating block included.

Example 11. 8.94̇78̇ + 2.346 + 4.7̇1̇. The finite 2.346 has 3 non-repeating digits, the most of the three, and its repeating part is 0. The blocks are made 6 digits long.
8.9478̇47847̇ + 2.3460̇00000̇ + 4.7171̇71717̇ = 16.011019564
The carry from past the line is 1: 16.011019564 + 1 (in the last place) = 16.011019565
So the sum is 16.0110̇19565̇.

The sum is a repeated decimal too, and its shape is known before you start. It has as many non-repeating digits as the longest non-repeating part, and as many repeating digits as the LCM of the block lengths. The same holds for a difference. That is why the answer to Example 11 has 3 non-repeating and 6 repeating digits.

4. Subtraction: the borrowed 1

Subtraction also starts by making them similar, then subtracting as finite decimals. This time the correction runs the other way. If, from where the repeating dot starts, the minuend's digits are smaller than the subtrahend's, so that past the line you would have to borrow 1, take 1 away from the last digit of the difference.

Example 12. 8.24̇3̇ − 5.246̇73̇. Two non-repeating digits; the LCM of 2 and 3 is 6.
8.243̇43434̇ − 5.246̇73673̇ = 2.99669761
The repeating parts start with 3 and 6, and 3 take away 6 needs a borrow, so 1 comes off the last digit: 2.99669761 − 1 (in the last place) = 2.99669760
So the difference is 2.996̇69760̇.

Example 13. 24.456̇45̇ − 16.4̇37̇. Bring the second to 2 non-repeating digits: 16.437̇43̇. Both blocks are then 3 digits long.
24.456̇45̇ − 16.437̇43̇ = 8.01902
645 is smaller than 743, so 1 comes off the last digit: 8.01902 − 1 (in the last place) = 8.01901
So the difference is 8.019̇01̇.

In a sum the carry is added, in a difference 1 comes off, and sometimes there is nothing to do. Here are five. For each, say the correction to yourself first, then tap to check.

5. The fraction route

There is another way: turn every number into a common fraction, add or subtract the fractions, then write the answer back as a decimal. It works; it just takes longer.
0.1̇ + 0.01̇ = 1/9 + 1/90 = 11/90
0.2̇ + 0.3̇ = 2/9 + 3/9 = 5/9 = 0.5̇, a rational number
0.7̇ + 0.15̇ = 7/9 + 14/90 = 84/90 = 0.93̇

6. Check yourself

Three questions. The first is about the correction in a sum.

The second is the last step of a subtraction.

The last is a whole sum.

Turn the cards over to test yourself.

Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Similar Repeated Decimals, Addition and Subtraction” part.