In the previous section we completed the topic of 'comparison of fractions'. In this section we will discuss about addition of fractions. Consider the following situation:
At a camp, bread loafs such as shown in the fig.5.27 below, are served. All loafs are of the same size. Each loaf is sliced into a number of pieces and kept on tables. Campers can walk around the tables and take as many slices as they wish.
One particular loaf was divided into 14 slices as shown in fig.5.28(a) below. Mr.A takes 3 slices (marked in green colour) from that loaf as shown in the fig.(b). So the portion taken by him is 3/14 of the ‘whole loaf’. Mr.B takes 4 slices (marked in yellow colour) from the same loaf. So the portion taken by him is 4/14 of the ‘whole loaf’. The total number of slices taken by A and B is 7.
So the portion taken by A and B together is 7/14 of the ‘whole loaf’. Mathematically we can write this as: 3/14 + 4/14 = 7/14. This is simple addition of numerators. We were able to simply add the numerators because, they are like fractions (denominators are the same). We will see some solved examples on this type of addition:
Solved example 5.22
Calculate the following:
(a) 2⁄7 + 4⁄7 (b) 8⁄13 + 4⁄13 (c) 14⁄25 + 7⁄25
Solution:
(a) 2⁄7 + 4⁄7 = 6⁄7 (b) 8⁄13 + 4⁄13 = 12⁄13 (c) 14⁄25 + 7⁄25 = 21⁄25
Solved example 5.23
Fill up the boxes in each of the following
Solution:
The given fractions are all like fractions. We can simply add the numerators. Thus we have:
(a) 2 + x = 7 ∴ x = 7 -2 = 5 (b) x + 8 = 14 ∴ x = 14 -8 = 6
Now let us see another type of addition:
We saw that Mr.A took 3 slices out of 14 from a loaf. After consuming the 3 slices, he went to the table again and took 2 slices from another loaf. This is shown in fig.5.29(b) below:
But this loaf was divided into 10 equal parts. (We must remember that, though different loafs are divided into different number of slices, all the loafs are of the same size). So the portion taken out by A this time is 2/10 of the ‘whole loaf’. What is the total portion consumed by A? For this we have to add the two portions. That is., 3/14 + 2/10. Here we cannot just add the numerators because the slices are of different sizes as shown in fig.(c).
We have to use another method. We have to find the equivalent fractions of both 3/14 and 2/10, which has a same denominator:
So our task is to add 3/14 and 2/10
■ Step 1: Find the suitable equivalent fraction for each given fractions
• Common multiple of 14 and 10.
There will be several common multiples for 14 and 10. Any one of them will serve our purpose. The easiest way to find one is to multiply them. Because the product of two numbers will obviously be a ‘common multiple’ of both.
So we have: common multiple of 14 and 10 = 14 × 10 = 140. This will be the denominator of both our ‘suitable equivalent fractions’
• Suitable equivalent fraction for 3/14:
♦ The denominator of the given fraction is 14.
♦ The denominator of the new equivalent fraction should be 140.
♦ We have seen above that, this 140 is obtained by multiplying 14 with 10
♦ So we have to multiply the numerator also by 10. The calculation steps and the result are given below:
♦ So the required equivalent fraction is 30⁄140
• Suitable equivalent fraction for 2/10:
♦ The denominator of the given fraction is 10.
♦ The denominator of the new equivalent fraction should be 140.
♦ We have seen above that, this 140 is obtained by multiplying 10 with 14
♦ So we have to multiply the numerator also by 14. The calculation steps and the result are given below:
■ Step 2: Write down the equivalent fractions
3⁄14 = 30⁄140 and 2⁄10 = 28⁄140
■ Step 3: Analyse the above result
• Taking 3 slices from a total of 14 equal slices is same as taking 30 slices from a total of 140 equal slices.
• Taking 2 slices from a total of 10 equal slices is same as taking 28 slices from a total of 140 equal slices.
So the slices have become equal in size. We can add them directly. Mr.A took a total of 58 slices out of 140 equal slices. Mathematically this can be written as:
30⁄140 + 28⁄140 = 58⁄140
58/140 can be reduced to the simplest form as 29/70. So Mr.A consumed 29/70 of one ‘whole loaf’.
Thus we have learned how to add unlike fractions. Let us see some solved examples in this category:
Solved example 5.24
Calculate the following:
(a) 2/5 + 1/3 (b) 3/7 + 5/9
Solution:
(a) Our task is to add 2/5 and 1/3
■ Step 1: Find the suitable equivalent fraction for each given fractions
• Common multiple of 5 and 3
There will be several common multiples for 5 and 3. Any one of them will serve our purpose. The easiest way to find one is to multiply them. Because the product of two numbers will obviously be a ‘common multiple’ of both.
So we have: common multiple of 5 and 3 = 5 × 3 = 15. This will be the denominator of both our ‘suitable equivalent fractions’
• Suitable equivalent fraction for 2/5:
♦ The denominator of the given fraction is 5.
♦ The denominator of the new equivalent fraction should be 15.
♦ We have seen above that, this 15 is obtained by multiplying 5 with 3
♦ So we have to multiply the numerator also by 3. The calculation steps and the result are given below:
♦ So the required equivalent fraction is 6⁄15
♦ This is same as multiplying both the numerator and the denominator by the other denominator
• Suitable equivalent fraction for 1/3:
♦ The denominator of the given fraction is 3.
♦ The denominator of the new equivalent fraction should be 15.
♦ We have seen above that, this 15 is obtained by multiplying 3 with 5
♦ So we have to multiply the numerator also by 5. The calculation steps and the result are given below:
♦ So the required equivalent fraction is 5⁄15
♦ This is same as multiplying both the numerator and the denominator by the other denominator
■ Step 2: Write down the equivalent fractions
2⁄5 = 6⁄15 and 1⁄3 = 5⁄15
■ Step 3: Analyse the above result
• Taking 2 parts from a total of 5 equal parts is same as taking 6 parts from a total of 15 equal parts
• Taking 1 part from a total of 3 equal parts is same as taking 5 parts from a total of 15 equal parts
So the parts have become equal in size. We can add them directly.
6⁄15 + 5⁄15 = 11⁄15
(b) Our task is to add 3/7 and 5/9. This can be done using the same procedure as above. So we will not write detailed steps. The required steps are shown below:
Solved example 5.25
Solve: (a) 3/10 + 7/15 (b) 4/21 + 3/19 (c) 8/30 + 26/45
Solution:
In the next section we will discuss about Mixed fractions.
At a camp, bread loafs such as shown in the fig.5.27 below, are served. All loafs are of the same size. Each loaf is sliced into a number of pieces and kept on tables. Campers can walk around the tables and take as many slices as they wish.
 |
| Fig.5.27 |
One particular loaf was divided into 14 slices as shown in fig.5.28(a) below. Mr.A takes 3 slices (marked in green colour) from that loaf as shown in the fig.(b). So the portion taken by him is 3/14 of the ‘whole loaf’. Mr.B takes 4 slices (marked in yellow colour) from the same loaf. So the portion taken by him is 4/14 of the ‘whole loaf’. The total number of slices taken by A and B is 7.
 |
| Fig.5.28 |
Solved example 5.22
Calculate the following:
(a) 2⁄7 + 4⁄7 (b) 8⁄13 + 4⁄13 (c) 14⁄25 + 7⁄25
Solution:
(a) 2⁄7 + 4⁄7 = 6⁄7 (b) 8⁄13 + 4⁄13 = 12⁄13 (c) 14⁄25 + 7⁄25 = 21⁄25
Solved example 5.23
Fill up the boxes in each of the following
The given fractions are all like fractions. We can simply add the numerators. Thus we have:
(a) 2 + x = 7 ∴ x = 7 -2 = 5 (b) x + 8 = 14 ∴ x = 14 -8 = 6
Now let us see another type of addition:
We saw that Mr.A took 3 slices out of 14 from a loaf. After consuming the 3 slices, he went to the table again and took 2 slices from another loaf. This is shown in fig.5.29(b) below:
 |
| Fig.5.29 |
We have to use another method. We have to find the equivalent fractions of both 3/14 and 2/10, which has a same denominator:
So our task is to add 3/14 and 2/10
■ Step 1: Find the suitable equivalent fraction for each given fractions
• Common multiple of 14 and 10.
There will be several common multiples for 14 and 10. Any one of them will serve our purpose. The easiest way to find one is to multiply them. Because the product of two numbers will obviously be a ‘common multiple’ of both.
So we have: common multiple of 14 and 10 = 14 × 10 = 140. This will be the denominator of both our ‘suitable equivalent fractions’
• Suitable equivalent fraction for 3/14:
♦ The denominator of the given fraction is 14.
♦ The denominator of the new equivalent fraction should be 140.
♦ We have seen above that, this 140 is obtained by multiplying 14 with 10
♦ So we have to multiply the numerator also by 10. The calculation steps and the result are given below:
3⁄14 = (3 × 10)⁄(14 × 10) = 30⁄140
♦ So the required equivalent fraction is 30⁄140
• Suitable equivalent fraction for 2/10:
♦ The denominator of the given fraction is 10.
♦ The denominator of the new equivalent fraction should be 140.
♦ We have seen above that, this 140 is obtained by multiplying 10 with 14
♦ So we have to multiply the numerator also by 14. The calculation steps and the result are given below:
2⁄10 = (2 × 14) ⁄ (10 × 14) = 28⁄140
♦ So the required equivalent fraction is 28⁄140■ Step 2: Write down the equivalent fractions
3⁄14 = 30⁄140 and 2⁄10 = 28⁄140
■ Step 3: Analyse the above result
• Taking 3 slices from a total of 14 equal slices is same as taking 30 slices from a total of 140 equal slices.
• Taking 2 slices from a total of 10 equal slices is same as taking 28 slices from a total of 140 equal slices.
So the slices have become equal in size. We can add them directly. Mr.A took a total of 58 slices out of 140 equal slices. Mathematically this can be written as:
30⁄140 + 28⁄140 = 58⁄140
58/140 can be reduced to the simplest form as 29/70. So Mr.A consumed 29/70 of one ‘whole loaf’.
Thus we have learned how to add unlike fractions. Let us see some solved examples in this category:
Solved example 5.24
Calculate the following:
(a) 2/5 + 1/3 (b) 3/7 + 5/9
Solution:
(a) Our task is to add 2/5 and 1/3
■ Step 1: Find the suitable equivalent fraction for each given fractions
• Common multiple of 5 and 3
There will be several common multiples for 5 and 3. Any one of them will serve our purpose. The easiest way to find one is to multiply them. Because the product of two numbers will obviously be a ‘common multiple’ of both.
So we have: common multiple of 5 and 3 = 5 × 3 = 15. This will be the denominator of both our ‘suitable equivalent fractions’
• Suitable equivalent fraction for 2/5:
♦ The denominator of the given fraction is 5.
♦ The denominator of the new equivalent fraction should be 15.
♦ We have seen above that, this 15 is obtained by multiplying 5 with 3
♦ So we have to multiply the numerator also by 3. The calculation steps and the result are given below:
2⁄5 = (2 × 3)⁄(5 × 3) = 6⁄15
♦ So the required equivalent fraction is 6⁄15
♦ This is same as multiplying both the numerator and the denominator by the other denominator
• Suitable equivalent fraction for 1/3:
♦ The denominator of the given fraction is 3.
♦ The denominator of the new equivalent fraction should be 15.
♦ We have seen above that, this 15 is obtained by multiplying 3 with 5
♦ So we have to multiply the numerator also by 5. The calculation steps and the result are given below:
1⁄3 = (1 × 5) ⁄ (3 × 5) = 5⁄15
♦ So the required equivalent fraction is 5⁄15
♦ This is same as multiplying both the numerator and the denominator by the other denominator
■ Step 2: Write down the equivalent fractions
2⁄5 = 6⁄15 and 1⁄3 = 5⁄15
■ Step 3: Analyse the above result
• Taking 2 parts from a total of 5 equal parts is same as taking 6 parts from a total of 15 equal parts
• Taking 1 part from a total of 3 equal parts is same as taking 5 parts from a total of 15 equal parts
So the parts have become equal in size. We can add them directly.
6⁄15 + 5⁄15 = 11⁄15
(b) Our task is to add 3/7 and 5/9. This can be done using the same procedure as above. So we will not write detailed steps. The required steps are shown below:
Solved example 5.25
Solve: (a) 3/10 + 7/15 (b) 4/21 + 3/19 (c) 8/30 + 26/45
Solution:
In the next section we will discuss about Mixed fractions.
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