Question 1: If , find:

Answer:

Question 2: If , find:

Answer:

Question 3: If , find:

Answer:

*Note: You could have also done this by applying componendo and dividendo.*

Question 4: Find the:

vi) Ratio compounded of the , the and the .

Answer:

vi) Ratio compounded of the , the and the .

Question 5: Find the value of , if:

.

.

.

Answer:

.

Question 6: What quantity must be added to each term of the ratio so that it may become equal to ?

Answer:

Let us add quantity

Therefore

Question 7: Two numbers are in the ratio . If is subtracted from each of them, the ratio between them becomes find the numbers.

Answer:

Let the two numbers be

Given

is subtracted from each of them, then

Substituting

Question 8: If , find ; if both are positive.

Answer:

Dividing both sides by , we get

Let

( not possible as both are positive)

Question 9: Find the:

i) Fourth Proportional to .

ii) Third proportional to

iii) Mean proportional to

Answer:

i) Fourth Proportional to .

Let the fourth proportional be

Therefore

ii) Third proportional to

Let the third proportional be

Therefore

Simplifying

iii) Mean proportional to

Let be the mean proportion.

Therefore

Simplifying

Question 10: Find two numbers such that the mean proportional between them is and third proportional to them is .

Answer:

Let the two numbers be .

Given, mean proportional between them is

Therefore

… … … … … … … … i)

Also given that third proportional to them is

Therefore

… … … … … … … … ii)

Solving i) and ii)

Substituting back in i),

Hence

Question 11: If be unequal is the , prove that is mean proportional between .

Answer:

Given

Question 12: If is the proportional between prove that:

Answer:

is the proportional between

Substituting

RHS

Hence proved.

Question 13: If are in continued proportion. Prove that:

Answer:

Therefore

LHS = RHS

.

Answer:

Therefore

Applying componendo and dividendo

Similarly Applying componendo and dividendo

Question 15: If , prove that: .

Answer:

Applying componendo and dividendo

Answer:

therefore

Hence, LHS = RHS

Answer:

Therefore

Hence LHS = RHS

Question 18: There are members in a student’s council in a school and the ratio of the number of boys to the numbers of girls is . How many more girls should be added to the council so that the ratio of the number of boys to the number of girls may be .

Answer:

Let the number of girls

The the number of boys

Therefore

Therefore Boys and Girls

Let n girls be added

Hence more girls should be added.

Answer:

Question 20: If , find the value of . hence, use componendo and dividendo to find the value of:

Answer:

Divide Numerator and Denominator by , we get

Divide Numerator and Denominator by , we get

i) .

Answer:

Applying componendo and dividendo

Applying componendo and dividendo

Answer:

If are in continued proportion, then

Applying componendo and dividendo

Squaring both sides

Substituting

Answer:

Applying componendo and dividendo

Simplifying

Square both sides

Applying componendo and dividendo

Simplifying

Answer:

Applying componendo and dividendo

Simplifying, we get

Applying componendo and dividendo

Question 25: Using componendo and dividendo, find the value of :

Answer:

Applying componendo and dividendo

Simplifying

Square both sides

Simplifying we get

Answer:

Applying componendo and dividendo

Simplify

Now square both sides

Simplifying