Question 1: If , find:

Answer:

Given

Question 2: If , find:

Answer:

Given

Question 3: If , find:

Answer:

or

or

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

Question 4: Find the:

i) Duplicate ratio of

ii) Triplicate ratio of

iii) Sub-duplicate ratio of

iv) Sub-triplicate ratio of

v) Reciprocal ratio of

vi) Ratio compounded of the duplicate ratio of , the reciprocal ratio of and the sub-duplicate ratio of .

Answer:

i) Duplicate ratio of

ii) Triplicate ratio of

iii) Sub-duplicate ratio of

iv) Sub-triplicate ratio of

v) Reciprocal ratio of

vi) Ratio compounded of the duplicate ratio of , the reciprocal ratio of and the sub-duplicate ratio of .

Duplicate ratio of

Reciprocal ratio of

Sub-duplicate ratio of

Compound ratio

Question 5: Find the value of , if:

i) is the duplicate ratio of .

ii) is the sub-duplicate ratio of .

iii) is the sub-triplicate ratio of .

Answer:

i) is the duplicate ratio of

ii) is the sub-duplicate ratio of .

iii) is the sub-triplicate ratio of

or

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

or

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

Therefore if is subtracted from each of them, then

Substituting

and

Hence

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

Answer:

Given

Dividing both sides by , we get

Let

( not possible as both are positive)

Hence

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

or … … … … … … … … i)

Also given that third proportional to them is

Therefore

or … … … … … … … … ii)

Solving i) and ii)

Substituting back in i),

Hence

Question 11: If be unequal and is the duplicate ratio of , prove that is mean proportional between .

Answer:

Given

or

Question 12: If is the proportional between prove that:

Answer:

Given is the proportional between

or

LHS

Substituting

RHS

Hence proved.

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

Answer:

Given

or

Therefore

Now, LHS

RHS

LHS = RHS

Question 14: If , find the value of: .

Answer:

Given

Therefore

Applying componendo and dividendo

Similarly Applying componendo and dividendo

Adding

or

Question 15: If , prove that: .

Answer:

Given

or

Applying componendo and dividendo

or

Question 16: If , show that

Answer:

Given

therefore

LHS

RHS

Hence, LHS = RHS

Question 17: If , prove that:

Answer:

Given

Therefore

LHS

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 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

Therefore

or

or

Hence more girls should be added.

Question 19: If , prove that:

Answer:

Given

LHS

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

i) ii)

Answer:

Given

or

i)

Divide Numerator and Denominator by , we get

ii)

Divide Numerator and Denominator by , we get

Question 21: If , use properties of proportion to find:

i) ii) .

Answer:

i) Given

Applying componendo and dividendo

ii)

Applying componendo and dividendo

Question 22: If are in continued proportion, prove that: . [2010]

Answer:

If are in continued proportion, then

Applying componendo and dividendo

Squaring both sides

Substituting

Question 23: Given . Use componendo and dividendo to prove that: . [2010]

Answer:

Given

Applying componendo and dividendo

Simplifying

Square both sides

Applying componendo and dividendo

Simplifying

Question 24: If , find:

i) ii) [2014]

Answer:

i) Given

Applying componendo and dividendo

Simplifying, we get

ii)

Applying componendo and dividendo

Question 25: Using componendo and dividendo, find the value of : [2011]

Answer:

Given

Applying componendo and dividendo

Simplifying

Square both sides

Simplifying we get

Question 26: If , using properties of proportion show that: [2012]

Answer:

Given

Applying componendo and dividendo

Simplify

Now square both sides

Simplifying

or