In the previous section we wrote 5⁄6 in the decimal form. In this section, we will solve the rest of the problems.
(ii) 3⁄11 : 1. We know that 3⁄11 = (3×10) ⁄(11×10).
2. Let us rearrange the right side: 3⁄11 = 3⁄10 × 10⁄11
3. In the above result, we cannot write 10⁄11 as a mixed fraction. So in this problem, we cannot use ten. We will start with 100
First cycle:
1. We know that 3⁄11 = (3×100) ⁄(11×100).
2. Let us rearrange the right side: 3⁄11 = 3⁄100 × 100⁄11
3. In the above result, we can write 100⁄11 as (9 + 1⁄11 )
4. So (2) becomes 3⁄11 = 3⁄100 × (9 + 1⁄11). So we get:
5. 3⁄11 = 27⁄100 + 3⁄1100
Second cycle:
1. We know that 3⁄11 = (3×1000) ⁄(11×1000).
2. Let us rearrange the right side: 3⁄11 = 3⁄1000 × 1000⁄11
3. In the above result, we can write 1000⁄11 as (90 + 10⁄11)
4. So (2) becomes 3⁄11 = 3⁄1000 × (90 + 10⁄11). So we get:
5. 3⁄11 = 270⁄1000 + 30⁄11000
Third cycle:
1. We know that 3⁄11 = (3×10000) ⁄(11×10000).
2. Let us rearrange the right side: 3⁄11 = 3⁄10000 × 10000⁄11
3. In the above result, we can write 10000⁄11 as (909 + 1⁄11 )
4. So (2) becomes 3⁄11 = 3⁄10000 × (909 + 1⁄11). So we get:
5. 3⁄11 = 2727⁄10000 + 3⁄110000
Based on the above results we can write:
■ 3⁄11 = 27⁄100 + 3⁄1100 = 0.27 + 3⁄1100
■ 3⁄11 = 270⁄1000 + 30⁄11000 = 0.27 + 30⁄11000
■ 3⁄11 = 2727⁄10000 + 3⁄110000 = 0.2727 + 3⁄110000
The fractions 27⁄100, 270⁄1000, 2727⁄10000, etc., gets closer and closer to 3⁄11 . Based on this, we can write the decimal form of 3⁄11 as:
In method 2, dots are given for both 2 and 7 because, they repeat as a pattern. For the same reason, the line is placed over both 2 and 7 in method 3. The result obtained when we divide 3 by 11 using a calculator is shown below:
(iii) 23⁄11 : This is an improper fraction. We must first convert it into the sum of a natural number and a proper fraction. It is cone as follows:
23⁄11 = 22⁄11 + 1⁄11 = 2 + 1⁄11
Now we will work with 1⁄11 and add the final result to 2
1. We know that 1⁄11 = (1×10) ⁄(11×10).
2. Let us rearrange the right side: 1⁄11 = 1⁄10 × 10⁄11
3. In the above result, we cannot write 10⁄11 as a mixed fraction. So in this problem, we cannot use ten. We will start with 100
First cycle:
1. We know that 1⁄11 = (1×100) ⁄(11×100).
2. Let us rearrange the right side: 1⁄11 = 1⁄100 × 100⁄11
3. In the above result, we can write 100⁄11 as (9 + 1⁄11 )
4. So (2) becomes 1⁄11 = 1⁄100 × (9 + 1⁄11). So we get:
5. 1⁄11 = 9⁄100 + 1⁄1100
Second cycle:
1. We know that 1⁄11 = (1×1000) ⁄(11×1000).
2. Let us rearrange the right side: 1⁄11 = 1⁄1000 × 1000⁄11
3. In the above result, we can write 1000⁄11 as (90 + 10⁄11)
4. So (2) becomes 1⁄11 = 1⁄1000 × (90 + 10⁄11). So we get:
5. 1⁄11 = 90⁄1000 + 10⁄11000
Third cycle:
1. We know that 1⁄11 = (1×10000) ⁄(11×10000).
2. Let us rearrange the right side: 1⁄11 = 1⁄10000 × 10000⁄11
3. In the above result, we can write 10000⁄11 as (909 + 1⁄11 )
4. So (2) becomes 1⁄11 = 1⁄10000 × (909 + 1⁄11). So we get:
5. 1⁄11 = 909⁄10000 + 1⁄110000
Based on the above results we can write:
■ 1⁄11 = 9⁄100 + 1⁄1100 = 0.09 + 1⁄1100
■ 1⁄11 = 90⁄1000 + 10⁄11000 = 0.09 + 10⁄11000
■ 1⁄11 = 909⁄10000 + 1⁄110000 = 0.0909 + 1⁄110000
The fractions 9⁄100, 90⁄1000, 909⁄10000, etc., gets closer and closer to 1⁄11 . Based on this, we can write the decimal form of 1⁄11 as: 1⁄11 = 0.0909...
But the problem was to convert 21⁄11. So we can write 21⁄11 = 2.0909...
(iv) 1⁄13 : 1. We know that 1⁄13 = (1×10) ⁄(13×10).
2. Let us rearrange the right side: 1⁄13 = 1⁄10 × 10⁄13
3. In the above result, we cannot write 10⁄13 as a mixed fraction. So in this problem, we cannot use ten. We will start with 100
First cycle:
1. We know that 1⁄13 = (1×100) ⁄(13×100).
2. Let us rearrange the right side: 1⁄13 = 1⁄100 × 100⁄13
3. In the above result, we can write 100⁄13 as (7 + 9⁄13 )
4. So (2) becomes 1⁄13 = 1⁄100 × (7 + 9⁄13). So we get:
5. 1⁄13 = 7⁄100 + 9⁄1300
Second cycle:
1. We know that 1⁄13 = (1×1000) ⁄(13×1000).
2. Let us rearrange the right side: 1⁄13 = 1⁄1000 × 1000⁄13
3. In the above result, we can write 1000⁄13 as (76 + 12⁄13)
4. So (2) becomes 1⁄13 = 1⁄1000 × (76 + 12⁄11). So we get:
5. 1⁄13 = 76⁄1000 + 12⁄13000
Third cycle:
1. We know that 1⁄13 = (1×10000) ⁄(13×10000).
2. Let us rearrange the right side: 1⁄13 = 1⁄10000 × 10000⁄13
3. In the above result, we can write 10000⁄13 as (769 + 3⁄13 )
4. So (2) becomes 1⁄13 = 1⁄10000 × (769 + 3⁄13). So we get:
5. 1⁄13 = 769⁄10000 + 3⁄130000
Fourth cycle:
1. We know that 1⁄13 = (1×100000) ⁄(13×100000).
2. Let us rearrange the right side: 1⁄13 = 1⁄100000 × 100000⁄13
3. In the above result, we can write 100000⁄13 as (7692 + 4⁄13 )
4. So (2) becomes 1⁄13 = 1⁄100000 × (7692 + 4⁄13). So we get:
5. 1⁄13 = 7692⁄100000 + 4⁄1300000
Fifth cycle:
1. We know that 1⁄13 = (1×1000000) ⁄(13×1000000).
2. Let us rearrange the right side: 1⁄13 = 1⁄1000000 × 1000000⁄13
3. In the above result, we can write 1000000⁄13 as (76923 + 1⁄13 )
4. So (2) becomes 1⁄13 = 1⁄100000 × (76923 + 1⁄13). So we get:
5. 1⁄13 = 76923⁄1000000 + 1⁄13000000
Based on the above results we can write:
■ 1⁄13 = 7⁄100 + 9⁄1300 = 0.07 + 3⁄1100
■ 1⁄13 = 76⁄1000 + 12⁄13000 = 0.076 + 12⁄13000
■ 1⁄13 = 769⁄10000 + 3⁄130000 = 0.0769 + 3⁄130000
■ 1⁄13 = 7692⁄100000 + 4⁄1300000 = 0.07692 + 4⁄1300000
■ 1⁄13 = 76923⁄1000000 + 1⁄13000000 = 0.076923 + 1⁄13000000
The fractions 7⁄100, 76⁄1000, 769⁄10000, 7692⁄100000, 76923⁄1000000 etc., gets closer and closer to 1⁄13 . Based on this, we can write the decimal form of 1⁄13 as: 1⁄13 = 0.076923...
In the next section we will see some more solved examples.
(ii) 3⁄11 : 1. We know that 3⁄11 = (3×10) ⁄(11×10).
2. Let us rearrange the right side: 3⁄11 = 3⁄10 × 10⁄11
3. In the above result, we cannot write 10⁄11 as a mixed fraction. So in this problem, we cannot use ten. We will start with 100
First cycle:
1. We know that 3⁄11 = (3×100) ⁄(11×100).
2. Let us rearrange the right side: 3⁄11 = 3⁄100 × 100⁄11
3. In the above result, we can write 100⁄11 as (9 + 1⁄11 )
4. So (2) becomes 3⁄11 = 3⁄100 × (9 + 1⁄11). So we get:
5. 3⁄11 = 27⁄100 + 3⁄1100
Second cycle:
1. We know that 3⁄11 = (3×1000) ⁄(11×1000).
2. Let us rearrange the right side: 3⁄11 = 3⁄1000 × 1000⁄11
3. In the above result, we can write 1000⁄11 as (90 + 10⁄11)
4. So (2) becomes 3⁄11 = 3⁄1000 × (90 + 10⁄11). So we get:
5. 3⁄11 = 270⁄1000 + 30⁄11000
Third cycle:
1. We know that 3⁄11 = (3×10000) ⁄(11×10000).
2. Let us rearrange the right side: 3⁄11 = 3⁄10000 × 10000⁄11
3. In the above result, we can write 10000⁄11 as (909 + 1⁄11 )
4. So (2) becomes 3⁄11 = 3⁄10000 × (909 + 1⁄11). So we get:
5. 3⁄11 = 2727⁄10000 + 3⁄110000
Based on the above results we can write:
■ 3⁄11 = 27⁄100 + 3⁄1100 = 0.27 + 3⁄1100
■ 3⁄11 = 270⁄1000 + 30⁄11000 = 0.27 + 30⁄11000
■ 3⁄11 = 2727⁄10000 + 3⁄110000 = 0.2727 + 3⁄110000
The fractions 27⁄100, 270⁄1000, 2727⁄10000, etc., gets closer and closer to 3⁄11 . Based on this, we can write the decimal form of 3⁄11 as:
(iii) 23⁄11 : This is an improper fraction. We must first convert it into the sum of a natural number and a proper fraction. It is cone as follows:
23⁄11 = 22⁄11 + 1⁄11 = 2 + 1⁄11
Now we will work with 1⁄11 and add the final result to 2
1. We know that 1⁄11 = (1×10) ⁄(11×10).
2. Let us rearrange the right side: 1⁄11 = 1⁄10 × 10⁄11
3. In the above result, we cannot write 10⁄11 as a mixed fraction. So in this problem, we cannot use ten. We will start with 100
First cycle:
1. We know that 1⁄11 = (1×100) ⁄(11×100).
2. Let us rearrange the right side: 1⁄11 = 1⁄100 × 100⁄11
3. In the above result, we can write 100⁄11 as (9 + 1⁄11 )
4. So (2) becomes 1⁄11 = 1⁄100 × (9 + 1⁄11). So we get:
5. 1⁄11 = 9⁄100 + 1⁄1100
Second cycle:
1. We know that 1⁄11 = (1×1000) ⁄(11×1000).
2. Let us rearrange the right side: 1⁄11 = 1⁄1000 × 1000⁄11
3. In the above result, we can write 1000⁄11 as (90 + 10⁄11)
4. So (2) becomes 1⁄11 = 1⁄1000 × (90 + 10⁄11). So we get:
5. 1⁄11 = 90⁄1000 + 10⁄11000
Third cycle:
1. We know that 1⁄11 = (1×10000) ⁄(11×10000).
2. Let us rearrange the right side: 1⁄11 = 1⁄10000 × 10000⁄11
3. In the above result, we can write 10000⁄11 as (909 + 1⁄11 )
4. So (2) becomes 1⁄11 = 1⁄10000 × (909 + 1⁄11). So we get:
5. 1⁄11 = 909⁄10000 + 1⁄110000
Based on the above results we can write:
■ 1⁄11 = 9⁄100 + 1⁄1100 = 0.09 + 1⁄1100
■ 1⁄11 = 90⁄1000 + 10⁄11000 = 0.09 + 10⁄11000
■ 1⁄11 = 909⁄10000 + 1⁄110000 = 0.0909 + 1⁄110000
The fractions 9⁄100, 90⁄1000, 909⁄10000, etc., gets closer and closer to 1⁄11 . Based on this, we can write the decimal form of 1⁄11 as: 1⁄11 = 0.0909...
But the problem was to convert 21⁄11. So we can write 21⁄11 = 2.0909...
(iv) 1⁄13 : 1. We know that 1⁄13 = (1×10) ⁄(13×10).
2. Let us rearrange the right side: 1⁄13 = 1⁄10 × 10⁄13
3. In the above result, we cannot write 10⁄13 as a mixed fraction. So in this problem, we cannot use ten. We will start with 100
First cycle:
1. We know that 1⁄13 = (1×100) ⁄(13×100).
2. Let us rearrange the right side: 1⁄13 = 1⁄100 × 100⁄13
3. In the above result, we can write 100⁄13 as (7 + 9⁄13 )
4. So (2) becomes 1⁄13 = 1⁄100 × (7 + 9⁄13). So we get:
5. 1⁄13 = 7⁄100 + 9⁄1300
Second cycle:
1. We know that 1⁄13 = (1×1000) ⁄(13×1000).
2. Let us rearrange the right side: 1⁄13 = 1⁄1000 × 1000⁄13
3. In the above result, we can write 1000⁄13 as (76 + 12⁄13)
4. So (2) becomes 1⁄13 = 1⁄1000 × (76 + 12⁄11). So we get:
5. 1⁄13 = 76⁄1000 + 12⁄13000
Third cycle:
1. We know that 1⁄13 = (1×10000) ⁄(13×10000).
2. Let us rearrange the right side: 1⁄13 = 1⁄10000 × 10000⁄13
3. In the above result, we can write 10000⁄13 as (769 + 3⁄13 )
4. So (2) becomes 1⁄13 = 1⁄10000 × (769 + 3⁄13). So we get:
5. 1⁄13 = 769⁄10000 + 3⁄130000
Fourth cycle:
1. We know that 1⁄13 = (1×100000) ⁄(13×100000).
2. Let us rearrange the right side: 1⁄13 = 1⁄100000 × 100000⁄13
3. In the above result, we can write 100000⁄13 as (7692 + 4⁄13 )
4. So (2) becomes 1⁄13 = 1⁄100000 × (7692 + 4⁄13). So we get:
5. 1⁄13 = 7692⁄100000 + 4⁄1300000
Fifth cycle:
1. We know that 1⁄13 = (1×1000000) ⁄(13×1000000).
2. Let us rearrange the right side: 1⁄13 = 1⁄1000000 × 1000000⁄13
3. In the above result, we can write 1000000⁄13 as (76923 + 1⁄13 )
4. So (2) becomes 1⁄13 = 1⁄100000 × (76923 + 1⁄13). So we get:
5. 1⁄13 = 76923⁄1000000 + 1⁄13000000
Based on the above results we can write:
■ 1⁄13 = 7⁄100 + 9⁄1300 = 0.07 + 3⁄1100
■ 1⁄13 = 76⁄1000 + 12⁄13000 = 0.076 + 12⁄13000
■ 1⁄13 = 769⁄10000 + 3⁄130000 = 0.0769 + 3⁄130000
■ 1⁄13 = 7692⁄100000 + 4⁄1300000 = 0.07692 + 4⁄1300000
■ 1⁄13 = 76923⁄1000000 + 1⁄13000000 = 0.076923 + 1⁄13000000
The fractions 7⁄100, 76⁄1000, 769⁄10000, 7692⁄100000, 76923⁄1000000 etc., gets closer and closer to 1⁄13 . Based on this, we can write the decimal form of 1⁄13 as: 1⁄13 = 0.076923...
In the next section we will see some more solved examples.
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