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