# UP Board Solutions for Class 10 Maths Chapter 1 Exercise 1.4 Real Numbers

## EXERCISE 1.4

### 1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion :

Sol. We know that if the denominator of a rational number has no prime factors other than 2 or 5, then it is expressible as a terminating, otherwise it, has non-terminating repeating decimal representation. Thus,- we will have to check the prime factors of the denominators of each of the given rational numbers.

### (i) In , the denominator is 3125.

We have, 3125 = 5 × 5 × 5 × 5 x 5

Thus, 3125 has 5 as the only

prime factor.
Hence, must have a
terminating decimal representation.

### (ii) In the denominator is 8

We have, 8=2 x 2 × 2
Thus, 8 has 2 as the only prime factor.
Hence, must have a terminating decimal representation.

### (iii) In , denominator is 455.

We have, 455 = 5 × 7 x 13
Clearly, 455 has prime factors
other than 2 and 5. So, it will not have
a terminating decimal representation.

### (iv) In , the denominator is 1600.

We have, 1600
= 2 x 2 x 2 × 2 x 2 × 2 × 5 × 5
Thus, 1600 has only 2 and 5 as prime factors.

Hence, must have a terminating decimal representation.

### (v) In ,  the denominator is 343.

We have, 343 = 7 × 7 × 7
Clearly, 343 has prime factors other than 2 and 5.
So, it will not have terminating decimal representation.

### (vi) In Clearly, the denominator has only 2 and 5 as prime factors.
Hence, must have a terminating decimal representation.

### (vii) In Clearly, the denominator has prime factors other than 2 and 5.

So, it will not have terminating decimal representation.

### (viii) In , we have 15 = 3 x5

Clearly, 15 has prime factors other than 2 and 5. So, it will not have terminating decimal representation.

### (ix) In , we have 50 = 2 × 5 x 5

The denominator has only 2 and 5 as prime factors.
Hence, must have a terminating decimal representation.

### (x) In , the denominator is 210.

We have, 210 = 2 × 3 × 5 x 7
Clearly, 210 has prime factors other than 2 and 5.
So, it will not have terminating decimal representation.

### 2. Write down the decimal expansion of those rational numbers in Question 1 above which have terminating decimal expansions.

(x) Non – terinatinating.