1. Number of students in institutes A and B were in the ratio of 7 : 15 respectively in 2012. In 2013, the number of students in institute A increased by 25% and number of students in institutes B increased by 26%, then what was the respective ratio between number of students in institutes A and B?

a) 25 : 56

b) 24 : 55

c) 24 : 53

d) 25 : 54

Explanation: Ratio of students in 2012 in institutes A and B = 7 : 15

Let number of students in institute A in 2012 = 700

And Number of students in institutes B in 2012 = 1500

25% increase in the number of students in 2013

Now, number of students in Institute A = 700 + 25% of 700 = 875

Number of students in B in 2013 as 26% students increased in B

= 1500 + 26% of 1500 = 1890

Current Ratio of the students,

$$ = \frac{{875}}{{1890}} = 25:54$$

2. A and B together have Rs. 1210. If $$\frac{4}{{15}}$$ of A's amount is equal to $$\frac{2}{{5}}$$ of B's amount, how much amount does B have?

a) Rs. 460

b) Rs. 484

c) Rs. 550

d) Rs. 664

Explanation:

$$\eqalign{ & \frac{4}{{15}}A = \frac{2}{5}B \cr & A = \left( {\frac{2}{5} \times \frac{{15}}{4}} \right)B \cr & A = \frac{3}{2}B \cr & \frac{A}{B} = \frac{3}{2} \cr & A:B = 3:2. \cr & {\text{B's}}\,{\text{share}} = Rs.\left( {1210 \times \frac{2}{5}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,484 \cr} $$

3. Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:

a) 2 : 5

b) 3 : 5

c) 4 : 5

d) 6 : 7

Explanation:

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{third}}\,{\text{number}}\,{\text{be}}\,x \cr & {\text{Then,}}\,{\text{first}}\,{\text{number}} \cr & = 120\% \,{\text{of}}\,x = \frac{{120x}}{{100}} = \frac{{6x}}{5} \cr & {\text{Second}}\,{\text{number}} \cr & = 150\% \,{\text{of}}\,x = \frac{{150x}}{{100}} = \frac{{3x}}{2} \cr & {\text{Ratio}}\,{\text{of}}\,{\text{first}}\,{\text{two}}\,{\text{numbers}} \cr & = {\frac{{6x}}{5}:\frac{{3x}}{2}} = 12x:15x = 4:5 \cr} $$

4.A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B's share?

a) Rs. 500

b) Rs. 1500

c) Rs. 2000

d) None of these

Explanation: Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively

Then, 4x - 3x = 1000

x = 1000

B's share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000

5. Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?

a) 2 : 3 : 4

b) 6 : 7 : 8

c) 6 : 8 : 9

d) None of these

Explanation: Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively

Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x)

$$ = \left( {\frac{{140}}{{100}} \times 5x} \right),$$ $$\left( {\frac{{150}}{{100}} \times 7x} \right)$$ and $$\left( {\frac{{175}}{{100}} \times 8x} \right)$$

$$\eqalign{ & = 7x,\,\,\frac{{21x}}{2}{\text{ and }}14x \cr & {\text{The required ratio}} = 7x:\frac{{21x}}{2}:14x \cr & = 14x:21x:28x \cr & = 2:3:4 \cr} $$

6. 8 litres are drawn from a cask filled with wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of the total solution is 16 : 81. How much wine did the cask hold originally?

a) 24 litres

b) 45 litres

c) 49 litres

d) 44 litres

Explanation: Let the quantity of the wine in the cask originally be x litres.

Final Amount of solute that is not replaced = Initial Amount × $${\left( {\frac{{{\text{Vol}}{\text{. after removal}}}}{{{\text{Vol}}{\text{. after replacing}}}}} \right)^{\text{N}}}$$

Where N = No. of operation done.

Then ratio of wine to total solution in cask after 4 operations,

$$\eqalign{ & 1 \times { {\left( {\frac{{x - 8}}{x}} \right)} ^4} = \frac{{16}}{{81}} \cr & 1 \times {\left\{ {\frac{{ {x - 8} }}{x}} \right\}^4} = {\left( {\frac{2}{3}} \right)^4} \cr & \frac{{ {x - 8} }}{x} = \frac{2}{3} \cr & 3x - 24 = 2x \cr & x = 24\,{\text{litres}} \cr} $$

7. The milk and water in a mixture are in the ratio 7 : 5. When 15 liters of water are added to it, the ratio of milk and water in the new mixture becomes 7 : 8. The total quantity of water in the new mixture is:

a) 35 litres

b) 40 litres

c) 60 litres

d) 96 litres

Explanation:

Milk | : | Water |

7 | : | 5 |

7 | : | 8 |

3 unit |

Remember water is added and not milk, so make milk equal but here milk is already equal

3 units = 15 litres

1 units = 5 litres

8 units = 40 litres

Total quantity of water in the new mixture = 40 litres

8. If x : y = 5 : 2, then (8x + 9y) : (8x + 2y) is :

a) 22 : 29

b) 26 : 61

c) 29 : 22

d) 61 : 26

Explanation:

$$\frac{{\text{x}}}{{\text{y}}} = \frac{5}{2}$$

Means x = 5, y = 2

Putting value of x and y in expression

8 × 5 + 9 × 2 = 58

8 × 5 + 2 × 2 = 44

58 : 44 = 29 : 22

9. Tom is chasing Jerry. In the same interval of time Tom jumps 8 times while Jerry jumps 6 times. But the distance covered by Tom in 7 Jumps is equal to the distance covered by Jerry in 5 Jumps. The ratio of speed of Tom and Jerry is:

a) 48 : 35

b) 28 : 15

c) 24 : 20

d) 20 : 21

Explanation: 7 jumps of Tom = 5 jumps of Jerry

$$\frac{{{\text{Tom}}}}{{{\text{Jerry}}}} = \frac{5}{7}$$

Let Jerry's 1 leap = 7 meter and Tom's 1 leap = 5 meter

Then, ratio of speed of Tom and Jerry

= $$\frac{{8 \times 5}}{{6 \times 7}}$$

= $$\frac{{40}}{{42}}$$

= 20 : 21

10. The ratio of ducks and frogs in a pond is 37 : 39 respectively. The average number of ducks and frogs in the pond is 152. What is the number of frogs in the pond ?

a) 148

b) 152

c) 156

d) 144

Explanation: Ratio of Ducks and Frogs in Pond = 37 : 39

Average of Ducks and Frogs in Pond = 152

So, total number of Ducks and Frogs in the Pond = 2 × 152 = 304

Number of Frogs = $$\frac{{304 \times 39}}{{76}}$$ = 156