1. If A : B = 3 : 4 and B : C = 8 : 9, then A : B : C is = ?
a) 8 : 6 : 9
b) 9 : 8 : 6
c) 6 : 8 : 9
d) 3 : 32 : 9
Explanation: A : B = 3 : 4
B : C = 8 : 9
A : B = 3 : 4 (multiply with 2)
i.e. A : B = 6 : 8 and B : C = 8 : 9
A : B : C = 6 : 8 : 9
2. If 2A = 3B = 4C, the A : B : C is = ?
a) 2 : 3 : 4
b) 4 : 3 : 2
c) 6 : 4 : 3
d) 3 : 4 : 6
Explanation:
$$\eqalign{ & \,\,\,{\text{A}}\,\,\,\,\,\,\,{\text{:}}\,\,\,\,\,\,{\text{B}}\,\,\,\,\,\,{\text{:}}\,\,\,\,\,{\text{C}} \cr & 3 \times 4:2 \times 4:2 \times 3 \cr & \,\,\,12\,\,\,\,:\,\,\,\,\,{\text{8}}\,\,\,\,\,:\,\,\,\,{\text{6}} \cr & \,\,\,\boxed{\,\,6\,\,\,\,:\,\,\,\,4\,\,\,\,\,\,:\,\,\,\,3\,\,} \cr} $$
3. x, y, z, u are real numbers such that x : y = y : z = z : u and x : u = 64 : 27. the value of x : z is -
a) 64 : 27
b) 16 : 9
c) 4 : 3
d) 3 : 4
Explanation:
$$\eqalign{ & {\text{Let}}\,\frac{x}{y} = \frac{y}{z} = \frac{z}{u} = k. \cr & {\text{Now, }}\frac{x}{u} = \frac{{64}}{{27}} \cr & \frac{x}{y} \times \frac{y}{z} \times \frac{z}{u} = \frac{{64}}{{27}} \cr & {k^3} = {\left( {\frac{4}{3}} \right)^3} \cr & k = \frac{4}{3}. \cr & {\text{So}},\,x:y = y:z = z:u = 4:3. \cr & \frac{x}{z} = \frac{x}{y} \times \frac{y}{z} = \frac{4}{3} \times \frac{4}{3} = \frac{{16}}{{9.}} \cr} $$
4. If p : q : r = 1 : 2 : 4, then $$\sqrt {5{p^2} + {q^2} + {r^2}} $$ is equal to -
a) 5
b) 2q
c) 5p
d) 4r
Explanation:
$$\eqalign{ & p = k,\,q = 2k,\,r = 4k \cr & \sqrt {5{p^2} + {q^2} + {r^2}} \cr & = \sqrt {5{k^2} + {{\left( {2k} \right)}^2} + {{\left( {4k} \right)}^2}} \cr & = \sqrt {5{k^2} + 4{k^2} + 16{k^2}} \cr & = \sqrt {25{k^2}} \cr & = 5k \cr & = 5p. \cr} $$
5. If A : B : C = 2 : 3 : 4, then the ratio $$\frac{{\text{A}}}{{\text{B}}}$$ : $$\frac{{\text{B}}}{{\text{C}}}$$ : $$\frac{{\text{C}}}{{\text{A}}}$$ is equal to -
a) 4 : 9 : 16
b) 8 : 9 : 24
c) 8 : 9 : 12
d) 8 : 9 : 16
Explanation:
$$\eqalign{ & {\text{let A}} = 2{\text{k}},\,{\text{B}} = 3{\text{k,}}\,{\text{C}} = 4{\text{k}} \cr & \Rightarrow \frac{{\text{A}}}{{\text{B}}} = \frac{{2{\text{k}}}}{{3{\text{k}}}} = \frac{2}{3}, \cr & \Rightarrow \frac{{\text{B}}}{{\text{C}}} = \frac{{3{\text{k}}}}{{4{\text{k}}}} = \frac{3}{4}, \cr & \Rightarrow \frac{{\text{C}}}{{\text{A}}} = \frac{{4{\text{k}}}}{{2{\text{k}}}} = 2 \cr & \frac{{\text{A}}}{{\text{B}}}:\frac{{\text{B}}}{{\text{C}}}:\frac{{\text{C}}}{{\text{A}}} = \frac{2}{3}:\frac{3}{4}:2 \cr & = 8:9:24 \cr} $$
6. If 8a = 9b then the ratio of $$\frac{{\text{a}}}{9}$$ to $$\frac{{\text{b}}}{8}$$ is
a) 1 : 1
b) 1 : 2
c) 2 : 1
d) 64 : 81
Explanation:
$$\eqalign{ & = {\text{8a}} = {\text{9b}} \Rightarrow a = \frac{9}{8}b \cr & \therefore \frac{{\text{a}}}{{\text{9}}}:\frac{{\text{b}}}{{\text{8}}} = \frac{{\left( {\frac{9}{8}b} \right)}}{9}:\frac{{\text{b}}}{{\text{8}}} \cr & = \frac{{\text{b}}}{{\text{8}}}:\frac{{\text{b}}}{{\text{8}}} = 1:1 \cr} $$
7. If x : y = 3 : 4, then (2x + 3y) : (3y - 2x) would be equal to
a) 2 : 1
b) 3 : 1
c) 3 : 2
d) 21 : 1
Explanation:
$$\eqalign{ & = \frac{x}{y} = \frac{3}{4} \cr & = \frac{{2x + 3y}}{{3y - 2x}} \cr & = \frac{{2\left( {\frac{x}{y}} \right) + 3}}{{3 - 2\left( {\frac{x}{y}} \right)}} \cr & = \frac{{2 \times \frac{3}{4} + 3}}{{3 - 2 \times \frac{3}{4}}} \cr & = \frac{9}{2} \times \frac{2}{3} = 3 \cr & = \left( {2x + 3y} \right):\left( {3y - 2x} \right) \cr & = 3:1 \cr} $$
8. Rs. 33630 are divided among A, B and C in such a manner that the ratio of the amount of A to that of B is 3 : 7 and the ratio of the amount of B to that of C is 6 : 5. The amount of money received by B is = ?
a) Rs. 14868
b) Rs. 16257
c) Rs. 13290
d) Rs. 12390
Explanation: A : B = 3 : 7
B : C = 6 : 5
A : B = 3 : 7 (multiply with 6) and
B : C = 6 : 5 (multiply with 7)
i.e. A : B = 18 : 42 and
B : C = 42 : 35
A : B : C = 18 : 42 : 35
18x + 42x + 35x = 33630
x = 354
Money received by B = 42 × 354 = Rs. 14868
9. 200 litres of a mixture contains milk and water in the ratio 17 : 3. After the addition of some more milk to it, the ratio of milk to water in the resulting mixture becomes 7 : 1. The quantity of milk added to it was =?
a) 20 Litres
b) 40 Litres
c) 60 Litres
d) 80 Litres
Explanation: Milk : water = 17 : 3 = 17x : 3x
17x + 3x = 200
⇒ x =10 litre
Milk = 170 litre and water = 30 litre in initial mixture.
Let 'y' litre of milk added in mixture
i.e. 170 + y : 30 = 7 : 1
$$\frac{{170 + y}}{{30}} = \frac{7}{1}$$
y = 210 - 170 = 40 litre
10. In an innings of a cricket match,three players A, B and C scored a total of 361 runs. If the ratio of the number of runs scored by A to that scored by B and also number of runs scored by B to that scored by C be 3 : 2, the number of runs scored by A was = ?
a) 171
b) 181
c) 185
d) 161
Explanation: Given total runs of A, B, C
(A + B + C = 361)
$$\eqalign{ & \Rightarrow {\text{A}}:{\text{B}}:{\text{C}} \cr & \,\,\,\,\,\,\,\,\,3:2 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3:2 \cr & \underline {\overline {\,\,\,\,\,\,\,\,\,\,9:6:4{\text{ }}} } \cr & 9x + 6x + 4x = 361 \cr & \Rightarrow 19x = 361 \cr & \Rightarrow x = 19 \cr & {\text{Runs scored by A }} = 9x \cr & = 9 \times 19 \cr & = 171 \cr} $$