Question 1: Find the fourth proportional to:

Answer:

i) Let the proportion be

Therefore

ii) Let the proportion be

Therefore

Question 2: Find the third proportional to:

Answer:

i) Let the proportion be

Therefore

ii) Let the proportion be

Therefore

Question 3: Find the mean proportional between:

Answer:

i) Let the mean proportional be

ii) Let the mean proportional be

iii) Let the mean proportional be

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

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

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

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

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

. Hence proved.

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

Answer:

Given is the mean proportional between

. 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

. Hence proved.

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

Answer:

Given are in proportion

To prove

. 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

If is the third proportion

Answer:

Let the third proportion by

Answer:

Given

Multiplying both sides by

Adding to both sides

or

Hence proved.

.

Answer:

Given

or

Answer:

Question 19: If are in proportion, prove that:

Answer:

Hence LHS = RHS.

Hence LHS = RHS.

Answer:

Hence LHS = RHS.