Ratio - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ &\,\,\,\, {\text{ }}a{\text{ }}:{\text{ }}b{\text{ }}:{\text{ }}c \cr &\,\,\,\, {\text{ }}3{\text{ }}:{\text{ }}4{\text{ }}:{\text{ }}7{\text{ }} \cr & \boxed{\,\,3x:4x:7x\,\,\,\,} \Rightarrow 14x \cr & \therefore a + b + c = 14x \cr & c = 7x \cr & \therefore \left( {a + b + c} \right):c \cr & = 14x:7x \cr & = 2:1 \cr} $$

Let the number of students in the classes be 2x, 3x and 4x respectively;
Total students = 2x + 3x + 4x = 9x
$$\eqalign{ & {\text{According}}\,{\text{to}}\,{\text{the}}\,{\text{question}}, \cr & \frac{{ {2x + 12} }}{{ {3x + 12} }} = \frac{8}{{11}} \cr & or,\,24x + 96 = 22x + 132 \cr & or,\,2x = 132 - 96 \cr & or,\,x = \frac{{36}}{2} = 18 \cr & {\text{Hence,}} \cr & {\text{Original}}\,{\text{number}}\,{\text{of}}\,{\text{students}}, \cr & 9x = 9 \times 18 \cr & \,\,\,\,\,\,\,\, = 162 \cr} $$

Respective ratio of the NUMBER of coins;
= 13 : 11 × 2 = 13 : 22
Hence, Number of 1 rupee coins;
= $$\frac{{13 \times 210}}{{13 + 22}}$$   = 78

$$\eqalign{ & {\text{According}}\,{\text{to}}\,{\text{thequestion,}} \cr & {\frac{{2A}}{3}} = {\frac{{75B}}{{100}}} = {\frac{{C \times 6}}{{10}}} \cr & {\text{Above}}\,{\text{relation}}\,{\text{gives}}; \cr & \frac{{A \times 2}}{3} = \frac{{B \times 3}}{4} \cr & \to \frac{A}{B} = \frac{9}{8} \cr & {\text{And}}, \cr & \frac{{B \times 3}}{4} = \frac{{C \times 3}}{5} \cr & \to \frac{B}{C} = 4:5 \cr & \to \frac{B}{C} = 8:10 {\text{ (multiple by 2)}} \cr & {\text{Thus,}} \cr & A:B:C = 9:8:10 \cr} $$

$$\eqalign{ & {\frac{A}{B}} \times {\frac{B}{C}} = {\frac{3}{4}} \times {\frac{{12}}{{13}}} \cr & Or,\,\frac{A}{C} = \frac{{36}}{{52}} = 9:13 \cr} $$

Let A's Salary = Rs. 100
Then, B's Salary = Rs. 300
And, C's Salary = Rs. 400
Salary has given in 1 : 3 : 4 ratio
Now,
5% increase in A's Salary,
A's new Salary = (100 + 5% of 100) = Rs. 105
B's Salary increases by 10%, Then,
B's new Salary = (300 + 10% of 300) = Rs. 330
C's Salary increases by 15%,
C's new Salary = (400 + 15% of 400) = Rs. 460
Then, ratio of increased Salary,
A : B : C = 105 : 330 : 460 = 21 : 66 : 92

Alternative
100 (A's salary) === 5%↑ ===> 105(A's increased salary)
300 (B's salary) === 10%↑ ===> 330 (B's increased salary)
400 (C's salary) === 15%↑ ===> 460 (C's increased salary)
Ratio of their increased salary = 105 : 330 : 460 = 21 : 66 : 92

$$\eqalign{ & \frac{{\text{A}}}{{\text{B}}} = \frac{2}{3} \cr & \frac{{\text{B}}}{{\text{C}}} = \frac{4}{5} \cr} $$
A : B : C = 2 × 4 : 3 × 4 : 3 × 5 = 8 : 12 : 15

$$\eqalign{ & 2A = 3B \cr & Or,\,B = \left( {\frac{2}{3}} \right)A;\,{\text{and}} \cr & 2A = 4C \cr & Or,\,C = \left( {\frac{1}{2}} \right)A; \cr & {\text{Hence}}, \cr & A:B:C = A:\frac{{2A}}{3}:\frac{A}{2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1:\frac{2}{3}:\frac{1}{2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 6:4:3 \cr} $$

Boys : girls = 8 : 5 (let the boys = 8x, girl = 5x)
Total strength = 286
8x + 5x = 286
13x = 286
Or, x = $$\frac{{286}}{{13}}$$ = 22
Boys = 176 and girls = 110
22 more girls get admitted then number of girls become
(5x + 22) = 110 + 22 = 132
Now, new ratio of boys and girls = 176 : 132 = 4 : 3

Let two numbers be 4x and 5x
Their LCM = 180 and HCF = x
Now,
1^st number × 2^nd number = LCM × HCF
Or, 4x × 5x = 180 × x
Or, 20x = 180
Or, x = 9
Then, the smaller number = 4 × 9 = 36