1. If a : b : c = 3 : 4 : 7, then the ratio (a + b + c) : c is equal to
a) 2 : 1
b) 14 : 3
c) 7 : 2
d) 1 : 2
Explanation:
$$\eqalign{ &\,\,\,\, {\text{ }}a{\text{ }}:{\text{ }}b{\text{ }}:{\text{ }}c \cr &\,\,\,\, {\text{ }}3{\text{ }}:{\text{ }}4{\text{ }}:{\text{ }}7{\text{ }} \cr & \boxed{\,\,3x:4x:7x\,\,\,\,} \Rightarrow 14x \cr & a + b + c = 14x \cr & c = 7x \cr & \left( {a + b + c} \right):c = 14x:7x \cr & = 2:1 \cr} $$
2. The number of students in 3 classes is in the ratio 2 : 3 : 4. If 12 students are increased in each class this ratio changes to 8 : 11 : 14. The total number of students in the three classes in the beginning was
a) 162
b) 108
c) 96
d) 54
Explanation: Let the number of students in the classes be 2x, 3x and 4x respectively;
Total students = 2x + 3x + 4x = 9x
$$\eqalign{ & \frac{{ {2x + 12} }}{{ {3x + 12} }} = \frac{8}{{11}} \cr & \,24x + 96 = 22x + 132 \cr & \,2x = 132 - 96 \cr & \,x = \frac{{36}}{2} = 18 \cr & {\text{Original}}\,{\text{number}}\,{\text{of}}\,{\text{students}}, \cr & 9x = 9 \times 18 \cr & \,\,\,\,\,\,\,\, = 162 \cr} $$
3. A box has 210 coins of denominations one-rupee and fifty paise only. The ratio of their respective values is 13 : 11. The number of one-rupee coin is
a) 65
b) 66
c) 77
d) 78
Explanation: Respective ratio of the NUMBER of coins;
= 13 : 11 × 2 = 13 : 22
Number of 1 rupee coins;
= $$\frac{{13 \times 210}}{{13 + 22}}$$ = 78
4.If $$\frac{2}{3}$$ of A=75% of B = 0.6 of C, then A : B : C is
a) 2 : 3 : 3
b) 3 : 4 : 5
c) 4 : 5 : 6
d) 9 : 8 : 10
Explanation:
$$\eqalign{ & {\frac{{2A}}{3}} = {\frac{{75B}}{{100}}} = {\frac{{C \times 6}}{{10}}} \cr & {\text{Above}}\,{\text{relation}}\,{\text{gives}}; \cr & \frac{{A \times 2}}{3} = \frac{{B \times 3}}{4} \cr & \to \frac{A}{B} = \frac{9}{8} \cr & {\text{And}}, \cr & \frac{{B \times 3}}{4} = \frac{{C \times 3}}{5} \cr & \to \frac{B}{C} = 4:5 \cr & \to \frac{B}{C} = 8:10 {\text{ (multiple by 2)}} \cr & A:B:C = 9:8:10 \cr} $$
5. If A and B are in the ratio 3 : 4, and B and C in the ratio 12 : 13, then A and C will be in the ratio
a) 3 : 13
b) 9 : 13
c) 36 : 13
d) 13 : 9
Explanation:
$$\eqalign{ & {\frac{A}{B}} \times {\frac{B}{C}} = {\frac{3}{4}} \times {\frac{{12}}{{13}}} \cr & \,\frac{A}{C} = \frac{{36}}{{52}} = 9:13 \cr} $$
6. In a bag, there are coins of 25 p, 10 p and 5 p in the ratio of 1 : 2 : 3. If there is Rs. 30 in all, how many 5 p coins are there?
a) 50
b) 100
c) 150
d) 200
Explanation: Let the number of 25 p, 10 p and 5 p coins be x, 2x, 3x respectively
Then, sum of their values
$$\eqalign{ & = Rs.\,\left( {\frac{{25x}}{{100}} + \frac{{10 \times 2x}}{{100}} + \frac{{5 \times 3x}}{{100}}} \right) \cr & = Rs.\,\frac{{60x}}{{100}} \cr & \frac{{60x}}{{100}} = 30 \Leftrightarrow x = \frac{{30 \times 100}}{{60}} = 50 \cr & {\text{Hence,}}\,{\text{the}}\,{\text{number}}\,{\text{of}}\,{\text{5p}}\,{\text{coins}} = \left( {3 \times 50} \right) \cr & = 150 \cr} $$
7. What is the ratio in Rs. 2.80 and 40 paise?
a) 1 : 7
b) 2 : 7
c) 7 : 1
d) 1 : 14
Explanation: Rs. 2.80 = 280 paise
Required ration = 280 : 40
= 7 : 1
8. A person spends Rs. 8100 in buying some tables at Rs. 1200 each and some chairs at Rs. 300 each. The ratio of the number of chairs to that of tables when the maximum possible number of tables is purchased,
a) 1 : 2
b) 1 : 4
c) 2 : 1
d) 5 : 7
Explanation: Maximum possible number of tables = 6
[1200 × 6 = 7200]
Number of chairs purchased
$$\eqalign{ & {\text{ = }}\frac{{{\text{8100}} - {\text{7200}}}}{{{\text{300}}}}{\text{ = }}\frac{{{\text{900}}}}{{{\text{300}}}}{\text{ = 3}}{\text{}} \cr & {\text{Required ratio}} = {\text{3}}:{\text{6}} \cr & {\text{ = 1}}:{\text{2}} \cr} $$
9. If $$x = \frac{1}{3}y$$ and $$y = \frac{1}{2}z{\text{,}}$$ then x : y : z is equal to = ?
a) 3 : 2 : 1
b) 1 : 2 : 6
c) 1 : 3 : 6
d) 2 : 4 : 6
Explanation:
$$\eqalign{ & x = \frac{1}{3}y\,\,{\text{and }}y = \frac{1}{2}z \cr & \frac{x}{y} = \frac{1}{3}\,{\text{and }}\frac{y}{z} = \frac{1}{2} \cr & x:y = 1:3 \cr & y:z = 1:2 \times 3\,({\text{multiply}}) \cr & i.e.\,y:z = 3:6 \cr & x:y:z = 1:3:6 \cr} $$
10. If x : y = 3 : 1, then x3 - y3 : x3 + y3 = ?
a) 13 : 14
b) 14 : 13
c) 10 : 11
d) 11 : 10
Explanation:
$$\eqalign{ & x:y = 3:1 \cr & \frac{x}{y} = \frac{3}{1} \cr & \frac{{{x^3} - {y^3}}}{{{x^3} + {y^3}}} \cr & \Rightarrow \frac{{{y^3}\left( {\frac{{{x^3}}}{{{y^3}}} - 1} \right)}}{{{y^3}\left( {\frac{{{x^3}}}{{{y^3}}} + 1} \right)}} \cr & {\text{taking }}{{\text{y}}^3}{\text{ common}} \cr & {\text{ = }}\frac{{\frac{{{x^3}}}{{{y^3}}} - 1}}{{\frac{{{x^3}}}{{{y^3}}} + 1}} \cr & \Rightarrow \frac{{27 - 1}}{{27 + 1}} \cr & \Rightarrow \frac{{26}}{{28}} \cr & \Rightarrow \frac{{13}}{{14}} \cr} $$