1. If a : b = b : c, then a4 : b4 would be equal to
a) ac : b2
b) a2 : c2
c) c2 : a2
d) b2 : ac
Explanation:
$$\eqalign{ & = \frac{a}{b} = \frac{b}{c} \cr & \Rightarrow {b^2} = ac \cr & \Rightarrow \frac{{{a^4}}}{{{b^4}}} = \frac{{{a^4}}}{{{{\left( {ac} \right)}^2}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{a^4}}}{{{a^2}{c^2}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{a^2}}}{{{c^2}}} \cr & {a^4}:{b^4} = {a^2}:{c^2} \cr} $$
2. If (4x2 - 3y2) : (2x2 + 5y2) = 12 : 19, then x : y is
a) 2 : 3
b) 1 : 2
c) 3 : 2
d) 2 : 1
Explanation:
$$\eqalign{ & {\text{ = }}\frac{{4{x^2} - 3{y^2}}}{{2{x^2} + 5{y^2}}} = \frac{{12}}{{19}} \cr & 19\left( {4{x^2} - 3{y^2}} \right) = 12\left( {2{x^2} + 5{y^2}} \right) \cr & 76{x^2} - 57{y^2} = 24{x^2} + 60{y^2} \cr & 52{x^2} = 117{y^2} \cr & 4{x^2} = 9{y^2} \cr & \frac{{{x^2}}}{{{y^2}}} = \frac{9}{4} \cr & {\left( {\frac{x}{y}} \right)^2} = {\left( {\frac{3}{2}} \right)^2} \cr & \frac{x}{y} = \frac{3}{2} \cr & x:y = 3:2 \cr} $$
3. If x : y = 3 : 4 and a : b = 1 : 2, then the value of $$\frac{{2xa + yb}}{{3yb - 4xa}}$$ is
a) $$\frac{5}{6}$$
b) $$\frac{6}{5}$$
c) $$\frac{6}{7}$$
d) $$\frac{7}{6}$$
Explanation:
$$\eqalign{ & = \frac{x}{y} = \frac{3}{4}{\text{ and }}\frac{a}{b} = \frac{1}{2} \cr & \Rightarrow \frac{{xa}}{{yb}} = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \cr & \frac{{2xa + yb}}{{3yb - 4xa}} \cr & = \frac{{2\left( {\frac{{xa}}{{yb}}} \right) + 1}}{{3 - 4\left( {\frac{{xa}}{{yb}}} \right)}} \cr & = \frac{{2 \times \frac{3}{8} + 1}}{{3 - 4 \times \frac{3}{8}}} \cr & = \frac{7}{4} \times \frac{2}{3} \cr & = \frac{7}{6} \cr} $$
4.Income of A and B are in the ratio 4 : 3 and their annual expenses are in the ratio 3 : 2. If each save Rs. 60000 at the end of the year, the annual income of A is = ?
a) Rs. 120000
b) Rs. 150000
c) Rs. 240000
d) Rs. 360000
Explanation:
| A | : | B | |
| Income | 4x | : | 3x |
| Expenses | 3y | : | 2y |
| Saving | 6000 | : | 6000 |
Income = Expenses + Saving
$$ \frac{{4x - 60000}}{{3x - 60000}} = \frac{3}{2}$$
8x - 120000 = 9x - 180000
x = 60000
Income of A = 4 × 60000 = 240000
5.The weight of Mr. Gupta and Mrs. Gupta are in the ratio 7 : 8 and their total weight is 120 kg. After taking a dieting course Mr. Gupta reduces by 6 kg and the ratio between their weights change to 5 : 6, so Mrs. Gupta has reduced by = ?
a) 2 kg
b) 4 kg
c) 3 kg
d) 5 kg
Explanation:
| Mr. | : | Mrs. | |
| Before | 7x | : | 8x |
| After | 5y | : | 6y |
Before 7x + 8x = 120
15x = 120
x = 8
Mr. Gupta = 7 × 8 = 56
Mrs. Gupta = 8 × 8 = 64
After losing 6kg by Mr. Gupta the ratio becomes 5 : 6
Let Mrs. Gupta loss x kg
$$\eqalign{ & \therefore \frac{{56 - 6}}{{64 - x}} \times \frac{5}{6} \cr & \left( {{\text{Cross multiplication}}} \right) \cr} $$
300 = 320 - 5x
5x = 20
x = 4 kg
6. The income of A, B and C are in the ratio 7 : 9 : 12 and their spending are in the ratio 8 : 9 : 15. If A saves $$\frac{1}{4}$$ th of his income then the savings of A, B and C are in the ratio of = ?
a) 56 : 99 : 69
b) 69 : 56 : 99
c) 99 : 56 : 69
d) 99 : 69 : 56
Explanation: Let income of A, B and C are 7x, 9x and 12x respectively
and expenditure of A, B and C are 8y, 9y and 15y respectively
$$\eqalign{ & \Rightarrow {\text{Income}}\,{\text{of,}} \cr & A \times \frac{1}{4} = {\text{Saving}}\,{\text{of}}\,{\text{A}}\,\left( {{\text{given}}} \right) \cr & 7x - 8y = 7x \times \frac{1}{4} \cr & 28x - 32y = 7x \cr & 21x = 32y \cr & x:y = 32:21 \cr} $$
The ratio of savings of A, B and C
⇒ (7x - 8y) : (9x - 9y) : (12x - 15y)
⇒ (7 × 32 - 8 × 21) : (9 × 32 - 9 × 21) : (12 × 32 - 15 × 21)
⇒ (224 - 168) : (288 - 189) : (384 - 315)
⇒ 56 : 99 : 69
7. What will be the simplest form of the ratio 3 hours : 1 day?
a) 1 : 3
b) 1 : 6
c) 1 : 8
d) 1 : 25
Explanation: 1 day = 24 hours.
Given ration = 3 : 24
= 1 : 8
8. In a proportion the product of 1st and 4th terms is 40 and that of 2nd and 3rd terms is 2.5x. Then the value of x is.
a) 16
b) 26
c) 75
d) 90
Explanation: Product of 1st and 4th terms (extremes) = product of 2nd and 3rd terms (means)
$$\eqalign{ & {\text{2}}{\text{.5}}x = {\text{40}} \cr & x = \frac{{40}}{{2.5}} = 16 \cr} $$
9. If 20% of A = 30% of B = $$\frac{1}{6}$$ of C, then A : B : C is -
a) 2 : 3 : 6
b) 3 : 2 : 16
c) 10 : 15 : 18
d) 15 : 10 : 18
Explanation:
$$\eqalign{ & 20\% \,{\text{of A}} = 30\% \,{\text{of B}} = \frac{1}{6}{\text{of C}} \cr & \Rightarrow \frac{{20{\text{A}}}}{{100}} = \frac{{30{\text{B}}}}{{100}} = \frac{{\text{C}}}{{\text{6}}} \cr & \Rightarrow \frac{{\text{A}}}{5} = \frac{{3{\text{B}}}}{{10}} = \frac{C}{6} = k(say) \cr & A = 5K,\,B = \frac{{10k}}{3},\,C = 6k \cr & {\text{A}}:{\text{B}}:{\text{C}} = 5k:\frac{{10k}}{3}:6k \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 5:\frac{{10}}{3}:6 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 15:10:18 \cr} $$
10. Two numbers are in the ratio 17 : 45, One - third of the smaller is less than $$\frac{1}{5}$$ of the bigger by 15. The smaller number is = ?
a) $$25\frac{1}{2}$$
b) $$67\frac{1}{2}$$
c) $$76\frac{1}{2}$$
d) $$86\frac{1}{2}$$
Explanation:
$$\eqalign{ & {\text{A}}:{\text{B}} \cr & 17:45 \cr & 17x:45x \cr & \Rightarrow 17x \times \frac{1}{3} + 15 = 45x \times \frac{1}{5} \cr & \Rightarrow \frac{{17x + 45}}{3} = 9x \cr & \Rightarrow 17x + 45 = 27x \cr & \Rightarrow 10x = 45 \cr & \Rightarrow x = \frac{{45}}{{10}} \cr & {\text{Smaller number is }} \cr & = \frac{{45}}{{10}} \times 17 \cr & = \frac{{765}}{{10}} \cr & = 76\frac{1}{2} \cr} $$