CH 6 : RATIO AND PROPORTION MATHEMATICS ICSE CLASS 10TH 2023-2034

CHAPTER 6
RATIO AND PROPORTION

QUESTION 121.
PREVIOUS EXAM QUESTIONS
Main 2020
2a + 1 + 2a - 1
123.If x = 2a + 1 - 2a - 1
, prove that x2 – 4ax + 1 = 0.
QUESTION 122. Comp 2020
124.If b is the mean proportional between a and c ,
prove that
a2 - b2 + c 2 = b 4
MULTIPLE CHOICE QUESTIONS a - b-2 + c-2
-2

QUESTION 123. Main 2019

QUESTION 114. Main 2023
116.The mean proportional between 4 and 9 is: 125.The numbers K + 3 , K + 2 , 3K - 7 and 2K - 3
(a) 4 (b) 6 are in proportion. Find K.
(c) 9 (d) 36 QUESTION 124. Main 2019
126.Using properties of proportion solve for x, given
QUESTION 115. Main 2021 Sem I 5x + 2x - 6
117.The mean proportional between 9 and 16 is : 5x - 2x - 6
= 4.
(a) 25 (b) 144 QUESTION 125. Comp 2019
(c) 7 (d) 12 127.If b is the mean proportion between a and c ,
show that :
a 4 + a2 b2 + b 4 = a2
b 4 + b2 c2 + c 4 c2
QUESTION 126. Main 2018
TWO MARKS QUESTIONS 128.Using properties of proportion, solve for x . Given
that x is positive.
QUESTION 116. Main 2002
118.If a | b = 5 | 3 , then find ^5a + 8b h | ^6a - 7b h . 2x + 4x2 - 1 = 4
2x - 4x2 - 1
QUESTION 127. Comp 2018
129.If a
b+c= c +b a = c
a+b , show that each ratio is equal
to 1
2or - 1.
THREE MARKS QUESTIONS QUESTION 128. Main 2017
130.If b is the mean proportional between a and c ,
QUESTION 117. Main 2023 4 2 2 4 2
119.What number must be added to each of the show that a 4 + a2 b2 + b4 = a2 .
b +b c +c c
numbers 4, 6, 8, 11 in order to get the four
QUESTION 129. Comp 2017
numbers in proportion? 3 3 2
3ab2 = x + 3xy , prove that x = y .
131.If a +2
3a b + b 3
3x y + y 3
2 a b
QUESTION 118. Main 2023
120.Using compodendo and dividendo solve for x: QUESTION 130. Main 2016
2x + 2 + 2x - 1 132.If ^3a + 2b h | ^5a + 3b h = 18 | 29 . Find a | b .
=3
2x + 1 - 2x - 1 QUESTION 131. Main 2016
QUESTION 119. y 3
y3 3
3xyz
133.If x = = z , show that x3 = 3 = z3 =
Comp 2023
121.If 6 is the mean proportion between two numbers .
a b c a b c abc
x and y and 48 is the third proportional to x and QUESTION 132. Comp 2016
y , find the numbers. 134.What quantity must be added to each term of the
ratio (a + 2b) : (a - 2b) to make it equal to the
QUESTION 120. Main 2020
122.Using properties of proportion find x : y, given duplicate ratio of (a + b) : (a - b)?
3
x2 + 2x = y + 3y QUESTION 133. Main 2015
2x + 4 3y + 9 135.If a , b and c are in continued proportion, prove
that

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ICSE CLASS 10TH 2023-2034 MATHEMATICS CH 6 : RATIO AND PROPORTION

^a + b + c h^a - b + c h = a + b + c QUESTION 145.

2 2 2
Main 2010
147.If x , y and z are in continued proportion, then
QUESTION 134. Main 2015 ^x + y h2 x
3
y3 + 27y prove that = .
136.Given x +2 12x = 2 , using componendo ^y + z h2 z
6x + 8 9y + 27
QUESTION 146. Main 2010
and dividendo, find x | y . a2 + b2 + a2 - b2 , show that
148.If x =
QUESTION 135. Main 2014 a2 + b2 - a2 - b2
137.If ^x - 9h | ^3x + 6h is the duplicate ratio of 4 | 9 b 2 x 2 - 2a 2 x + b 2 = 0 .
, then find the value of x .
QUESTION 147. Comp 2010
QUESTION 136. Main 2014 149.If a, b, c, d are in continued proportion, prove
2 2
x +y
138.If = 17 , then find the value of that:
x2 - y2 8
x3 + y3 (a + b + c) (b + c + d) = ab + bc + cd
(i) x | y (ii) 3 .
x - y3
QUESTION 137. Main 2013, Main 2005 QUESTION 148. Main 2009
139.What number must be added to each of 150.What least number must be added to each of the
the numbers 6, 15, 20 and 43 to make them numbers 5, 11, 19 and 37, so that they are in
proportional? proportion?
QUESTION 138. Main 2013 QUESTION 149. Main 2009
140.Using the properties of proportion, solve for x , 3 2
4 151.Given that a3 + 3ab = 63 . Using componendo
given x +2 1 = 17 . b + 3a b 62
2

2x 8 and dividendo, find a | b .

QUESTION 139. Comp 2013
141.Solve the following equation for x, using properties QUESTION 150. Comp 2009
4 152.If a, b, c, d are in continued proportion, prove
of proportion: x +2 1 = 17
2x 8 that:
QUESTION 140. Main 2012 (b - c) 2 + (c - a) 2 + (d - b) 2 = (a - d) 2
142.The monthly pocket money of Ravi and Sanjeev
are in the ratio 5 | 7 . Their expenditures are in
QUESTION 151. Main 2008
the ratio 3 | 5 . If each saves Rs. 80 every month,
153.If 8a - 5b = 8a + 5b , then prove that a = c .
then find their monthly pocket money. 8c - 5d 8c + 5d b d
Since each saves money Rs 80 and QUESTION 152. Main 2006
3x + 5y 7
154.If = , then find x | y .
Expenditure = Pocket money - Saving money 3x - 5y 3
Expenditure of Ravi = 5x - 80 QUESTION 153. Comp 2008
155.If a, b, c, d are in continued proportion, prove
and Expenditure of Sanjeev = 7x - 80 that:
According to the question, their expenditures are (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd) 2
in the ratio 3 | 5 . Therefore
QUESTION 154. Main 2007
QUESTION 141. Comp 2012 a + 3b + a - 3b , show that
1/3
(p + 1) + (p - 1) 1/3 156.If y =
143.If y = , find the value of a + 3b - a - 3b
(p + 1) 1/3 - (p - 1) 1/3
3by2 - 2ay + 3b = 0
y3 - 3py2 + 3y - p
QUESTION 155. Comp 2007
QUESTION 142. Main 2011 157.If b + c - a = c + a - b = a + b - c , then show
144.Using componendo and dividendo, find the value y+z-x z+x-y x+y-z
of x . that each ratio is equal to a =b =c.
x y z
QUESTION 143. Main 2011 QUESTION 156. Main 2004
145.6 is the mean proportion between two numbers x 158.What number should be subtracted from each of
and y and 48 is the third proportional of x and y the following numbers 23, 30, 57 and 78, so that
. Find the numbers. the remainders are in proportion?
QUESTION 144. Comp 2011
146.If a, b, c, d are in continued proportion, prove
that:
ab - bc + cd = (a - b + c) (b - c + d)
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CH 6 : RATIO AND PROPORTION MATHEMATICS ICSE CLASS 10TH 2023-2034

QUESTION 157. Main 2001 QUESTION 168. Main 2003

159.Two numbers are in the ratio of 3 | 5 . If 8 is 171.Solve the following equation for x, using properties
added to each number, then the ratio becomes 1 - px 1 + qx
of proportion: =1
2 | 3 . Find the numbers. 1 + px 1 - qx
QUESTION 169. Comp 2003
QUESTION 158. Comp 2000 172.Find the two numbers such that their mean
160.If x and y be unequal and x : y is the duplicate proportional is 24 and the third proportional is
ratio of x + z and y + z, prove that z is mean 1536.
proportional between x and y.
161.Given, a = c , prove that 3a - 5b = 3c - 5d . QUESTION 170.
173.Find the numbers such that the mean proportional
Main 2001
b d 3a + 5b 3c + 5d
between them is 14 and third proportional to
them is 112.
QUESTION 171. Comp 2002
FOUR MARK QUESTIONS 174.If a, b, c are in continued proportion, prove that:
(a + b + c) 2 : (a2 + b2 + c2) : : (a + b + c) : (a - b + c)
QUESTION 159. Main 2017
162.If 7m + 2n = 5 , use properties of proportion to QUESTION 172. Main 2002
7m - 2n 3 175.Solve the following equation for x, using properties
find
(i) m | n . (ii) m2 + n2 . of proportion: x + 4 + x - 10 = 5
x + 4 - x - 10 2
m2 - n2
QUESTION 160. Comp 2015 END
163.If b + c - a = c + a - b = a + b - c , then prove
y+z-x z+x-y x+y-z
that each ratio is equal to a = b = c .
x y z
QUESTION 161. Comp 2014

164.If x = 4 6 , find the value of

x + 2 2 2++x + 32 3
x-2 2 x-2 3
QUESTION 162. Main 2013
165.Solve the following equation for x, using properties
of proportion: x + 1 + x - 1 = 4x - 1
x+1- x-1 2
QUESTION 163. Main 2012

166.If x = a + 1 + a - 1 , then using the properties

a + 1 -show
of proportion, a -that
1 x2 - 2ax + 1 = 0 .

QUESTION 164. Main 2007

167.If x = a + 3b + a - 3b , then prove that

3bx2 - 2axa++33bb=-0 . a - 3b
QUESTION 165. Comp 2006
by + cz cz + ax = ax + by , then show that
168.If 2 2 =
b +c c2 + a2 a2 +
yb
2

each ratio is equal to x = = z .

a b c
QUESTION 166. Comp 2005
169.If a, b, c are in continued proportion, prove that:
(a + b + c) (a - b + c) = a2 + b2 + c2

QUESTION 167. Comp 2004

170.If a, b, c are in continued proportion, prove that:
(a + b) : (b + c) : : a2 (b - c) : b2 (a - b)

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