Question 1:

(i) lf find

(ii) If find the value of

Answer:

(i)

[ICSE Board 2002]

Answer:

Question 3: Two numbers are in the ratio 3 : 5. If 8 is added to each number, the ratio becomes 2 : 3. Find the numbers.

Answer:

Since, the ratio between the numbers is

Therefore if one number is the other number is

Question 4:

(i) What quantity must be added to each term of the ratio 8 : 15 so that it becomes equal to 3:5?

(ii) What quantity must be subtracted from each term of the ratio a : b so that it becomes c : d?

Answer:

(i) Let be added to each term of the ratio

(ii) Let be subtracted, then :

Question 5: The work done by men in days and the work done by men in days are in the ratio Find the value of

Answer:

Assuming that all the men do the same amount of work in one day and one day work of each man = 1 unit; we get :

Amount of work done by men in days

= amount of work done by men in one day

units of work.

Similarly, amount of work done by men in days.

= amount of work done by men in one day.

units of work.

According to the given statement:

Therefore

Question 6: When the fare of a certain journey by an airliner was increased in the ratio 5:7 the cost of the ticket for the journey became Rs. 1421. Find the increase in the fare.

Answer:

According to the given statement :

Question 7: In a regiment, the ratio of number of officers to the number of soldiers was 3 : 31 before a battle. In the battle 6 officers and 22 soldiers were killed. The ratio between the number of officers and the number of soldiers now is 1 : 13. Find the number of officers and soldiers in the regiment before the battle. [ICSE Board 1992]

Answer:

Before the battle:

Let the number of officers be the number of soldiers

After the battle:

The number of officers and the number of soldiers

Given

Therefore the number of officers before the battle

And the number of soldiers before the battle

Answer:

Answer:

Question 10: Find the compound ratio of :

Answer:

Question 11: Find the ratio compounded of the duplicate ratio of 5 : 6, the reciprocal ratio of 25 : 42 and the sub-triplicate ratio of 216 : 343.

Answer:

Question 12: Find:

(i) the fourth proportional to 3, 6 and 4.5

(ii) the mean proportional between 6.25 and 0.16

(iii) the third proportional to 1.2 and 1.8

Answer:

(i) Let the fourth proportional to and be

(ii) Let the mean proportional between and be be

Question 13: Quantities and are in continues proportion; find the values of a and b.

Answer:

Question 14: What number should be subtracted from each of the numbers 23, 30, 57 and 78; so that the remainders are in proportion. [ICSE Board 2004 ]

Answer:

Let the number subtracted be

Question 15: What should be added to each of the numbers 13, 17 and 22 so that the resulting numbers are in continued proportion.

Answer:

Let the number added be

Question 16: If and are in continued proportion: prove that and are in proportion.

Answer:

Given and are in continued proportion.

Therefore and are in continued proportion.

Answer:

Question 18: If and is the duplicate ratio of and prove that is the mean proportion between and

Answer:

is mean proportion between

Answer:

and are in proportion

Question 20: If q is the mean proportional between p and r, prove that:

Answer:

Since is the mean proportional between and

Answer:

Question 22: If a, b, c and d are in proportion, prove that :

Answer:

Hence LHS = RHS. Hence proved.

Hence LHS = RHS. Hence proved.

Question 23: 6 is the mean proportion between two numbers and and 48 is third proportion to and Find the number. [ICSE Board 2011]

Answer:

Since, is mean proportional between and

and, is third proportional to and

The required nos. are and

Answer:

Applying componendo and dividendo:

Answer:

Applying componendo and dividendo:

Answer:

Applying componendo and dividendo:

Answer:

Now apply componendo and dividendo

Again

Now apply componendo and dividendo

Question 28: If prove that:

Answer:

Hence proved.

Question 29: If and are in continued proportion, prove that :

Answer:

and are in continued proportion

Also,

Therefore

Question 30: Using the properties of proportion, solve the following equation for

Answer:

Applying componendo and dividendo, we get :

Again, applying componendo and dividendo, we get :

Answer:

Given:

Applying componendo and dividendo, we get :

Squaring both sides, we get

Applying componendo and dividendo, we get :

Hence proved.