Question 1: If are in continued proportion, prove that:
. [2010]
Answer:
If are in continued proportion, then
Applying componendo and dividendo
Squaring both sides
Substituting
Question 2: 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 3: If
, find:
i) ii)
[2014]
Answer:
i) Given
Applying componendo and dividendo
Simplifying, we get
ii)
Applying componendo and dividendo
Question 4: Using componendo and dividendo, find the value of :
[2011]
Answer:
Given
Applying componendo and dividendo
Simplifying
Square both sides
Simplifying we get
Question 5: If
, using properties of proportion show that:
[2012]
Answer:
Given
Applying componendo and dividendo
Simplify
Now square both sides
Simplifying
or
Question 6: Given,
, prove that:
[2000]
Answer:
Given
By componendo and dividendo
By Alternendo
Question 7: If , prove that:
. [2007]
Answer:
Given
Applying componendo and dividendo
Squaring both sides
Applying componendo and dividendo once again
simplifying
Hence proved.
Question 8: Using the properties of proportion, solve for Given:
. [2013]
Answer:
Given
Applying componendo and dividendo
Taking the square root of both sides
Question 9: 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 10: The monthly pocket money of Ravi and Sanjeev are in the ratio of 5:7 Their expenditures are in the ratio of 3:5. If each saves Rs. 80 per month, find their monthly pocket money. [2012]
Answer:
Let monthly pocket of Rave and Sanjeev by respectively.
Substituting
Substituting
Question 11: If is the triplicate ratio of
, find
. [2014]
Answer:
Question 12: If , find
. [2002]
Answer:
Given
or
Now substituting
Hence
Question 13: The work done by men in
days and the work done by
men in
days are in the ratio
. Find the value of
. [2003]
Answer:
Amount of work done by men in
days
Similarly, amount of work done by men in
days
Given
Simplifying
Therefore
Question 14: What number should be subtracted from each of the numbers ; so that the ratios are in proportion. [2004]
Answer:
Let the number subtracted
Therefore
Simplifying
Question 15: is the mean proportion between two numbers
and
and
is the third proportion to
and
. Find the numbers. [2011]
Answer:
Given is the mean proportion between two numbers
Therefore
… … … … … … i)
Also given is the third proportion to
Therefore
… … … … … … ii)
Solving i) and ii)
Hence
Hence the numbers are .
Question 16: if
, prove that
. [2008]
Answer:
Given
or
Applying Componendo and Dividendo
or
. Hence proved.