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Question 1: Find the fourth proportional to:

i)

ii)

Answer:

i) Let the proportion be

Therefore

Hence

ii) Let the proportion be

Therefore

Hence

Question 2: Find the third proportional to:

i)

ii)

Answer:

i) Let the proportion be

Therefore

Hence

ii) Let the proportion be

Therefore

Hence

Question 3: Find the mean proportional between:

i)

ii)

iii)

Answer:

i) Let the mean proportional be

Therefore

ii) Let the mean proportional be

Therefore

iii) Let the mean proportional be

Therefore

Question 4: If is the means proportion between ; find the value of .

Answer:

Given is the means proportion between

Therefore

Question 5: what least number must be added to each of the numbers so that the resulting numbers are in proportion?

Answer:

Let the number added be

Therefore

Question 6: What least number must be added to each of the numbers to make them proportional. [2005, 2013]

Answer:

Let the number added be

Therefore

Question 7: What number must be added to each of the numbers so that the resulting numbers may be in continued proportion?

Answer:

Let the number added be

Therefore

Question 8: What least number must be subtracted from each of the numbers so that the remainders are in continued proportion?

Answer:

Let the number subtracted be

Therefore

Question 9: If is the mean proportional between ; show that is the mean proportional between .

Answer:

Since is the mean proportional between

Let the mean proportional between by

Therefore

. Hence proved.

Question 10: If is the mean proportional between . show that: .

Answer:

Given is the mean proportional between

Therefore

. Hence proved.

Question 11: If three quantities are in continued proportion; show that the ratio of the first to the third is the duplicate ratio of the first to the second.

Answer:

Let the three quantities by

If they are in proportion, then we have

Now we have to prove that

Substituting . Hence proved.

Question 12: If is the mean proportional between , prove that: .

Answer:

Given is the mean proportional between

Therefore

. Hence proved.

Question 13: Give four quantities are in proportion. Show that:

Answer:

Given are in proportion

To prove

LHS

RHS

Hence . Hence proved.

Question 14: Find two numbers such that the mean proportional between them is and the third proportional to them is .

Answer:

Let the two numbers be

Therefore

If is the third proportion

Hence

Question 15: Find the third proportional to

Answer:

Let the third proportion by

Therefore

Question 16: If ; then show that:

Answer:

Given

Therefore

Multiplying both sides by

Adding to both sides

or

Hence proved.

Question 17: If ; then prove that: .

Answer:

Given

Given

or

Question 18: If , prove that each of the given ratio is equal to:

i)

ii)

iii)

iv)

Answer:

Given

i)

ii)

iii)

iv)

Question 19: If are in proportion, prove that:

i)

ii)

Answer:

Given

i)

LHS

RHS

Hence LHS = RHS.

ii)

LHS

RHS

Hence LHS = RHS.

Question 20: If , prove that:

Answer:

Given

Therefore

LHS

RHS

Hence LHS = RHS.