**Exercise 1.4**

**Q1. 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:**

**(i) 13/3125**

** (ii) 17/8**

** (iii) 64/455**

** (iv) 15/1600**

** (v) 29/343**

** (vi) 23/23 × 52**

** (vii) 129/22 × 57 × 75**

** (viii) 6/15**

** (ix) 35/50**

** (x) 77/210**

**Answer**

(i) 13/3125

Factorize the denominator we get

3125 =5 × 5 × 5 × 5 × 5 = 55

So denominator is in form of 5m so it is terminating .

(ii) 17/8

Factorize the denominator we get

8 =2 × 2 × 2 = 23

So denominator is in form of 2m so it is terminating .

(iii) 64/455

Factorize the denominator we get

455 =5 × 7 × 13

There are 7 and 13 also in denominator so denominator is not in form of 2m × 5n . so it is not terminating.

(iv) 15/1600

Factorize the denominator we get

1600 =2 × 2 × 2 ×2 × 2 × 2 × 5 × 5 = 26 × 52

so denominator is in form of 2m × 5n

Hence it is terminating.

(v) 29/343

Factorize the denominator we get

343 = 7 × 7 × 7 = 73

There are 7 also in denominator so denominator is not in form of 2m × 5n

Hence it is non-terminating.

(vi) 23/(23 × 52)

Denominator is in form of 2m × 5n

Hence it is terminating.

(vii) 129/(22 × 57 × 75 )

Denominator has 7 in denominator so denominator is not in form of 2m × 5n

Hence it is none terminating.

(viii) 6/15

divide nominator and denominator both by 3 we get 2/5

Denominator is in form of 5m so it is terminating.

(ix) 35/50 divide denominator and nominator both by 5 we get 7/10

Factorize the denominator we get

10=2 × 5

So denominator is in form of 2m × 5n so it is terminating.

(x) 77/210

simplify it by dividing nominator and denominator both by 7 we get 11/30

Factorize the denominator we get

30=2 × 3 × 5

Denominator has 3 also in denominator so denominator is not in form of 2m × 5n

Hence it is none terminating.

**Q2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.**

**Answer**

(i) 13/3125 = 13/55 = 13×25/55×25 = 416/105 = 0.00416

(ii) 17/8 = 17/23 = 17×53/23×53 = 17×53/103 = 2125/103 = 2.125

(iv) 15/1600 = 15/24×102 = 15×54/24×54×102 = 9375/106 = 0.009375

(vi) 23/2352 = 23×53×22/23 52×53×22 = 11500/105 = 0.115

(viii) 6/15 = 2/5 = 2×2/5×2 = 4/10 = 0.4

(ix) 35/50 = 7/10 = 0.7.

**Q3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p , q you say about the prime factors of q?**

** (i) 43.123456789**

** (ii) 0.120120012000120000…**

** (iii) 43.123456789**

**Answer:**

(i) Since this number has a terminating decimal expansion, it is a rational number of the form p/q, and q is of the form 2m × 5n.

(ii) The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii) Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form p/q, and q is not of the form 2m × 5n.