Tuesday, July 19, 2016

Chapter 5.16 - Fractions between fractions

In the previous section we saw some properties of unequal fractions which are helpful for their comparisons. In this  section, we will see a more advanced case.

Consider the following example:
• We have two fractions 12 and 34 in hand
• Out of the two, 12 is lesser. That is., 12 < 34  (∵ × 4   <   3 × 2)
• We are going to make a new fraction from the two fractions
• For that, we add the numerators and denominators
• So the new fraction is (1+3)(2+4) = 46 = 23
• This new fraction 23 is greater than 12That is., 12 < 23  (∵ × 3   <   2 × 2)
• At the same time, this new fraction is less than 34That is., 23 < 34  (∵ × 4   <   3 × 3) 
• So we can write: 12 < 23 < 34
• That means., the new fraction 23 lies in between the original two fractions 12 and 34
• We have seen how to represent fractions on a number line.
• The three fractions are marked on a number line in fig.5.34 below
A new fraction obtained by adding the numerators and denominators of two fractions will lie in between the two.
Fig.5.34
We find that the new fraction 23 lies in between the original two fractions 12 and 34 

Another example:
• We have two fractions 57 and 611 in hand
• Out of the two, 611 is lesser. That is., 611 < 57  (∵ × 7   <   5 × 11  42  <  55)
• We are going to make a new fraction from the two fractions
• For that, we add the numerators and denominators
• So the new fraction is (6+5)(11+7) = 1118
• This new fraction 1118 is greater than 611That is., 611 < 1118  (∵ × 18   <   11 × 11  108 < 121)
• At the same time, this new fraction is less than 57That is., 1118 < 57  (∵ 11 × 7   <   5 × 18  77 < 90) 
• So we can write: 611 < 1118 < 57
• That means., the new fraction 1118 lies in between the original two fractions 611 and 57
• The three fractions are marked on a number line in fig.5.35
Fig.5.35
We find that the new fraction 1118 lies in between the original two fractions 611 and 47 

Another example:
• We have two fractions 85 and 97 in hand
• Out of the two, 97 is lesser. That is., 97 < 85  (∵ × 5   <   8 × 7  45  <  56)
• We are going to make a new fraction from the two fractions
• For that, we add the numerators and denominators
• So the new fraction is (9+8)(7+5) = 1712
• This new fraction 1712 is greater than 97That is., 97 < 1712  (∵ × 12   <   17 × 7  108 < 119)
• At the same time, this new fraction is less than 85That is., 1712 < 85  (∵ 17 × 5   <   8 × 12  85 < 96) 
• So we can write: 97 < 1712 < 85
• That means., the new fraction 1712 lies in between the original two fractions 97 and 85
• The three fractions are marked on a number line in fig.5.36
Fig.5.36
We find that the new fraction 1712 lies in between the original two fractions 97 and 85 

One more example:
• We have two fractions 35 and 1512 in hand
• Out of the two, 35 is obviously lesser. That is., 35 < 1512  (∵ 3is a proper fraction and 1512 is an improper fraction)
• We are going to make a new fraction from the two fractions
• For that, we add the numerators and denominators
• So the new fraction is (3+15)(5+12) = 1817
• This new fraction 1817 is greater than 35That is., 35 < 1817  (∵ 3is a proper fraction and 1817 is an improper fraction)
• At the same time, this new fraction is less than 1512That is., 1817 < 1512  (∵ 18 × 12   <   15 × 17  216 < 255) 
• So we can write: 35 < 1817 < 1512
• That means., the new fraction 1817 lies in between the original two fractions 35 and 1512
• The three fractions are marked on a number line in fig.5.37
Fig.5.37
We find that the new fraction 1817 lies in between the original two fractions 35 and 1512 

Based on the above examples we can write: A new fraction (a+p)(b+q) obtained by adding numerators and denominators of two original fractions ab and pq will lie in between the two fractions. Not just 'in between' the two fractions. It follows a strict rule:
• We have two fractions ab and pq in hand
• One of them will be lesser than the other
• The new fraction will be greater than the 'lesser original fraction'
• The new fraction will be lesser than the 'greater original fraction'
 So the 'lesser original fraction' will lie on the extreme left
■ The 'greater original fraction' will lie on the extreme right
■ The new fraction will lie in between the two

But we must show the proof for all the above:
If  ab <  pq, Prove that ab <  (a+p)(b+q) < pq
1. We have ab < pq. From this we get aq < pb
2. We have to prove that ab < (a+p)(b+q) < pq 
3. Consider the first two terms: ab < (a+p)(b+q)
4. If (3) is true, then a(b+q) < b(a+p)  (ab + aq) < (ba + bp)
5. ab is same as ba. That means we have 'one term same' on both sides in (4). They will cancel out each other
6. So the superiority or inferiority of the left side and right side in (4) is decided by aq and bp
8. It is given in (1) that aq < pb. So we find that left side of (4) is indeed inferior.
9. Hence (4) is proved, and consequently, (3) is established
10. Consider the last two terms in (2): (a+p)(b+q) < pq
11. If (10) is true, then (a+p)q < p(b+q)  (aq + pq) < (pb + pq)
12. We have 'one term same' on both sides in (11). The term is pq. They will cancel out each other
13. So the superiority or inferiority of the left side and right side in (11) is decided by aq and pb
14. It is given in (1) that aq < pb. So we find that left side of (11) is indeed inferior
15. Hence (11) is proved, and consequently, (10) is established
16. Taking (3) and (10) together we get: ab < (a+p)(b+q) < pq  



So we learned the method to obtain a fraction between 'any two fractions'. Concentrate on the words: 'any two fractions'. The new fraction that we obtain by the above method can become one of the 'any two fractions'. For example, in fig.5.34 above, we obtained 23 in between 12 and 34. The fig. is shown again below:

• Now, 12 and 23 can be considered as 'any two fractions'. 
• Let us add the numerators and denominators: (1+2)(2+3) = 35
• This 35 will lie in between 12 and 23 . This is shown in the fig.5.38 below:
[The reader is advised to check and confirm whether 35 indeed lies in between 12 and 23]
Fig.5.38
• 23 and 34 can be considered as 'any two fractions'. 
• Let us add the numerators and denominators: (2+3)(3+4) = 57
• This 57 will lie in between 23 and 34 . This is also shown in the fig.5.38 above
• So altogether we get:
12 35  < 23 < 57  < 34  
3and 5are the two new fractions that we obtained
• We can continue like this for any number of times. For example, we can take 12 and 35 as 'any two fractions'

Now we will see some solved examples:

Solved example 5.37
(i) Find 3 fractions which are larger than 13 and smaller than 12
(ii) Find 3 fractions, all with denominator 24, which are larger than 13 and smaller than 12
(iii) Find 3 fractions, all with numerator 4, which are larger than 13 and smaller than 12
Solution:
(i) We know that 13 < 12
1. We can use the property: If  ab <  pq, Then ab <  (a+p)(b+q) < pq
2. Let us add the numerators and denominators: (1+1)(3+2) = 25
3. This 25 will lie in between 13 and 12 
4. So we can write: 13 25  < 12 
5. Take 13 and 25
6. Let us add the numerators and denominators: (1+2)(3+5) = 38
7. This 38 will lie in between 13 and 25
8. So we can write: 13 38  < 212  
9. Take 25 and 12
10. Let us add the numerators and denominators: (2+1)(5+2) = 37
11. This 37 will lie in between 25 and 12
12. So we can write: 13 38  < 2312  
13. Thus we get 3 fractions: 3825 and 37 between 13 and 12

(ii)  1. First we write 13 and 12 as fractions with denominator 24:
1(1×8)(3×8) = 824 and  12 = (1×12)(2×12) = 1224
2. So we get two fractions: 824 and 1224
3. Now we use the property of fractions with the same denominators:
'When the denominators are the same, the fraction with the larger numerator will be the larger'
4. So the three required fractions are: 924 , 1024 and 1124
5. So we can write: 824 924  < 1024 1124 1224   
6. This is same as:  13 924  < 1024 1124 12    

(iii) 1. First we write 13 and 12 as fractions with numerator 4:
1(1×4)(3×4) = 412 and  12 = (1×4)(2×4) = 48
2. So we get two fractions: 412 and 48
3. Now we use the property of fractions with the same numerators:
'When the numerators are the same, the fraction with the smaller denominator will be the larger'
4. So the three required fractions are: 411 , 410 and 49
5. So we can write: 412 411  < 410 49 48   
6. This is same as: 13 411  < 410 49 12   

In the next section we will see addition and subtraction with fractions.

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