#### Exercise 4.2

*Q 1 . Express each of the following as rational number with positive denominator :*

*(i) . \(\frac{-15}{-28}\)*

*(ii) . \(\frac{6}{-9}\)*

*(iii) . \(\frac{-28}{-11}\)*

*(iv) . \(\frac{19}{-7}\)*

**SOLUTION :**

Rational number with positive denominators :

(i) Multiplying the number by -1, we get : -15-28 = -15 x -1-28 x-1 = 1528

(ii) Multiplying the number by -1, we get : Â 6-9 = 6x-1-9x-1 = -69

(iii) Multiplying the number by -1, we get : Â -28-11 = -28 x-1-11 x-1=2811

(iv) Multiplying the number by -1, we get : 19-7 = 19 x-1-7 x-1 = -197

**Q 2 . Express \(\frac{3}{5}\) as a rational number with numerator :**

**(i) 6**

**(ii) -15**

**(iii) 21**

**(iv) -27**

**SOLUTION :**

Rational number with numerator :

(i) 6 is :

\(\frac{3\times 2}{5\times 2}\)

(ii) -15 is :

\(\frac{3\times -5}{5\times -5}\)

(iii) 21 is :

\(\frac{3\times 7}{5\times 7}\)

(iv) -27 is :

\(\frac{3\times -9}{5\times -9}\)

*Q 3 . Express \(\frac{5}{7}\) as a rational number with denominator :*

*(i) -14*

*(ii) 70*

*(iii) -28*

*(iv) -84*

**SOLUTION :**

\(\frac{5}{7}\)

(1) -14 is :

\(\frac{5\times -2}{7\times -2}\)

(ii) 70 is :

\(\frac{5\times 10}{7\times 10}\)

(iii) -28 is :

\(\frac{5\times -4}{7\times -4}\)

(iv) -84 is :

\(\frac{5\times -12}{7\times -12}\)

*Q 4 . Express \(\frac{3}{4}\) as a rational number with denominator :*

*(i) 20*

*(ii) 36*

*(iii) 44*

*(iv) -80*

**SOLUTION :**

\(\frac{3}{4}\)

(i) 20 is :

\(\frac{3\times 5}{4\times 5 }\)

(ii) 36 is :

\(\frac{3\times9 }{4\times 9}\)

(iii) 44 is :

\(\frac{3\times 11}{4\times 11}\)

(iv) -80 is :

\(\frac{3\times -20}{4\times -20}\)

**Q 5 . Express \(\frac{2}{5}\) as a rational number with numerator :**

**(i) -56**

**(ii) 154**

**(iii) -750**

**(iv) -80**

**SOLUTION :**

2/5 as a rational number with numerator :

(i) . -56 is :

\(\frac{2\times -28}{5\times-28 }\)

(ii) 154 is :

\(\frac{2\times 77}{5\times 77}\)

(iii) -750 is :

\(\frac{2\times -375}{5\times -375}\)

(iv) 500 is :

\(\frac{2\times 250}{5\times 250}\)

**Q 6 . Express \(\frac{-192}{108}\) as a rational number with numerator :**

**(i) 64**

**(ii) -16**

**(iii) 32**

**(iv) -48**

**SOLUTION :**

Rational number with numerator :

(i) Â 64 as numerator :

-192/-3 & 108/-3 =64/-36 (Dividing the numerator and denomintor by -3)

(ii) -16 as numerator :

-192/12 & 108/12 = -16/9 (Dividing the numerator and denomintor by 12)

(iii) 32 as numerator :

-192/-6 & 108/-6 = 32/-18 (Dividing the numerator and denomintor by -6)

(iv) -48 as numerator :

-192/4 & 108/4 = -48/27 (Dividing the numerator and denomintor by 4)

**Q 7 .Express \(\frac{168}{-294}\) as a rational number with denominator :**

**(i) 14**

**(ii) -7**

**(iii) -49**

**(iv) 1470**

**SOLUTION :**

Rational number with denominator:

(i) 14 as denominator :

168/-21 & -294/-21 = -8/14 (Dividing the numerator and denomintor by -21)

(ii) -7 as denominator :

168/42 & -294/42 = 4/-7 (Dividing the numerator and denomintor by 42)

(iii) -49 as denominator :

168/6 & -294/6 = 28/-49 (Dividing the numerator and denomintor by 6)

(iv) 1470 as denominator :

\(\frac{168\times -5}{-294\times -5}\)

*Q 8 . Write \(\frac{-14}{42}\) in a form so that numerator is equal to :*

*(i) -2*

*(ii) Â 7*

*(iii) 42*

*(iv) -70*

**SOLUTION :**

Rational number with numerator :

(i) -2 is :

-14/7 & 42/7 = -26 ( Dividing numerator and denominator by 7)

(ii) 7 is :

-14/-2 & 42/-2 = 7/-21 ( Dividing numerator and denominator by -2)

(iii) 42 is :

-14x-3 & 42x-3 = 42/-126 ( Multiplying numerator and denominator by -3)

(iv) -70 is :

-14×5 & 42×5 = -70/210 ( Multiplying numerator and denominator by 5)

*Q 9 . Select those rational numbers which can be written as a rational number with numerator 6 :*

*\(\frac{1}{22}\) , \(\frac{2}{3}\) , \(\frac{3}{4}\) , \(\frac{4}{-5}\) , \(\frac{5}{6}\) , \(\frac{-6}{7}\) , \(\frac{-7}{8}\)*

**SOLUTION :**

Given rational numbers that can be written as a rational number with numerator 6 are :

1/22 (On multiplying by 6) = 6/132 , 2/3 (On multiplying by 3) = 6/9 , 3/4 (On multiplying by 2) = 6/8 , -6/7 (On multiplying by -1) = 6/-7

*QÂ 10 . Select those rational numbers which can be written as a rational number with denominator 4 :*

*\(\frac{7}{8}\) , \(\frac{64}{16}\) , \(\frac{36}{-12}\) , \(\frac{-16}{17}\) , \(\frac{5}{-4}\) , \(\frac{-140}{28}\).*

**SOLUTION :**

Given rational numbers that can be written as a rational number with denominator 4 are :

7/8 (On dividing by 2) = 3.5/4 ,

64/16 (On dividing by 4) = 16/4 ,

36/-12(On dividing by 3) = 12/-4 = -12/4 ,

16/17 can’t be expressed with a denominator 4.

5/- 4(On multiplying by -1) =-5/4

140/28(On dividing by 7) =20/4

*Q 11 . In each of the following , find an equivalent form of the rational number having common denominator :*

*(i) \(\frac{3}{4}\) and \(\frac{5}{12}\)*

*(ii) \(\frac{2}{3}\) , \(\frac{7}{6}\) and \(\frac{11}{12}\)*

*(iii) \(\frac{5}{7}\) , \(\frac{3}{8}\) , \(\frac{9}{14}\) and \(\frac{20}{21}\)<*

**SOLUTION :**

Equivalent forms of the rational number having common denominator are :

(i) 3/4 = (3×3)/(4×3) = 9/12 and 512.

(ii) 2/3 = (2×4)/(3×4) = 8/12 and 7/6 = (7×2)/(6×2) = 14/12 and 11/12

Forms are 8/12 , 14/12 and 11/12

(iii) 5/7 = (5×24)/(7×24) = 120/168 , 3/8 = (3×21)/(8×21) = 63/168 , 9/14 = (9×12)/(14×12) = 108/168 and 20/21 = (20×8)/(21×8) = 160/168

Forms are 120/168 , 63/168 , 108/168 and 160/168.