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.