Chapter No 2: Ratio and Proportion What is ratio? ✓ A ratio is the comparison of two or more quantities of the same kinds. ✓ A ratio may be written as “a : b” or “a/b” or “a is to b” or “a to b”. ✓ Ratios are represented mathematically by separating each quantity with a colon(:) for example, the ratio 2:3, which is read as the ratio "two to three". ✓ The quantities separated by colons are sometimes called terms. ✓ The quantities to be compared must be measured in same units. ✓ The order of ratio is important. ✓ A ratio has no units. Topic No 1. Express the following Ratio in the simplest form: Conversion scale: ✓ 1 hour = 60 minutes, ✓ 1 m = 100 c.m, ✓ 1 minute = 60 seconds ✓ 1 feet = 12 inches ✓ 1 kg. = 1000 gm, ✓ 1 year = 365 days, ✓ Rs 1 = 100 paisa, ✓ 1 month = 30 days ✓ 1 dozen = 12 pieces ✓ 1 day = 24 hours, Simplify the following ratios: Q no 1. 10:20 Q no 2. 240:320 Q no 3. 15:35:45 Q no 4. 120:220:360 Q no 5. 480000:360000 Q no 6. Rs 3 : Rs 9 Q no 7. Rs 2 : 40 paisa Q no 8. RS 2 and 40 paisa : 60 paisa Q no 9. 2 eggs : 2 dozen eggs Q no 10. 4 bananas: 4 dozen bananas Q no 11. 2 Kg : 25 gm Q no 12. 2.5 Kg : 500 gm Q no 13. 50 grams: 20 gm: 2 Kg Q no 14. 4 weeks : 4 days Q no 15. 4 months : 12 months Q no 16. 2 months : 20 days Q no 17. 3 feet: 4 inches Q no 18. 1.3 : 3.9 Q no 19. 2.5 : 7.5 Q no 39. *2:1, *2:3 Q no 40. *2:45 Q no 41. *3:2 Q no 42. *3:4 *1:4 Q no 43. *2:1 Q no 44. *1:3 *1:5 Q no 45. *20:1 Q no 46. *5:20 Q no 47. 3:85 Q no 48. 1:4 Q no 49. *1:15 Q no 50. *1:15 Q no 51. *2:9 *9:2 Q no 20. Q no 21. Q no 22. Q no 23. Q no 24. Q no 25. Q no 26. Q no 27. Q no 28. Q no 29. Q no 30. Q no 31. Q no 32. Q no 33. Q no 34. Q no 35. Q no 36. Q no 37. Q no 38. 3 months : 2 years 4 m : 40 cm 5 cm : 2 m 3 m and 15 cm : 15 cm 1/3: 4/3:5/3 1/5:2/3:3/4 77 days : 2 weeks 250 gm : 45kg: 450 gm 3/2:1/4:3/8 12 hours : 36 hours 20% : 60% 2 hours : 20 minutes 396000:150000:75000 15% to 135% 4 min : 3 hours 2100 : 3500 9/81 5min : 30 sec 30 min to 3 hours & 30 sec. 5 Years MCQs The ratio of 30 inches to 5 feet is 2018 *1:2 *3:60 The ratio of 2 hours to 45 minutes is: 2017, 2012P *60:45 *8:3 *4:3 In a certain college, there are 360 boys and 240 girls. The ratio of boys to girls is: *2:3 *5:3 *3:5 2014P The ratio of 12 days to 8 weeks 2 *3:14 *None of these 2013R The ratio of 5 feet to 30 inches is: *1:2 *1:6 *6:1 2013P, 2007 The ratio of 30 Min to 1.5 hours *1:7 *1:9 2012R The ratio of 1 Kg to 50 gm 2011R *1;20 *1:50 *51:1 The ratio of 50 gm to 2KG is 2008 *1:40 *50:2 *none The ratio of 85 Paisa to Rs 3 2011P 17:60 60:17 85:3 The ratio of 50 Paisa to Rs 2: 2009 2:50 4:1 50:2 The ratio of 5 Rupees and 75 paisa is: 2006 *15:1 *3:20 *20:3 The ratio of 3 Rupees and 80 paisa is: 2003R *15:4 *3:20 *20:3 The ratio of 40 minutes to 2 hours and 20 minutes is 2003P *4:2 *none Prepared By: Syed Adnan 1 AIMS Topic No 2. Continues Ratio Example If a:b = 3:4 and b:c 2:3 then find A:B:C A:B:C 3 4 2 3 6 8 12 Then Divide by 2 6 3:8 4 :12 6 3:4:6 ans Find the ratio i.e a:b:c when Q no 52. a:b = 3:4 and b:c 2:3 Q no 53. a:b = 4:6 and b:c 3:7 Q no 54. a:b 12:4 and b:c 3:6 Q no 55. a:c = 5:3 and a:b = 7:2 Q no 56. a:c = 9:7 and a:b 3:5 Q no 57. a:c = 2:8 and a:b = 1:3 Q no 58. X:Y =4:5 and Y:Z = 2:3 Q no 59. X:Z = 5: 4 and X:Z = 3: 6 Q no 60. Q no 61. *3:7:5 Q no 62. Q no 63. Find the ratio Q no 64. Q no 65. Q no 66. Q no 67. Q no 68. 2014R If a:b = 13:4 and b:c = 5:6 find a:b:c 2017, 2013P If A: B = 3:5 and B:C is 5: 7 then A:B:C is *7:5:3 *3:5:7 *1:1:1 2011R If A: B = 5:7 and B:C is 3: 14 then A:B:C is 2007 If A: B = 5:6 and B:C is 8: 15 then A:B:C is a:b:c:d when a:b = 2:7 and b:c 2:4 and b:d= 2:8 a:b = 1:5 and b:c 3:8 and b:d= 4:9 a:c = 4:8 and a:b = 1:5 and a:d 2:5 a:c = 3:5 and a:b = 1:5 and b:d 5:15 a:c = 2:4 and a:b = 3:6 and c:d 7:9 Prepared By: Syed Adnan 2 AIMS Topic No 3. Distribution through ratio Example Distribute Rs 9,000 between Adnan and Salman in the Ratio of 1:2 Solution Adnan : Salman 1 : 2 Sum of Ratio = 1+ 2 = 3 Adnan’s Share = Total Amount x Adnan’s Share = 9000 x Ratio . Sum of Ratio 1. 3 Adnan’s Share = Rs 3,000 Salman’s Share = 9000 x 2 . 3 Salman’s Share = Rs 6,000 2014R Distribute an a amount of Rs 68,000 among three person in the ratio of 1:5:4. 2012R Q no 70. Distribute an a amount of Rs 7800 among three person in the ratio of 1:5:4. 2011P Q no 71. Distribute an a amount of Rs 50,000 among three person A,B & C in the ratio of 10:5:1. 2004 Q no 72. A, B and C are business partners. They earn a profit of Rs 24,000. Find the shares of each partners in the profit if the profit can be shared in the ratio 2:5:3. 2008 Q no 73. Divide Rs. 100, 000 between Noman and Atif such that Noman gets double of what Atif gets. 2018 Q no 74. Distribute profit of Rs 50,000 among three partners Jawad, Areeba and Hamza such that jawwad and Areeba Get the profit in the ratio of 5:3 whereas Areeba and Hamza get 3:2 2017 Q no 75. Distribute Rs 93,000 among three friends A,B,C such that A and B Get the profit in the ratio of 1:3 whereas B and C get 4:5. 2014P Q no 76. Distribute Rs 36000 profit among three businessmen A, B, C such that A:B= 3:2 and B:C = 4:5. 2013R Q no 77. Three bankers A, B, C invest Rs 20,000 ; 25,000 and Rs 30,000 respectively. Find the share of each partner if they got profit of Rs 6,000/= 2003R Q no 78. A, B and C invested jointly a sum of Rs 40,000, Rs 50,000 and Rs 10,000 respectively in a computer business. If the profit was Rs 35,000 at the end of year, Find the share of the profit of each investor. 2003P Q no 79. Three men took part in a business. They put in Rs 2,000, Rs 4000 and Rs 1500 as their capitals. The profit was divided among them in the same ratio as the amount invested by them. If the profit was Rs 300 what did each get? 2002P Q no 80. A, B and C invested jointly a sum of Rs 7,000, Rs 14,000 and Rs 9,000 respectively in a business. If the profit was Rs 9900 at the end of year, Find the share of the profit of each investor. 2013P Q no 81. Distribute an amount of Rs 2400 among three persons, A,B and C in the ratio of 1/10, 2/5, ½ 2011R Q no 82. Distribute an amount of Rs 1600 among three persons, A,B and C in the ratio of 1/10, 2/5, ½ 2007,2002R Q no 83. Distribute an amount of Rs 5100 among three persons, A,B and C in the ratio of 1/2, 2/3, ¼ Q no 69. Prepared By: Syed Adnan 3 AIMS What is proportion? A proportion is a statement that two ratios are equal. Or equality of two ratios is called proportion. a:b::c:d where a:d is called extreme and b:c is called means. There are three types of proportion 1. Direct Proportion When one variable increase and another is also increase or when one variable decreases and other also decreases then it is called direct proportion. In this type of proportion, both variables moves in same direction. 2. Indirect or inverse Proportion When one variable increases but other decreases then its called Inverse proportion. In this type of proportion, both variables moves in opposite direction. 3. Combined/ Compound proportion In this type of proportion there are more than two variables and all variables are may be direct and indirect proportion Topic No 4. Direct Proportion Example The Cost of 5 pen is Rs 60 then find the cost of 10 pen Pen 5 10 Cost 60 𝑥 a : b :: c : d 5: 60 :: 10 : 𝑥 Formula a.d = b.c 5 x 𝑥 = 60 x 10 5𝑥 = 600 𝑥 = 600/5 𝑥 = Rs 120 . 2010 The Cost of 10 chairs is Rs 3250. Find the cost of 7 chairs. 2002 Q no 85. 125 Kg of rice cost Rs 3500 Kg of rice can be bought for Rs 2380 2006 Q no 86. A typist can type 350 words in 7 minutes. Then the number of words typed in 5 minutes will be? 2005 Q no 87. At present the ages of khalid and sohail are in the ratio of 4:3. in 6 years times khalid will be 26 years old. How old is sohail at present. Q no 84. Q no 88. If 12 boxes of soap weight is 36 Kg. what will be weight of 6 boxes. Q no 89.At present the ages of adnan and salman are in the ratio of 1:4. in 7 years adnan will be 47 years old. How old is salman (a) at present (b) in 7 years. Prepared By: Syed Adnan 4 AIMS Topic No 5. Indirect Proportion OR Inverse proportion Example 5 men paint a building in 6 days. How long would 10 men take. Men 5 10 Days 6 𝑥 a : b :: c : d 5: 6 :: 10 : 𝑥 Formula a.b = c.d 5 x 6 = 𝑥 x 10 30 = 10𝑥 30/10 = 𝑥 𝑥 = 3 days 2014R 20 men paint a building in 5 days. How long would 8 men take. 2012P Q no 91. 19 men paint a building in 5 days. How long would 9 men take. 2011P Q no 92. Nine men paint a building in 21 days. How long would 7 men take 2002P Q no 93. In a garrison, there is provision for 30 days for 14 soldiers. After 5 days 4 soldiers left. For how many days will be the provision last. Q no 90. Q no 94. In a camp has enough food to last 30 days for his 60 soldiers. If 10 soldiers are also joint this company. How long can the same foods last? Q no 95. Ten persons can dig a water tank in 4 days. How long will 5 persons take to dig the same tank? Q no 96. A train reaches from Hyderabad to Karachi in 2 hours at a speed of 200 Km per hour. If the train is late due to some technical problem and reached in 3 hours. What would be the speed of the train? Q no 97. A construction company estimates that he needs 50 workers to complete a work in 25 days. If he asked to complete the work in 15 days. Find the additional number of workers he has to employed? Q no 98. 36 workers build a house in 20 days. How many days will 18 worker working at the same rate take to build the same house Q no 99. At a scouts camp there is sufficient food to last 72 days. If 18 scouts leave the camp, how much longer can the food last. Q no 100. If takes 3 taps to fill a swiming pool in 8 hours. How long would it take to fill 5 taps. Q no 101. In a graze yard there is a feed for 40 cows for 20 days. How many cows have to leave the graze yards so that the same feeds can be used for 30 days. Q no 102. In a warehouse there was a food storage for 90 days for 400 peoples. After 20 days 50 people left, now how long would it take to finish the food. Q no 103. A flyover is constructed by 84 workers in 300 days. If 15 more workers joined them, now how long would it take to complete the flyover. Q no 104. 4 men can plough a filed in 15 hours. How long would it take 6 men to do the work. Prepared By: Syed Adnan 5 AIMS Topic No 6. Compound Proportion Example 5 Men build 20 rooms of equal size in 9 days. How long will 10 men takes to complete 40 rooms? Men 5 10 5 10 = Rooms 20 40 20 40 0.5 = x Days 9 𝑥 𝑥 9 20𝑥 360 0.5 x 360 = 20 𝑥 180 180 20 = 20 𝑥 = 𝑥 𝑥 = 9 𝑑𝑎𝑦𝑠 2018 Q no 105. 30 Men build 15 rooms of equal size in 18 days. How long will 20 men takes to complete 10 rooms? 2016 Q no 106. 30 labors construct 15 rooms of equal size in 18 days. In How many days can 12 labor construct 12 rooms of the same size? 2004 Q no 107. 25 labors construct 15 rooms of equal size in 18 days. In How many days can 10 labor construct 10 rooms of the same size? 2017 Q no 108. 40 machines produce 1000 screws in 100 hours. How many screws can 24 machines produce in 60 hours. 2015 Q no 109. 195 Men working 10 hours a day can finish a job in 20 days. How many men should be employed to finish the job in 15 days works 13 hours a day. 2013R Q no 110. If 15 men drink 60 gallons of water in 20 days, how many gallons will 25 men drink in 5 days, 2012R, 2005 Q no 111. if 9 men consume 120 kg of rice in 20 days. In how many days will 15 men consume 250 kg of rice? 2012P, 2003R Q no 112. 5 photocopiers can produce 90000 copies in 6 hours a day. How many photocopiers will be required to produce 16800 copies working 8 hours a day? 2011R Q no 113. If 1400 men make 11 miles of railroad in 12 weeks, how long will it take 2400 men to make 27.5 miles of such road. 2006 Q no 114. If 500 men make 11 miles of railroad in 13 weeks, how long will it take 2000 men to make 27.5 miles of such road. 2010 Q no 115. Eight machines can produce 560 units per day working 6 hours a day. How many machines are required to produce 1680 units per day working 9 hours a day. 2009 Q no 116. If 9 men are paid Rs 15,120 for 7 day’s work. What will be the wages of 8 men for 5 days 2008, 2002R Q no 117. 20 men can build a road 15 km long in 18 days how long will 25 men take to complete a 10 km long road. Prepared By: Syed Adnan 6 AIMS 2007 Q no 118. A soap factory makes 600 units in 9 days with the help of 20 machines. How many units can be made in 12 days with the help of 18 machines? Topic No 7. Find the value of x in each problem. Example If 𝑥 : 2 ::8 : 𝑥 then value of x is By applying direct proportion a : b ::c : d 𝑥: 2:: 8 : 𝑥 𝑥 x𝑥 =2x8 𝑥2 = 16 Taking square roots both side √𝑥2 = √16 𝑥=±4 Q no 119. Q no 120. Q no 121. Q no 122. *16 Q no 123. *20 Q no 124. *2 Q no 125. *36 Q no 126. *4 Q no 127. *±2 Q no 128. Q no 129. Q no 130. *3 Q no 131. Q no 132. *±2 x : 15 :: 24 : 30 24 : 30 :: x : 26 X : 26 :: 24 : 30 The value of x in 4:3::x:12 is *±6 *-6 *36 The value of x in x:4::10:2 is *21 *22 *23 If x : 15 :: 24 : 30 then value of x is: *10 *12 *16 If 4 : x :: x : 9 then value of x is *-6 *6 *±6 3 : 4 :: 6 : x *24 *8 *12 9:x::x:4 *±3 *±4 *±6 3 : 4 :: 6 : x 6 : 3 :: 4 : x In 10:6::8:x the value of x is *6 *4 *none of these X : 4 :: 3 : 2 2 : x :: x : 18 *±18 *±6 *6 2012R 2012P 2011R 2018 2016 2015 2014P, 2010, 2007 2013R 2013P 2011P 2009 2008 2004 2003R Topic No 8. Find the equivalent ratio 2003P Which ratios are equal 28:36; 72:27; 21:27; 16:21 Q no 133. 2: 3, 20: 25, 12:18, 4: 7. Q no 134. 48:24, 36:39, 6: 3, 21: 28. Q no 135. 21:35, 18:27, 48: 64: 39: 65 Q no 136. 4:16, 21:35, 9:15, ¼:1, 1:4, 3:5 Q no 137. 24:28, 7:6, 30:35, 1/7:1/6, 6:14, 18:21. Q no 138. 16:21, ¾:1, 15:20, 12:16, 4:6, 3:4. Prepared By: Syed Adnan 7 AIMS 2016 Q no 139. If x:y = 3:7 then find 𝟓𝐱−𝐲 𝟑𝐱+𝟐𝐲 If x : y = 3 : 7 then 𝐱 𝟑 = 𝐲 𝟕 𝐱= 𝟑𝐲 𝟕 Put in equation 𝟑𝐲 𝟓 𝟕 −𝐲 𝟑𝐲 𝟑 𝟕 + 𝟐𝐲 𝟏𝟓𝐲 𝟕 −𝐲 𝟗𝐲 𝟕 + 𝟐𝐲 𝟏𝟓𝐲 − 𝟕𝐲 𝟗𝐲 + 𝟏𝟒𝐲 𝟖𝐲 𝟐𝟑𝐲 𝟖 𝟐𝟑 Q no 140. Find the ratio in the following equation 6x2 -23xy +20y2 0 Prepared By: Syed Adnan 8 AIMS Home Work Q no 141. Divide Rs 10,000 in the ratio of 2:3. Q no 142. dived Rs 12,000 in the ratio of 1:3 Q no 143. Distribute Rs 45500 in two parts of 1:4. Q no 144. Divide 900 in the ratio of 1: 2: 3 Q no 145. Distribute Net Income of Rs 9000 in two partners fifty-50( equally). Q no 146. Distribute net loss of Rs 10500 in three partners equally. Q no 147. A profit of Rs 24000 is to be divided among A,B and C in the ratio of 1/3,1/4 and 2/5. Q no 148. A profit of Rs 48 000 is to be divided among A, B and C in the ratio of 2/3,1/5 and 3/4. Q no 149. Divide Rs 20000 in the ratio of 20%: 30% and 50%. Q no 150. Divide Rs 75000 in the ratio of 10% to A , 60% to B and balance to C. Q no 151. A,B and C are partners having capital of RS 200000, 400000 and 600000 respectively. It they earned net profit of Rs 24000. find the share of each partner. Q no 152. Adnan , Hussain and Umer are partners having capital of RS 36000, 48000 and 42000 respectively. It they earned net income of Rs 21000. find the share of each partner. Q no 153. A,B and C are equal partners having capital of RS 100000, 300000 and 500000 respectively. It they earned net profit of Rs 9000. find the share of each partner. Q no 154. The ratio of A and B is 1:3 while the ratio of B and C is 2 : 4. divide 12500 among them.The ratio of X and Y is 2:5 while the ratio of X and Z is 3 : 4. divide 2000 among X,Y,Z. Q no 155. Find total profit If the ratio of A:B is 2:3 and A gets Rs 300. Q no 156. Divide 40000 in such a way that shabir gets double of zaki. Q no 157. divide 600 in three parts so than 4 times the first, 3 times the second and times the third are all mutually equal. Q no 158. The Cost of 10 books is Rs 750. How many books can be purchased in Rs 15000. Q no 159. A carpenter makes 10 tables in 4 days. In how many days will he take to make 30 such tables. Q no 160. If Ali can read 30 pages of a book in 3 days, then by the same rate how many pages will be read in 11 days. Q no 161. A lift can carry goods upto 1600 Kg. how many boxes can be loaded on it if each box weight 40 Kg. Q no 162. A child can solve 100 problems in 40 minutes. How much time will he take to solve 150 such problems? Q no 163. A man covers a distance of 150 km in 5 days. Find the number of days that he will cover a distance of 450Km with same speed. Q no 164. A computer operator types 300 words in 5 minutes. How many words he can types in 10 minutes? Q no 165. Aslam saves Rs 450 in 2 months. How much time he will take to save Rs 1800? Q no 166. ¾ of a property value is Rs. 615000. Find the value of the half property. Q no 167. If 15 boxes of juices weight is 2.25 Kg. what will be the eight of 5 boxes. Q no 168. A 6 ft man has a 4 ft shadow, hen a tree casts a 22 ft shadow. How tall is the tree? Q no 169. 45 men can paint 24 apartments in 2 weeks and 4 days. How long will 90 men take to complete the 10 apartment in 5 days? Q no 170. A person travels 120 kilometers in 5 days walking 10 hours a day. How many kilometers will he walk in 15 days of 8 hours days. Q no 171. If 235 men consume 517 Kilogram of wheat in 5 1/2 months, how many men will consume 182 Kilograms of wheat in 7 months? Q no 172. 10 machines produce 5000000 screws in 200 hours. How many screws can 24 machines produce in 60 hours. Q no 173. 25 laborers can construct 15 rooms in 18 days in how many days can 10 laborers complete 10 rooms of the same size. Q no 174. If 8 men earn Rs 9600 in 5 days. How much will 9 men earn in 7 days. Q no 175. 10 photocopiers can produce 30,000 copies in 2 days and 4 hours a day. How many photocopiers will be required to produce 15000 copies working 4 days and 4 hours a day? Q no 176. A garments factory makes 8150 shirts in 30 days with the help of 10 machines. How many shirts can be made in 15 days with the help of 8 machines? Q no 177. A biscuit factory makes 1000000 packets of biscuits in 2 months, 3 weeks and 5 days with the help of 10 machines. How many biscuits can be made in 6 months, 4 weeks and 3 days with the help of 20 machines? Prepared By: Syed Adnan 9 AIMS