Page 41 - SM inner class 7.cdr

Ex am ple 14. Prove that: So lu tion :
1 æ 9 1ö æ 1 9 ö æ 1 1ö 2 4
(i) ´ ç + ÷ = ç ´ ÷ + ç ´ ÷ (i) ,
5 è 10 2 ø è 5 10 ø è 5 2 ø 7 7
1 æ 1 1ö æ 1 1ö æ 1 1ö It can be seen that
(ii) ´ ç - ÷ = ç ´ ÷ - ç ´ ÷
4 è 2 6 ø è 4 2 ø è 4 6 ø 2 < 4
2 4
So lu tion : So, <
7 7
1 æ 9 1ö æ 1 9 ö æ 1 1ö
(i) ´ ç + ÷ = ç ´ ÷ + ç ´ ÷ -1 -5
5 è 10 2 ø è 5 10 ø è 5 2 ø (ii) ,
6 6
L.H.S. R.H.S. It can be seen that
1 æ 9 1 ö æ1 9 ö
= ´ ç + ÷ = ç ´ ÷ - > -5
1
5 è10 ø 2 è 5 10 ø
-1 -5
1 æ 9 + ö 5 æ1 1 ö So, >
= ´ ç ÷ + ç ´ ÷ 6 6
5 è 10 ø è 5 ø 2
1 - 3
(iii) ,
9 1
= + 4 4
50 10
It can be seen that
1 14 7 9 + 5 14 7
= ´ = = = = 1 > - 3
5 10 25 50 50 25
1 - 3
So, > -
L.H.S. = R.H.S.
4 4
1 é 1 1ù é 1 1ù é 1 1ù † Case II: Dif fer ent De nom i na tor
(ii) ´ - = ´ - ´
ú
ú
ú
4 ê 2 6 û ê 4 2 û ê 4 6 û Ex am ple 16. Put the correct sign > or <
ë
ë
ë
between the following pair of rational
R.H.S.
L.H.S.
num bers.
1 é 1 1ù
´ - é 1 1ù é 1 1ù 1 3 9 - 41
4 ê ë 2 6û ú ê 4 ´ 2û ú - ê 4 ´ 6û ú (i) , (ii) - ,
1 é 3 - ù 1 ë ë 2 5 11 121
= ´ ê ú 1 1
4 ë 6 û = - So lu tion : (i) Write other two ra tional
8 24
numbers from the given rational numbers
1 2 3 - 1 such that their de nom i na tors must be equal.
= ´ =
´
4 6 24 1 1 5 5
= =
1 2 1 2 2 ´ 5 10
= = Þ
12 24 12
3 3 ´ 2 6
= =
L.H.S. = R.H.S. 5 5 ´ 2 10
Comparison of Rational Numbers Now compare the numerators of rational
We have studied the comparison of integers numbers with the same denominators.
and fractions in our previous class. Similarly, 5 < 6
we can compare the rational numbers by using 5 6
the same rules for comparison. We shall make 10 < 10
it clear with examples. 1 3
Thus, <
† Case I: Same De nom i na tor 2 5
Ex am ple 15. Compare the following pairs of 9 - 14
,
ra tio nal num bers. (ii) - 11 121
2 4 -1 -5
(i) , (ii) , By making their denominators equal
7 7 6 6 9 9 ´ - 11) - 99
(
1 - 3 = =
(iii) , - 11 - 11 ´ - 11) 121
(
4 4
41
Mathematics In Focus - 7