Question 1:
(i) lf find
(ii) If find the value of
Answer:
(i)
[ICSE Board 2002]
Answer:
Question 3: Two numbers are in the ratio 3 : 5. If 8 is added to each number, the ratio becomes 2 : 3. Find the numbers.
Answer:
Since, the ratio between the numbers is
Therefore if one number is the other number is
Question 4:
(i) What quantity must be added to each term of the ratio 8 : 15 so that it becomes equal to 3:5?
(ii) What quantity must be subtracted from each term of the ratio a : b so that it becomes c : d?
Answer:
(i) Let be added to each term of the ratio
(ii) Let be subtracted, then :
Question 5: The work done by men in
days and the work done by
men in
days are in the ratio
Find the value of
Answer:
Assuming that all the men do the same amount of work in one day and one day work of each man = 1 unit; we get :
Amount of work done by men in
days
= amount of work done by men in one day
units of work.
Similarly, amount of work done by men in
days.
= amount of work done by men in one day.
units of work.
According to the given statement:
Therefore
Question 6: When the fare of a certain journey by an airliner was increased in the ratio 5:7 the cost of the ticket for the journey became Rs. 1421. Find the increase in the fare.
Answer:
According to the given statement :
Question 7: In a regiment, the ratio of number of officers to the number of soldiers was 3 : 31 before a battle. In the battle 6 officers and 22 soldiers were killed. The ratio between the number of officers and the number of soldiers now is 1 : 13. Find the number of officers and soldiers in the regiment before the battle. [ICSE Board 1992]
Answer:
Before the battle:
Let the number of officers be the number of soldiers
After the battle:
The number of officers and the number of soldiers
Given
Therefore the number of officers before the battle
And the number of soldiers before the battle
Answer:
Answer:
Question 10: Find the compound ratio of :
Answer:
Question 11: Find the ratio compounded of the duplicate ratio of 5 : 6, the reciprocal ratio of 25 : 42 and the sub-triplicate ratio of 216 : 343.
Answer:
Question 12: Find:
(i) the fourth proportional to 3, 6 and 4.5
(ii) the mean proportional between 6.25 and 0.16
(iii) the third proportional to 1.2 and 1.8
Answer:
(i) Let the fourth proportional to and
be
(ii) Let the mean proportional between and
be be
Question 13: Quantities and
are in continues proportion; find the values of a and b.
Answer:
Question 14: What number should be subtracted from each of the numbers 23, 30, 57 and 78; so that the remainders are in proportion. [ICSE Board 2004 ]
Answer:
Let the number subtracted be
Question 15: What should be added to each of the numbers 13, 17 and 22 so that the resulting numbers are in continued proportion.
Answer:
Let the number added be
Question 16: If and
are in continued proportion: prove that
and
are in proportion.
Answer:
Given and
are in continued proportion.
Therefore and
are in continued proportion.
Answer:
Question 18: If and
is the duplicate ratio of
and
prove that
is the mean proportion between
and
Answer:
is mean proportion between
Answer:
and
are in proportion
Question 20: If q is the mean proportional between p and r, prove that:
Answer:
Since is the mean proportional between
and
Answer:
Question 22: If a, b, c and d are in proportion, prove that :
Answer:
Hence LHS = RHS. Hence proved.
Hence LHS = RHS. Hence proved.
Question 23: 6 is the mean proportion between two numbers and
and 48 is third proportion to
and
Find the number. [ICSE Board 2011]
Answer:
Since, is mean proportional between
and
and, is third proportional to
and
The required nos. are and
Answer:
Applying componendo and dividendo:
Answer:
Applying componendo and dividendo:
Answer:
Applying componendo and dividendo:
Answer:
Now apply componendo and dividendo
Again
Now apply componendo and dividendo
Question 28: If prove that:
Answer:
Hence proved.
Question 29: If and
are in continued proportion, prove that :
Answer:
and
are in continued proportion
Also,
Therefore
Question 30: Using the properties of proportion, solve the following equation for
Answer:
Applying componendo and dividendo, we get :
Again, applying componendo and dividendo, we get :
Answer:
Given:
Applying componendo and dividendo, we get :
Squaring both sides, we get
Applying componendo and dividendo, we get :
Hence proved.