In the previous section we learned how to add two fractions. We learned it by analysing the number of 'slices of bread' consumed by two participants Mr. A and Mr. B at a camp. Let us explore some more possibilities:
We saw that the campers can take as many number of slices as they wish. Let us leave A and B, and concentrate on a fat student. This fat student took
■ 9 slices out of 14 equal slices of a loaf.
After consuming it, he again went to the table and took
■ 8 slices out of 14 equal slices of another loaf.
This is shown in the fig.5.30(a) below.
What fraction of the 'whole loaf' did he consume?
• First he took 9 out of 14. So the fraction is 9⁄14
• Next he took 8 out of 14. So the fraction is 8⁄14
• So the total fraction is 9⁄14 + 8⁄14
We can easily add these fractions because the slices are of equal size. (This is also indicated by the same denominators)
• Adding we get 9⁄14 + 8⁄14 = 17⁄14
We have not seen such a fraction (with a numerator greater than the denominator) so far in our discussions. All the fractions that we saw so far were proper fractions. Let us analyse the reason for obtaining such a fraction in this case:
We obtained it by adding two proper fractions: 9⁄14 and 8⁄14
We can obtain the same result 17⁄14, by adding another two fractions:
14⁄14 + 3⁄14 = 17⁄14. So it is same as taking
• 14 slices out of 14 slices at the first time he went to the table, and
• 3 slices out of 14, the second time.
This is shown in fig.5.30(b). But taking 14 slices out of 14 slices is same as taking one 'whole loaf'. After that he took 3 slices out of 14 from another loaf.
That means what he consumed is not a 'fraction of a loaf'. But one 'whole loaf and a little (3 slices) more'. He took
• Different 'fractions' of different loafs at different times.
• But in effect, he consumed one 'whole loaf' plus 3 slices more
He consumed 'more than one whole loaf'. But he did not consume 'two whole loafs'. It is between one and two. In such cases we can write 1 plus a fraction. The fraction after 1 indicates that, 'one whole' is taken completely, and in addition, a fraction of the whole is taken. We write this as 13⁄14. It consists of a 'whole part' and a 'proper fraction'. Such fractions are called mixed fractions. We can also write it as an improper fraction as 17⁄14. We call a fraction an improper fraction when the numerator is greater than the denominator.
To get the mixed fraction 13⁄14 from 17⁄14, we wrote
17⁄14 as 14⁄14 + 3⁄14 = 1 + 3⁄14.
But there is also another way. It is the 'long division' method. The long division of 17 by 14 is shown below:
The quotient is 1. So there is one 'whole loaf'. The remainder is 3. It becomes the numerator of the fraction part. The divisor is 14. It becomes the denominator of the fraction part. This is shown below:
Thus we get 17⁄14 = 13⁄14.
In the above example, the fat student took so many slices that, there was one whole loaf included in what he consumed. In addition to one full loaf, there was also three slices more. In some cases, there may be two or more whole loafs included in a share. Let us see an example:
Among the campers, there were a group of 5 students. These 5 students are friends. They decide to eat together. So they go to the table together and take as many slices as they want. Let us suppose, they took 47 slices altogether. All the slices were taken from loafs which are divided into 14 equal parts. So the fraction they took is 47⁄14. But
47⁄14 = 14⁄14 + 14⁄14 + 14⁄14 + 5⁄14 = 1 + 1 + 1 + 5⁄14 = 3 + 5⁄14 = 35⁄14
So together, the group of 5 students consumed 3 whole loafs and 5⁄14 of a loaf. Thus we can write 47⁄14 = 35⁄14. This same result can be obtained using long division also:
In this way we can write any improper fraction as a mixed fraction. The reverse is also possible: If we get a mixed fraction, we can convert it into an improper fraction. For that we can do the following steps:
Step 1: Multiply the whole number part with the denominator
Step 2: To the above product, add the numerator
Step 3: Write the sum obtained in step 2 as the numerator of the improper fraction
Step 4: Write the denominator of the mixed fraction itself as the denominator of the improper fraction
Let us convert 35⁄14 back to the improper fraction using the above steps:
Step 1: Multiply the whole number part with the denominator: 3 × 14 = 42
Step 2: To the above product, add the numerator: 42 + 5 = 47
Step 3: Write the sum obtained in step 2 as the numerator of the improper fraction: 47⁄
Step 4: Write the denominator of the mixed fraction itself as the denominator of the improper fraction: 47⁄14
Thus we have learned the basics about mixed fractions. In our day to day life we encounter mixed fractions quite often. It indicates that, some 'whole portions' are taken, and in addition, some fractions are also taken. For example, if a person bought 31⁄4 cakes, it means, he bought 3 full cakes and 1⁄4 of a fourth cake.
We saw that the campers can take as many number of slices as they wish. Let us leave A and B, and concentrate on a fat student. This fat student took
■ 9 slices out of 14 equal slices of a loaf.
After consuming it, he again went to the table and took
■ 8 slices out of 14 equal slices of another loaf.
This is shown in the fig.5.30(a) below.
 |
| Fig.5.30 |
• First he took 9 out of 14. So the fraction is 9⁄14
• Next he took 8 out of 14. So the fraction is 8⁄14
• So the total fraction is 9⁄14 + 8⁄14
We can easily add these fractions because the slices are of equal size. (This is also indicated by the same denominators)
• Adding we get 9⁄14 + 8⁄14 = 17⁄14
We have not seen such a fraction (with a numerator greater than the denominator) so far in our discussions. All the fractions that we saw so far were proper fractions. Let us analyse the reason for obtaining such a fraction in this case:
We obtained it by adding two proper fractions: 9⁄14 and 8⁄14
We can obtain the same result 17⁄14, by adding another two fractions:
14⁄14 + 3⁄14 = 17⁄14. So it is same as taking
• 14 slices out of 14 slices at the first time he went to the table, and
• 3 slices out of 14, the second time.
This is shown in fig.5.30(b). But taking 14 slices out of 14 slices is same as taking one 'whole loaf'. After that he took 3 slices out of 14 from another loaf.
That means what he consumed is not a 'fraction of a loaf'. But one 'whole loaf and a little (3 slices) more'. He took
• Different 'fractions' of different loafs at different times.
• But in effect, he consumed one 'whole loaf' plus 3 slices more
He consumed 'more than one whole loaf'. But he did not consume 'two whole loafs'. It is between one and two. In such cases we can write 1 plus a fraction. The fraction after 1 indicates that, 'one whole' is taken completely, and in addition, a fraction of the whole is taken. We write this as 13⁄14. It consists of a 'whole part' and a 'proper fraction'. Such fractions are called mixed fractions. We can also write it as an improper fraction as 17⁄14. We call a fraction an improper fraction when the numerator is greater than the denominator.
To get the mixed fraction 13⁄14 from 17⁄14, we wrote
17⁄14 as 14⁄14 + 3⁄14 = 1 + 3⁄14.
But there is also another way. It is the 'long division' method. The long division of 17 by 14 is shown below:
The quotient is 1. So there is one 'whole loaf'. The remainder is 3. It becomes the numerator of the fraction part. The divisor is 14. It becomes the denominator of the fraction part. This is shown below:
In the above example, the fat student took so many slices that, there was one whole loaf included in what he consumed. In addition to one full loaf, there was also three slices more. In some cases, there may be two or more whole loafs included in a share. Let us see an example:
Among the campers, there were a group of 5 students. These 5 students are friends. They decide to eat together. So they go to the table together and take as many slices as they want. Let us suppose, they took 47 slices altogether. All the slices were taken from loafs which are divided into 14 equal parts. So the fraction they took is 47⁄14. But
47⁄14 = 14⁄14 + 14⁄14 + 14⁄14 + 5⁄14 = 1 + 1 + 1 + 5⁄14 = 3 + 5⁄14 = 35⁄14
So together, the group of 5 students consumed 3 whole loafs and 5⁄14 of a loaf. Thus we can write 47⁄14 = 35⁄14. This same result can be obtained using long division also:
In this way we can write any improper fraction as a mixed fraction. The reverse is also possible: If we get a mixed fraction, we can convert it into an improper fraction. For that we can do the following steps:
Step 1: Multiply the whole number part with the denominator
Step 2: To the above product, add the numerator
Step 3: Write the sum obtained in step 2 as the numerator of the improper fraction
Step 4: Write the denominator of the mixed fraction itself as the denominator of the improper fraction
Let us convert 35⁄14 back to the improper fraction using the above steps:
Step 1: Multiply the whole number part with the denominator: 3 × 14 = 42
Step 2: To the above product, add the numerator: 42 + 5 = 47
Step 3: Write the sum obtained in step 2 as the numerator of the improper fraction: 47⁄
Step 4: Write the denominator of the mixed fraction itself as the denominator of the improper fraction: 47⁄14
Thus we have learned the basics about mixed fractions. In our day to day life we encounter mixed fractions quite often. It indicates that, some 'whole portions' are taken, and in addition, some fractions are also taken. For example, if a person bought 31⁄4 cakes, it means, he bought 3 full cakes and 1⁄4 of a fourth cake.
We can now think about adding mixed fractions. We can follow the same steps that we use for adding proper fractions. We can use any one of the two methods:
(1) Add the 'whole parts' and 'fraction parts' separately
(2) Convert each mixed fraction into an improper fraction and then add.
We will now see some solved examples below. We will use both the methods and see if we get the same answers.
Solved example 5.26
Solve: (a) 121⁄5 + 82⁄5 (b) 36⁄11 + 54⁄11
Solution:
(a) First method: We will add the 'whole parts' and 'fraction parts' separately:
• 12 + 8 = 20 • 1⁄5 + 2⁄5 = 3⁄5
∴ 121⁄5 + 82⁄5 = 203⁄5
Second method: We will convert each fraction into an improper fraction, and then add
• 121⁄5 = 61⁄5 • 82⁄5 = 42⁄5
∴ 121⁄5 + 82⁄5 = 61⁄5 + 42⁄5 = 103⁄5 = 203⁄5
This is same as the previous answer
(b) First method: We will add the 'whole parts' and 'fraction parts' separately:
• 3 + 5 = 8 • 6⁄11 + 4⁄11 = 10⁄11
∴ 36⁄11 + 54⁄11 = 810⁄11
Second method: We will convert each fraction into an improper fraction, and then add
• 36⁄11 = 39⁄11 • 54⁄11 = 59⁄11
∴ 36⁄11 + 54⁄11 = 39⁄11 + 59⁄11 = 98⁄11 = 810⁄11
This is same as the previous answer
Solved example 5.27
Solve: (a) 124⁄5 + 83⁄5 (b) 38⁄11 + 510⁄11
Solution:
(a) First method: We will add the 'whole parts' and 'fraction parts' separately:
• 12 + 8 = 20 • 4⁄5 + 3⁄5 = 7⁄5
• But 7⁄5 is an improper fraction. We have to convert it into a proper fraction.
We have 7⁄5 = 12⁄5
∴ 124⁄5 + 83⁄5 = 20 + 12⁄5 = 212⁄5
Second method: We will convert each fraction into an improper fraction, and then add
• 124⁄5 = 64⁄5 • 83⁄5 = 43⁄5
∴ 124⁄5 + 83⁄5 = 64⁄5 + 43⁄5 = 107⁄5 = 212⁄5
This is same as the previous answer
(b) First method: We will add the 'whole parts' and 'fraction parts' separately:
• 3 + 5 = 8 • 8⁄11 + 10⁄11 = 18⁄11
• But 18⁄11 is an improper fraction. We have to convert it into a proper fraction.
We have 18⁄11 = 17⁄11
∴ 38⁄11 + 510⁄11 = 8 + 17⁄11 = 97⁄11
Second method: We will convert each fraction into an improper fraction, and then add
• 38⁄11 = 41⁄11 • 510⁄11 = 65⁄11
∴ 38⁄11 + 510⁄11 = 41⁄11 + 65⁄11 = 106⁄11 = 97⁄11
This is same as the previous answer
Solved example 5.28
Solve: (a) 122⁄7 + 83⁄5 (b) 38⁄11 + 510⁄17
Solution:
(a) Here we have unlike fractions. In such cases, it is convenient to use the second method. We will convert each fraction into an improper fraction, and then add
• 122⁄7 = 86⁄7 • 83⁄5 = 43⁄5
• 122⁄7 = 86⁄7 • 83⁄5 = 43⁄5
The remaining steps and the result are shown below:
(b) Here also we use the same steps:
• 38⁄11 = 41⁄11 • 510⁄17 = 95⁄17
The remaining steps and the result are shown below:
So we have learned about the mixed fractions. We have also seen how to add them. In the next section we will see how mixed fractions are used in the measurements of distances.
• 38⁄11 = 41⁄11 • 510⁄17 = 95⁄17
The remaining steps and the result are shown below:
So we have learned about the mixed fractions. We have also seen how to add them. In the next section we will see how mixed fractions are used in the measurements of distances.
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