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