RS Aggarwal Solutions: Exercise 1C - Real Numbers | Extra Documents, Videos & Tests for Class 10 PDF Download

 Page 1


Exercise - 1C 
1. Without actual division, show that each of the following rational numbers is a terminating
decimal. Express each in decimal form.
(i)   (ii)     (iii)   (iv)   (v)  
(vi)  
Answer:
(i) = = = 0.115 
We know either 2 or 5 is not a factor of 23, so it is in its simplest form 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating. 
(ii) = = = = 0.192 
We know 5 is not a factor of 23, so it is in its simplest form. 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating. 
(iii) = = = = 0.21375 
We know either 2 or 5 is not a factor of 171, so it is in its simplest form. 
Page 2


Exercise - 1C 
1. Without actual division, show that each of the following rational numbers is a terminating
decimal. Express each in decimal form.
(i)   (ii)     (iii)   (iv)   (v)  
(vi)  
Answer:
(i) = = = 0.115 
We know either 2 or 5 is not a factor of 23, so it is in its simplest form 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating. 
(ii) = = = = 0.192 
We know 5 is not a factor of 23, so it is in its simplest form. 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating. 
(iii) = = = = 0.21375 
We know either 2 or 5 is not a factor of 171, so it is in its simplest form. 
 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(iv)  = = = = 0.009375
We know either 2 or 5 is not a factor of 15, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(v)  = = = = 0.053125
We know either 2 or 5 is not a factor of 17, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(vi)  = = = = 0.00608
We know either 2 or 5 is not a factor of 19, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
2. Without actual division show that each of the following rational numbers is a non-
terminating repeating decimal.
(i) (ii) (iii) (iv) 
(v) (vi) (vii) (viii)
Answer:
(i)
We know either 2 or 3 is not a factor of 11, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non terminating repeating decimal.
(ii)
We know 2, 3 or 5 is not a factor of 73, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(iii)
We know 2, 5 or 7 is not a factor of 129, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(iv) =
We know either 5 or 7 is not a factor of 9, so it is in its simplest form.
Moreover, (5 × 7) 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
Page 3


Exercise - 1C 
1. Without actual division, show that each of the following rational numbers is a terminating
decimal. Express each in decimal form.
(i)   (ii)     (iii)   (iv)   (v)  
(vi)  
Answer:
(i) = = = 0.115 
We know either 2 or 5 is not a factor of 23, so it is in its simplest form 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating. 
(ii) = = = = 0.192 
We know 5 is not a factor of 23, so it is in its simplest form. 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating. 
(iii) = = = = 0.21375 
We know either 2 or 5 is not a factor of 171, so it is in its simplest form. 
 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(iv)  = = = = 0.009375
We know either 2 or 5 is not a factor of 15, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(v)  = = = = 0.053125
We know either 2 or 5 is not a factor of 17, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(vi)  = = = = 0.00608
We know either 2 or 5 is not a factor of 19, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
2. Without actual division show that each of the following rational numbers is a non-
terminating repeating decimal.
(i) (ii) (iii) (iv) 
(v) (vi) (vii) (viii)
Answer:
(i)
We know either 2 or 3 is not a factor of 11, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non terminating repeating decimal.
(ii)
We know 2, 3 or 5 is not a factor of 73, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(iii)
We know 2, 5 or 7 is not a factor of 129, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(iv) =
We know either 5 or 7 is not a factor of 9, so it is in its simplest form.
Moreover, (5 × 7) 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
 
(v) = = =
We know 2, 3 or 5 is not a factor of 11, so is in its simplest form.
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(vi) =
We know either 3 or 7 is not a factor of 32, so it is in its simplest form.
Moreover, (3 × 7
2 m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(vii) =
We know 7 is not a factor of 29, so it is in its simplest form.
Moreover, 7
3 m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(viii) =
We know 5, 7 or 13 is not a factor of 64, so it is in its simplest form.
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
3. Express each of the following as a rational number in its simplest form:
(iii) (iv) (v)  (vi)
Answer:
(i) Let x = 
x = 0.888
10x = 8.888
On subtracting equation (1) from (2), we get
9x = 8 x = 
0.8 = 
(ii) Let x = 
x = 2.444
10x = 24.444
On subtracting equation (1) from (2), we get
9x = 22 x = 
2.4 = 
(iii) Let x = 
x = 0.2424
100x = 24.2424
Page 4


Exercise - 1C 
1. Without actual division, show that each of the following rational numbers is a terminating
decimal. Express each in decimal form.
(i)   (ii)     (iii)   (iv)   (v)  
(vi)  
Answer:
(i) = = = 0.115 
We know either 2 or 5 is not a factor of 23, so it is in its simplest form 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating. 
(ii) = = = = 0.192 
We know 5 is not a factor of 23, so it is in its simplest form. 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating. 
(iii) = = = = 0.21375 
We know either 2 or 5 is not a factor of 171, so it is in its simplest form. 
 
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(iv)  = = = = 0.009375
We know either 2 or 5 is not a factor of 15, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(v)  = = = = 0.053125
We know either 2 or 5 is not a factor of 17, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
(vi)  = = = = 0.00608
We know either 2 or 5 is not a factor of 19, so it is in its simplest form.
Moreover, it is in the form of (2
m
× 5
n
).
Hence, the given rational is terminating.
2. Without actual division show that each of the following rational numbers is a non-
terminating repeating decimal.
(i) (ii) (iii) (iv) 
(v) (vi) (vii) (viii)
Answer:
(i)
We know either 2 or 3 is not a factor of 11, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non terminating repeating decimal.
(ii)
We know 2, 3 or 5 is not a factor of 73, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(iii)
We know 2, 5 or 7 is not a factor of 129, so it is in its simplest form.
Moreover, 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(iv) =
We know either 5 or 7 is not a factor of 9, so it is in its simplest form.
Moreover, (5 × 7) 
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
 
(v) = = =
We know 2, 3 or 5 is not a factor of 11, so is in its simplest form.
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(vi) =
We know either 3 or 7 is not a factor of 32, so it is in its simplest form.
Moreover, (3 × 7
2 m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(vii) =
We know 7 is not a factor of 29, so it is in its simplest form.
Moreover, 7
3 m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
(viii) =
We know 5, 7 or 13 is not a factor of 64, so it is in its simplest form.
m
× 5
n
)
Hence, the given rational is non-terminating repeating decimal.
3. Express each of the following as a rational number in its simplest form:
(iii) (iv) (v)  (vi)
Answer:
(i) Let x = 
x = 0.888
10x = 8.888
On subtracting equation (1) from (2), we get
9x = 8 x = 
0.8 = 
(ii) Let x = 
x = 2.444
10x = 24.444
On subtracting equation (1) from (2), we get
9x = 22 x = 
2.4 = 
(iii) Let x = 
x = 0.2424
100x = 24.2424
 
On subtracting equation (1) from (2), we get
99x = 24 x = 
0.24 = 
(iv) Let x = 
x = 0.1212
100x = 12.1212
On subtracting equation (1) from (2), we get
99x = 12 x = 
0.12 = 
(v) Let x = 
x = 2.2444
10x = 22.444
100x = 224.444
On subtracting equation (2) from (3), we get
90x = 202 x = =
=
(vi) Let x = 
x = 0.3656565
10x = 3.656565
1000x = 365.656565
On subtracting equation (2) from (3), we get
990x = 362 x = =
=