Ratio, Proportion & Variation PDF

RATIO, PROPORTION AND VARIATION 3.5 ❐ Inverse Variation ❐ Joint Variation A quantity A is said to vary inversely as another quantity If there are three quantities A, B and C such that A B if the two quantities depend...

RATIO, PROPORTION AND VARIATION 3.5 ❐ Inverse Variation ❐ Joint Variation A quantity A is said to vary inversely as another quantity If there are three quantities A, B and C such that A B if the two quantities depend upon each other in such varies with B when C is constant and varies with C a manner that if B is increased in a certain ratio, A gets when B is constant, then A is said to vary jointly with decreased in the same ratio and if B is decreased in a B and C when both B and C are varying. i.e., A ∝ B certain ratio, then A gets increased in the same ratio. when C is constant and A ∝ C when B is a constant; It is the same as saying that A varies directly with ⇒ A ∝ BC 1/B. It is denoted as A ∝ 1/B, i.e., A = k/B where k is k A ∝ BC ⇒ A = kBC where k is the constant of pro- the constant of proportionality. portionality. For example, as the number of men doing a certain In these types of problems on variation, there are work increases, the time taken to do the work decreas- typically three parts: es and conversely, as the number of men decreases, the 1. The relationship between different variables is time taken to do the work increases. defined to frame an equation involving the vari- From the definition of inverse variation, we can see ables and the constant of proportionality. that when two quantities A and B vary inversely with 2. One set of values of all the values of all the vari- each other, then AB = a constant, i.e., the product of ables is given to enable us to find the value of the the two quantities is a constant. Conversely, if the prod- constant of proportionality. uct of two quantities is a constant, we can conclude that 3. The values of all but one variable of a second set they vary inversely with each other. are given and we are asked to find the value of the If X varies inversely with Y and we have two sets of one variable whose value is not given. values of X and Y – X1 corresponding to Y1 and X2 The problems involving ratio and proportion are just corresponding to Y2, then since X and Y are inversely different forms of the models of the basic problems related to each other, we can write down we saw above. For example, in place of variables, the problems can be reframed using mangoes, apples, FORMULA number of days worked, work done etc. Here, practice and perseverance pay you a lot. In entrance exams, there Y will be either direct problems on ratio, proportion and X1 X1Y1 = X2Y2 or = 2 variation or indirect problems of application of these X2 Y1 concepts just discussed to areas like time and work or time and distance. SOLVED EXAMPLES 1.01. The sum of two numbers is 84. If the two numbers Required expression (7a + 9b):(4a + 5b) are in the ratio 4 : 3, then find the two numbers. = [(7 × 3k) + (9 × 4k)] : [(4 × 3k) + (5 × 4k)] Sol: As the two numbers are in the ratio 4 : 3, let their = (21k + 36k):(12k + 20k) actual values be 4x and 3x. = 57k : 32 k = 57 : 32 As the sum of two numbers is 84, we have 4x + 3x = 84. 1.03. The number of red balls and green balls in a bag ⇒ 7x = 84 are in the ratio 16 : 7. If there are 45 more red ⇒ x = (84/7) = 12 balls than green balls, find the number of green Hence, 4x = 48 and 3x = 36. balls in the bag. Alternatively, the two numbers are (4/7) × 84 Sol: Since the ratio of number of red and green balls and (3/7) × 84, i.e., 48 and 36, respectively since is 16 : 7, let the number of red balls and green the ratio of the two numbers is 4 : 3. balls in the bag be 16x and 7x. So, the difference 1.02. If 4a = 3b, then find (7a + 9b):(4a + 5b). of red and green balls is 9x. Sol: It is given that 4a = 3b. 16x – 7x = 9x = 45 ⇒ x = 5 Hence, (a/b) = (3/4) ⇒ Hence, the number of green balls ⇒ a = 3k and b = 4k, where k is the common fac- = 7x, i.e., 35 tor of a and b. Alternatively, 7x = (7/9) (9x) Chapter 1.indd 5 7/13/2018 8:32:58 PM  3.6 UNIT 3 – CHAPTER 1 = (7/9) (45) = 35. 1.07. If x : y = 4 : 3, y : z = 2 : 3, find x : y : z. Hence, there are 45 green balls in the bag. Sol: As y is common to both the ratios, make y in 1.04. What least number must be added to each of a both ratios equal. This is done by making y have pair of numbers that are in the ratio 7 : 16 so that the value equal to the L.C.M of the two parts the ratio between the terms becomes 13 : 22? corresponding to y in the two ratios, i.e., L.C.M Sol: Let the number to be added to each number be (3, 2) = 6. a. Let the actual values of the numbers be 7x and If y = 6, x = (4/3) × 6 = 8, z = (3/2) × 6 = 9 16x, since their ratio is 7 : 16. Hence x : y : z = 8 : 6 : 9. Given that, a 4 2a 2 + 3b 1.08. If = , then find. 7x + a 13 b 5 7a + 6b 2 = Sol: It is given that (a/b) = (4/5). 16x + a 22 Hence a and b can be taken as 4k and 5k, where k ⇒ 154x + 22a = 208x + 13a ⇒ 9a = 54x is the common factor of a and b. ⇒ a = 6x. When x = 1, a is the least number Substituting the values in given expression, the required and is equal to 6. expression is (2a 2 + 3b):(7a + 6b 2) 1.05. A number is divided into four parts such that 4 [2(4k)2 + 3(5k)]:[7(4k) + 6(5k)2] times the first part, 3 times the second part, 6 (32k 2 + 15k):(28k + 150k 2) times the third part, and 8 times the fourth part k (32k + 15): k (28 + 150k) are all equal. In what ratio is the number divided? (32k + 15):(150k + 28) Sol: Let the four parts into which the number is As the value of k is not known, the value of the divided be a, b, c, and d. required expression cannot be determined. 4a = 3b = 6c = 8d. Let the value of each of these 1.09. Two numbers are in the ratio 4 : 5. If 7 is added equal to e. to each, the ratio between the numbers becomes e e e e a = , b = , c = , and d =. 5 : 6. Find the numbers. 4 3 6 8 Sol: Let the numbers be x and y. e e e e x 4 4 Hence, a : b : c : d = : : : = ⇒x = y 4 3 6 8 y 5 5 6 8 4 3 x +7 5 ⎛4 ⎞ = : : : = ⇒ 6 ⎜ y + 7⎟ = 5(y + 7) 24 24 24 24 y+7 6 ⎝5 ⎠ (where 24 is the L.C.M of the denominators) (on cross multiplication and substituting for x) = 6 : 8 : 4 : 3. 24 ⇒ y + 42 = 5 y + 35 Hence, the ratio of the parts into which the num- 5 ber is divided is 6 : 8 : 4 : 3. y ⇒ = 7 ⇒ y = 35 1.06. Divide 3150 into four parts such that half of the 5 first part, a third of the second part, a fourth 4 of the third part is equal to one-twelfth of the x = y = 28. 5 fourth part. Alternative Method: Sol: Let the four parts into which 3150 is divided be Let the numbers be 4k and 5k, where k is the a, b, c, and d. common factor. Given that, 4k + 7 5 a b c d = = = = ; 5k + 7 6 2 3 4 12 24k + 42 = 25k + 35 Let each of the above equal k. ⇒ k = 7. Then, a = 2k, b = 3k, e = 4k and d = 12k The numbers are 4k = 28 and 5k = 35. As a + b + c + d = 3150, the equation becomes, (2k 1.10. The scores of Mohan and Sohan in a test are + 3k + 4k + 12k) = 3150; in the ratio 5 : 4. If their total score is 135, find ⇒ 21k = 3150 ⇒ k = 150. Mohan’s score. Hence, the four parts in the order, are: Sol: As their scores are in the ratio of 5 + 4, let their 300, 450, 600 and 1800. scores be 5k and 4k. Chapter 1.indd 6 7/13/2018 8:33:02 PM  RATIO, PROPORTION AND VARIATION 3.7 Given that the sum of their scores = 5k = 4k = 135 a = 6d, b = 4d, c = 3d ⇒ 9k = 135 Given, a + b + c + d = 1400 ⇒ k = 15 ∴ 6d + 4d + 3d + d = 1400 ∴ Mohan’s score = 5k = 5 × 15 = 75 ⇒ d = 100 Alternative Method: ∴ a = 600, b = 400, c = 300 5 1.15. 1400 is divided into 4 parts such that half of Mohan’s score = (135) = 75 the first part, one third of the second part, one 5+4 1 fourth of the third part and th of the last part 1.11. If a : b = 3 : 4, find 3a + 4b : 4a + 5b. 12 Sol: 3a + 4b : 4a + 5b are all equal. Find the 4 parts. 3a + 4b Sol: Let the first part, second part, third part, and 3a + 4b fourth part be a, b, c, and d, respectively. = = b 4a + 5b 4a + 5b 1 1 1 1 a= b= c= d b 2 3 4 12 ⎛a⎞ ⎛ 3⎞ 3 d = 6a, b = a , c = 2a 3⎜ ⎟ + 4 3⎜ ⎟ + 4 ⎝b⎠ ⎝ 4⎠ 25 2 = = = Given, a + b + c + d = 1400 ⎛ ⎞ a ⎛ ⎞ 3 32 4⎜ ⎟ + 5 4⎜ ⎟ + 5 ⎝b⎠ ⎝ 4⎠ 3 ⇒ a + a + 2a + 6a = 1400 2 1.12. The ratio of the number of marbles with Ram and Shyam is 19 : 13. If Ram gives Shyam 30 mar- 400 ⇒a = bles, both will have equal number of marbles. 3 Find the number of marbles with Ram. 800 ∴ b = 200, c = , d = 800 Sol: Let the number of marbles with Ram and Shyam 3 be 19x and 13x, respectively. Total number of 1.16. If a : b = b : c = 2 : 3, find a:b:c marbles with them = 32x a b 2 2 2 If Ram gives Shyam 30 marbles each will have Sol: = = ⇒ a = b and b = c b c 3 3 3 32x = 16x marbles. 2⎛2 ⎞ 4 2 ∴ a = ⎜ c⎟ = c 3⎝3 ⎠ 9 ∴ 19x − 16x = 30 x = 10 4 2 ∴a : b : c = c : c : c = 4 : 6 : 9 19x = 190 9 3 Alternative Method: 1.13. Two numbers are in the ratio 3 : 4. What part of As b is common to both ratios and since it is divis- the larger number must be added to each num- ible by 3 (from the first ratio) and it is divisible by ber so that their ratio becomes 5 : 6? 2 (from the second ratio), it is divisible by L.C.M Sol: Let the two numbers be 3x and 4x. (3, 2), i.e., 6. Hence, if b = 6, a = 4, and c = 9 3x + k 5 ∴ a:b:c = 4:6:9 = 4x + k 6 1.17. There are 2 classes A and B. If 10 students leave 18x + 6k = 20x + 5k class A and join class B, then the ratio of the k = 2x number of students in class A and class B would ∴ Half of the larger number must be added to reverse. Find the difference in the numbers of each number. students in class A and class B. Sol: Let the numbers of students in class A and class B 1.14. 1400 is divided into 4 parts such that twice the be ax and bx, respectively. first part, thrice the second part, 4 times the ax − 10 b third part and 12 times the last part are all equal. Given, = bx + 10 a Find the 4 parts. Sol: Let the first part, second part, third part, and last a 2x − 10a = b 2x + 10b ⇒ a 2x – b 2x – 10a – 10b = 0 part be a, b, c, and d, respectively. ⇒ (ax – bx – 10) (a + b) = 0 2a = 3b = 4c = 12d ∴ ax – bx = 10 Chapter 1.indd 7 7/13/2018 8:33:06 PM  3.8 UNIT 3 – CHAPTER 1 1.18. A husband’s age exceeds that of his wife by 6 1.22. X varies directly with Y 2 + 18. When Y = 18, years. Before 10 years, the ratio of their ages was X = 18. Find Y when X = 1. 5 : 4. Find the present age of the husband. X 1 Y12 + 18 Sol: Let the present age of the husband be x years. Sol: = ⇒ Present age of the wife = (x − 6) years. X 2 Y22 + 18 10 years ago, the ages of the husband and the wife will be (x − 10) years and (x − 16) years, 18 182 + 18 = respectively. 1 Y22 + 18 5 Given x − 10 = (x − 16 ) 4 Y22 + 18 = 19 ∴ x = 40 Y2 = ± 1 Alternative Method: Let the age of the husband 10 years ago be 5x 1.23. The curved surface area of a cylinder jointly var- years. Age of his wife at that time = 4x years. ies directly with the height and the radius. When The husband would then also be 6 years older the height of the cylinder is 36 cm and the radius than his wife. of the cylinder is 10 cm, the curved surface area ∴ 5x = 4x + 6 ⇒ x = 6 of the cylinder is 720p cm2. Find the curved sur- Hence, the present age of the husband = 5x + 10, face area of the cylinder when the height of the i.e., 40 years. cylinder is 54 cm and the radius of the cylinder is 1.19. Find x, if x + 2 : 4x + 1::5x + 2 : 13x + 1. 15 cm. Sol: In a proportion, product of means = product of Sol: Let the curved surface area of the cylinder be extremes denoted by s. Let the radius and height of the (x + 2) (13x + 1) = (4x + 1) (5x + 2) cylinder be denoted by r and h. ⇒ 13x 2 + x + 26x + 2 = 20x 2 + 8x + 5x + 2 s ∝ rh. Hence, s = c r h where c is a constant. ⇒ 13x 2 + 27x + 2 = 20x 2 + 13x + 2 c = s/rh ⇒ 7x 2 – 14x = 0 ⇒ 7x (x – 2) = 0 When s = 720p sq.cm r = 36 cm and h = 10 cm. ⇒ x = 0 or 2. 720p Hence, c = = 2p. y4 1.20. If x varies directly as + 9 and x = 3 when y = 3, 36 × 10 find x when y = 9. Surface area of the cylinder when r = 48 cm and Sol: x ∝ (y4 + 9). h = 15 cm is 2p × 54 × 15 = 1620p cm2. Hence, x = c (y4 + 9) where c is a constant. Alternative Method: x As both radius and height become 3/2 times c= 4. y +9 their original values, the curved surface area, 2 when x = 3, y = 3 (given) ⎛ 3⎞ being proportional to rh, becomes ⎜ ⎟ , i.e., 3 3 1 ⎝ 2⎠ Hence, c = 4 = = ; (9/4) times its original value. 3 +9 90 30 Hence, it is (9/4) × 720p = 1620p cm2. 1 4 and x = (y + 9) 1.24. The total monthly sales of two companies A and 30 B are in the ratio 2 : 3 and their total monthly When y = 9 expenditures are in the ratio 3 : 4. Find the ratio 1 4 1 of the profits of the two companies given that x= (y + 9) = (6561 + 9) = 219. 30 30 company A’s profit is equal to a fifth of its sales. Sol: Let the total monthly sales of companies A and 2x + 5 x + 2 1.21. Find the value(s) of x if =. B be 2x and 3x and their total monthly expendi- x +1 x −1 tures be 3y and 4y. Sol: (2x + 5) (x − 1) = (x + 2) (x + 1) 1 2x 2 + 5x − 2x − 5 = x 2 + 2x + x + 2 Given that A’s profit = of sales = (2x/5). 5 ⇒ x2 = 7 1 ∴ 2x – 3y = (2x ) ∴x=± 7 5 Chapter 1.indd 8 7/13/2018 8:33:10 PM  RATIO, PROPORTION AND VARIATION 3.9 4 whose base area is A cm2 and height is 10 cm is ⇒ (2x ) = 3y 270 cm3. Find the volume of a cone whose base 5 8 area is 2A cm2 and height is 30 cm. ⇒y= x Sol: V ∝ A h 15 Profit of company B V1 A1 h1 = ⎛ 8 ⎞ 13x V 2 A2 h2 = 3x – 4y = 3x – 4 ⎜ x ⎟ = ⎝ 15 ⎠ 15 270 ⎛ A ⎞ 10 =⎜ Hence, the ratio of the profits of the two compa- V2 ⎝ 2A ⎟⎠ 30 2 13x V2 = 1620 cm3 nies are x : = 6 : 13 5 15 Note that there should be consistency of the 1.25. Given that x varies directly as y, verify whether units used for the variables, i.e., whatever be the (x + y)3 varies directly with (x – y)3. units used to express the variables when the con- Sol: This is a model of a problem where a certain rela- stant of proportionality is being calculated, the tionship is given, and we are asked to check the same units should be used for different variables relationship between different forms of combi- later on also when finding the value of the vari- nations of the two variables. able which we are asked to find out. As x varies directly with y. 1.28. The ratio of the monthly incomes of A and B is x = Ry where R is a constant. 4 : 3. The ratio of their monthly expenditures is 5 : 4. If A saves one fourth of his monthly income, (x + y)3 (Ry + y)3 = find the ratio of their monthly savings. (x − y)3 (Ry − y )3 Sol: Let the monthly incomes of A and B be `4x and `3x, respectively. (R + 1)3 = Let the monthly expenditures of A and B be `5y (R − 1)3 and `4y respectively. Monthly savings of A = `(4x As R.H.S. of above equation is also a constant, − 5y). Monthly savings of B = `(3x − 4y). (x + y)3 varies directly with (x – y)3. 1 Given that 4x − 5y = (4x ) 1.26. A part of the monthly expenses of Amar, a mar- 3x = 5y. 4 keting executive is fixed, and the remaining part ∴ Monthly saving of B = 3x – 4y = 5y – 4y, i.e., `y. varies with the distance travelled by him. If he 1 travels 200 km in a month, his total expenditure Required ratio = (4x ) : y = x : y = 5:3 4 is ≠3300. If he travels 500 km in a month, his to- tal expenditure is ≠3900. Find his total expendi- 1.29. If x varies directly with y, check whether x 3 + y 3 ture, if he travels 800 km in a month. varies directly with x 3 − y 3. x Sol: Let the total expenses be T, F be the fixed part Sol: Let = K , where K is a constant. and V be the variable part. Given that, T = F + V. y As V varies with the distance travelled, if distance x=Ky travelled is denoted by d, x 3 + y 3 = y 3 (K 3 + 1) V = Rd where ‘R ’ is the proportionality constant. x 3 − y 3 = y 3 (K 3 − 1) Hence, T = F + Rd x 3 + y3 K3 + 1 From the given data, = a constant 3300 = F + 200 R (1) x 3 − y3 K3 − 1 3900 = F + 500 R (2) ∴ (x 3 + y 3)varies directly with (x 3 − y 3) Subtracting (1) from (2) 1.30. The monthly expenses of Raja on his car are 600 = 300 R. (⇒ R = 2) partly constant and partly vary with the number Total expenditure if he travels 800 km of kilometres he travels in a month. If he travels = F + 800 R = F + 500 R + 300 R 100 km in a month his total car expenses will be = 3900 + 600 = `4500 `3500. If he travels 200 km in a month, his total 1.27. The volume of a cone varies jointly as the area car expenses will be `4000. If he travels 250 km of its base and its height. The volume of a cone in a month, what will be his total car expenses. Chapter 1.indd 9 7/13/2018 8:33:14 PM  3.10 UNIT 3 – CHAPTER 1 Sol: Let his total car expenses be `T. Let the fixed Given that expense be `F. Let the variable expense be `V. 3500 = F + 100K (A) T=F +V 4000 = F + 200K (B) V Solving (A) and (B), If he travels D km in a month, = K , where K D F = 3000 and K = 5. is a constant. Total car expenses if he travels 250 km ∴T=F+KD = F + 250K = `4250. EXERCISE-1 Directions for questions 1 to 50 : For the multiple choice ques- (A) 48, 60 (B) 60, 48 tions, select the correct alternative from the given choices. (C) 30, 24 (D) 40, 32 For the non-multiple choice questions, write your answer in 8. Quantities a and b are inversely proportional to each the space provided. other. When a = 8, b = 240, find b when a = 6 _______. 1. Raja divided 35 sweets among his daughters Rani and Sita 9. Calculate the fourth proportional to the numbers 0.8, in the ratio 4 : 3. How many sweets did Rani get? _______ 1.6, and 1.6 2. The force of attraction between two objects varies directly (A) 32.4 (B) 2.34 with the product of their masses and inversely with the (C) 3.2 (D) 25.6 square of the distance between them. When the product X +Y 4 2X + Y of the masses (taken in kg) of two objects is 12 and the 10. If = , then find. distance between them is 2 m, the force of attraction 2X + Y 5 3X + Y between them is 18 Newtons. The force of attraction 4 5 between two objects whose product of masses (taken in kg) (A) (B) 5 6 Difficulty Level-1: Foundation is 18 and whose separation is 3 m is _______ Newtons. 6 3 3. The area of a circle varies as the square of its radius. (C) (D) 7 4 Given that the area of a circle whose radius is 7 ft is 196 sq ft, then the area (in sq ft) of a circle whose radius is 11. An article worth `6400 breaks into two pieces whose 8 ft is _______. weights are in the ratio 3 : 5. If the value of the article is 4. Find the numbers that are in the ratio 3 : 2 : 4 such that proportional to the square of its weight, the loss incurred the sum of the first and the second numbers added to the due to the breakage is ` _______. difference of the third and the second numbers is 21. 12. x varies directly as the square of y. When y = 8, x = 192. (A) 12, 8, 16 (B) 6, 4, 8 Find x when y = 10. (C) 9, 6, 24 (D) 9, 6, 12 (A) 100 (B) 30 q + 6p (C) 300 (D) 200 5. If 2.7p = 0.09q, = q − 6p 13. The marks obtained by Raju in Maths, Physics, and 4 3 Chemistry are in the ratio 2 : 3 : 4. If the total marks that (A) (B) Raju obtained in these three subjects is 189, how many 3 2 marks did Raju score in Maths? 5 6 (C) (D) (A) 21 (B) 42 4 5 (C) 63 (D) 84 6. A varies directly with B when C is constant and inversely with C when B is constant. A is 16, when B is 28 and C is 14. If P : Q = 3 : 4, find 5P : 7Q. 7. Find the value of A, when B is 9 and C is 6. 20 3 (A) (B) (A) 6 (B) 7 37 4 (C) 8 (D) 9 15 20 (C) (D) 7. The number of men and women in a conference hall 28 37 are in the ratio 5 : 4. If three men and six women join the 15. An amount of `1560 was divided among A, B and C, in conference, then the number of men and women in the 1 1 1 conference hall will be in the ratio 7 : 6. Find the original the ratio : :. The share of C is _______. number of men and women in the conference hall. 2 3 4 Chapter 1.indd 10 7/13/2018 8:33:19 PM  RATIO, PROPORTION AND VARIATION 3.11 16. A fort had provisions for 150 men for 45 days. After ten (A) 6 (B) 12 days, 25 men left the fort. How long will the food last at (C) 18 (D) 24 the same rate for the remaining men? a − 3b (A) 40 days (B) 28 days 29. If a : b = 4 : 1, find. 2a − b 2 (C) 50 days (D) 42 days 2 1 17. In a class of 30 students, which of the following can’t be (A) (B) 7 7 the ratio of boys and girls? (A) 2 : 3 (B) 1 : 5 3 (C) (D) Cannot be determined (C) 4 : 5 (D) 2 : 1 7 18. The sum of the present ages of Anil and his wife is 88 30. If a + b : b + c : c + a = 3 : 4 : 5, find a : b : c. years. After 8 years, the ratio of their ages will be 7 : 6. (A) 1 : 2 : 3 (B) 2 : 1 : 3 Find Anil’s present age (in years). (C) 2 : 3 : 1 (D) 1 : 3 : 2 (A) 48 (B) 46 31. If a : b = 7 : 3, find a + b :a − b. (C) 50 (D) 52 (A) 5 : 2 (B) 2 : 5 19. Ratio of two numbers is 3 : 5 and their sum is 40. Find the (C) 7 : 3 (D) 3 : 7 smaller of the two numbers _______. 32. The ratio of the ages of Arun, Brahma, and Chari is 20. x varies directly with the square of y. When y is 12, x is 5 : 4 : 3. If Brahma’s age is 28 years, then the sum of the 452. Find x when y is 18 _______. ages of the three persons is _______. 21. x varies inversely with the square of y. When y is 2, x is 48. 33. If P 2 : Q = R : P and 27Q 2 = R, find P : Q. Find x when y is 4 _______. (A) 1 : 9 (B) 9 : 1 1 2 22. If x + y + z = 120 and x = y and y = z , find z _______. (C) 3 : 1 (D) 1 : 3 2 3 34. At a party, there are total of 28 adults. If x ladies join the 23. Salary of A in a month varies directly with the number of party, the ratio of the number of ladies to that of gents working days in the month. A got a salary of `10,000, in will change from 3 : 4 to 5 : 4. Find x _______. a month which had 25 working days. What salary will he Difficulty Level-1: Foundation get in a month which has 26 working days? 35. There are total of 30 employees in a company. The ratio (A) `10,100 (B) `10,200 of male employees to the female employees is 8:7. How (C) `10,300 (D) `10,400 many female employees have to be recruited so that the ratio becomes 1:1? 24. The extension of a spring is directly proportional to the (A) 5 (B) 2 force applied. The extension is 3 cm when the force (C) 3 (D) 4 applied is 30 N. Find the extension when the force applied is 20 N. 4 x +5 y 36. If x : y = 4 : 9, = (A) 2 cm (B) 1.5 cm 6 x +7 y (C) 1 cm (D) 3 cm 21 19 25. If a : b = 2 : 3 and b:c = 5 : 7, then find a : b : c. (A) (B) 31 29 (A) 10 : 15 : 21 (B) 10 : 21 : 15 (C) 9 : 12 : 14 (D) 12 : 7 : 18 27 23 (C) (D) 37 33 26. If P, Q, R and S are in proportion, which of the following follows? 37. Quantity A varies directly with the sum of the quantities (A) S, R, Q and P are in proportion. B and C. If B increases by 2 and C increases by 4, by how (B) Q, S, P and R are in proportion. much does A increase? (C) Both (A) and (B) (A) 2 (B) 4 (D) Neither (A) nor (B) (C) 6 (D) Cannot be determined 27. Two numbers A and B are in the ratio 10 : 13. If 90 is 38. Three numbers are in the ratio of 2 : 3 : 5. Given subtracted from each, the resulting numbers will be in that the product of the extremes is 90. The difference the ratio 7 : 10. Find A. between the largest and the smallest of them is _______. (A) 290 (B) 280 39. Find the following: (C) 310 (D) 300 (a) Duplicate ratio of 3 : 4. 28. There are two variables x and y, where x varies directly as (A) 3 : 8 (B) 6 : 4 the cube root of y. When y = 8, x = 2. Find x when y = 216. (C) 5 : 7 (D) 9 : 16 Chapter 1.indd 11 7/13/2018 8:33:21 PM  3.12 UNIT 3 – CHAPTER 1 (b) Triplicate ratio of 2 : 3. (C) Both (A) and (B) (A) 6:9 (B) 3 : 2 (D) Neither (A) nor (B) (C) 8 : 27 (D) 5 : 8 44. If three numbers are in the ratio 1 : 3 : 5 and their sum is (c) Sub-duplicate ratio of 16 : 9. 108, then the largest number is _______. (A) 2:3 (B) 4 : 3 45. The monthly salaries of X and Y are in the ratio 3 : 4. The (C) 4:9 (D) 8 : 3 monthly expenditures of X and Y are in the ratio 4 : 5. (d) Mean proportional of 16 and 4. Find the ratio of the monthly savings of X and Y. (A) 64 (B) 16 (A) 5 : 3 (B) 4 : 7 (C) 8 (D) 14 (C) 3 : 5 (D) Cannot be determined 40. If x : y = 3 : 7 and y : z = 7 : 4, find x : z. 46. If a = 50b and b = 25c , then find a:c. (A) 4:3 (B) 3 : 2 100 100 (C) 2:3 (D) 3 : 4 (A) 1 : 2 (B) 1 : 4 41. If a + b : a − b = 3 : 2, find a : b. (C) 3 : 4 (D) 1 : 8 (A) 5 : 1 (B) 1 : 5 47. There are two positive numbers in the ratio 5 : 8. If the (C) 3 : 5 (D) 5 : 3 larger number exceeds the smaller by 15, the smaller number is _______. 42. Quantity A varies directly with the product of B and C. A = 300 when B = 20 and C = 50. What is the value of B 48. Quantity P varies inversely with the product of Q and R. when A = 900 and C = 60? When Q = 6 and R = 12, P = 75. Find P when Q = 5 and R (A) 40 (B) 45 = 10 _______. (C) 50 (D) 60 49. The mean proportional of 43. Which of the following must be true? (a) 6 and 24 is _______. (A) If x varies directly with y , x 2 varies directly with y. (b) 50 and 512 is _______. 1 50. The present ages of Rohit and Sunil are in the ratio 3 : 5. (B) If x varies inversely with , y varies inversely with y 10 years hence, the ratio of their ages will be 4 : 5. Find 1 the present age of Rohit. (in years) _______. x Difficulty Level-2: Moderate EXERCISE-2 Directions for questions 1 to 34: For the multiple choice ques- His marks in these papers were in the ratio of tions, select the correct alternative from the given choices. 12 : 13 : 14 : 15 : 16 : 17. Then, the number of papers in For the non-multiple choice questions, write your answer in which he got more than 55% is the space provided. (A) 6 (B) 5 (C) 4 (D) 3 1. Eighteen men take 30 days to complete a piece of work working 10 hours a day. Find the time taken by 4. The cost of a precious stone varies directly as the square 25 men to complete five times as much work working 12 root of its weight. A certain precious stone broke into 3 hours a day. pieces whose weights are in the ratio 1 : 4 : 4. As a result, (A) 30 days (B) 36 days its value went up by `12000. Find its initial value. (C) 72 days (D) 90 days (A) `9000 (B) `12000 (C) `15000 (D) `18000 2. The cost of 3 kg tea powder is equal to the cost of 6 kg of sugar, the cost of 24 kg of sugar is equal to the cost 5. Nine farmers take 6 hours to plough 18 acres of land. of 21 kg of oil, the cost of 56 kg of oil is equal to the cost How many acres of land can 16 farmers plough in 21 of 12 kg of ghee. What is the cost of 3 kg of ghee if the hours? (Assume that the rate of work of each farmer is cost of 1 kg tea powder is `72? the same) _______ (A) `576 (B) `288 4a + 3b 6. If a : b = 4 : 5 then find the value of. (C) `192 (D) `436 3a + 2b 3. A student took 6 papers in an examination, where the 22 maximum marks were the same for each paper. In all papers 3x 2 + 4 y 2 7. If x : y = 6 : 7 then find the value of. together, the candidate obtained 58% of the total marks. 2x + 3 y Chapter 1.indd 12 7/13/2018 8:33:23 PM  RATIO, PROPORTION AND VARIATION 3.13 304 152 A body fell 95 m in the 10th second. Find the distance (in (A) (B) m) it fell in the 14th second _______. 33 33 608 17. There are five identical glasses containing milk in the (C) (D) Cannot be determined ratio 3 : 4 : 5 : 6 : 7. How many glasses are at least half full 33 of milk if the total volume of milk in the glasses is three- 8. In a school, there are 650 students. The ratio of the num- fifth of the total volume of the five glasses? _______ ber of boys to that of the girls is 8 : 5. How many more girls should join the school so that the ratio becomes 18. A string is cut into two parts such that the ratio of the 4 : 3? _______ lengths of the complete string and the smaller part is 20 times the ratio of the lengths of the smaller part and the 9. If heat radiated by a certain body per unit time varies larger part. Find the ratio of the length of the string and directly with the square root of the excess of the the square of the length of the smaller part (taken in cm) temperature of the body over the ambient temperature. if the longer part is 4 cm long The heat radiated by the body in 1 second is 12 joules, (A) 5 : 3 (B) 5 : 4 when the temperature of the body is 34°C. Find the (C) 5 : 2 (D) 5 : 1 temperature of the body when the heat radiated in 1 second was 20 joules. (Assume the ambient temperature 19. A purse contains 72 coins comprising one rupee, 50 to be 25°C). paise and 25 paise coins, their values being in the ratio (A) 60°C (B) 45°C 10 : 15 : 8. Find the number of 50 paise coins _______. (C) 40°C (D) 50°C 20. A machine with a power of 18 units can lift an object of 10. Find a : b : c, given 3a + 2b = 7c and b = a + c. a maximum weight of 9 units. If the power of a machine varies directly as the square root of the maximum weight (A) 3 : 8 : 5 (B) 1 : 6 : 5 of an object that it can lift, then the machine with 24 (C) 1 : 2 : 1 (D) 3 : 10 : 7 units of power can lift an object weighing a maximum of 11. A purse contains 10 paise, 20 paise, and 50 paise coins _______ units. in the ratio 5 : 2 : 1. The total value of all the coins in the purse is `140. How many 20 paise coins are in the purse? 21. What number must be subtracted from both the 22 _______. numerator and denominator of the fraction so that it becomes 2 : 7? _______ 37 12. The total income and the total expenditure of two Difficulty Level-2: Moderate persons A and B are in the ratio 22 : 17. The incomes of A 22. Ten years ago, the ratio of the ages of a woman and her and B are in the ratio 5 : 6 and their expenses are in the daughter was 3:2. Which of the following cannot be the ratio 7 : 10. What is the ratio of the savings of A and B? ratio of their ages 5 years from now? (A) 3 : 2 (B) 2 : 3 (A) 6 : 5 (B) 7 : 3 (C) 7 : 12 (D) 10 : 9 (C) 8 : 7 (D) 11 : 9 p +q q +r p +r 23. If a, b, c and d are in proportion, then which of the 13. If = = = k , then find k. r p q following is equal to (a − b )(a − c )/ a ? (A) 1 (B) –1 (A) a + b + d – c (B) a − b − c − d (C) 2 (D) Either (B) or (C) (C) a + d − b − c (D) a + b − c + d 14. In a shop, the quantities of three kinds of commodities a 3 b c 4 d e 1 def sold on a particular day were in the ratio 3 : 4 : 5. The 24. If = , = 5, = , = 2 and = then = b 4 c d 3 e f 4 abc total sales proceeds was `4,000. If the prices of the three commodities were also in the same ratio, what was the 40 400 (A) (B) total amount received by selling the commodity which 9 9 fetched the maximum sales revenue? 9 9 (A) `2,500 (B) `2,000 (C) (D) (C) `3,000 (D) `1,500 40 400 25. A writer gets a fixed amount for his book apart from the 15. A man divided `62500 among his four sons such that four royalty he gets per book sold. He gets `22000 and `46000 times the share of his first son, three times the share of for 6000 books sold and 18000 books sold, respectively. his second son, two times the share of his third son and Find his income per book when 25000 books are sold the share of his fourth son are all equal. Find the share _______. of each of his sons. (in `) _______, _______, _______, _______ p q r 26. If = = , each of these equals q + r − p p + r −q p +q −r 16. The distance travelled by a freely falling body is directly proportional to the square of the time for which it falls. _______ or _______. Chapter 1.indd 13 7/13/2018 8:33:28 PM  3.14 UNIT 3 – CHAPTER 1 27. A piece of land is to be divided between two men in the There are two colleges in the town – college A and college ratio of 5 : 11. Instead, if the land is divided in the ratio B. There are 500 more students in college A than in college 11 : 5, what fraction of the whole piece of land does the B. The ratio of the boys to that of the girls in college A is second man lose? 3 : 2 and that in college B is 4 : 1. The ratio of the number of (A) 3/8 (B) 6/11 Science, Humanities, and Commerce students in college A (C) 5/6 (D) 10/11 and college B are 2 : 5 : 3 and 2 : 3 : 3, respectively. The number 28. The volume of a cylinder varies jointly as its height and of Commerce students in both the colleges is the same. the area of its base. When the area of the base is 64 sq.ft. 35. How many students are there in college A? _______ and the height is 10 ft, the volume is 640 cu.ft. What is the height of the cylinder (in ft), whose volume is 360 36. How many girls are there in the two colleges together? cu.ft and the area of the base is 72 sq.ft.? _______ _______ 29. The time taken (T) by on automobile to cover a certain Directions for questions 37 to 50: For the multiple choice ques- distance (D) varies directly as the distance when its tions, select the correct alternative from the given choices. engine capacity (C) is constant. T varies inversely as C For the non-multiple choice questions, write your answer in when D is constant. P and Q are two automobiles. The the space provided. engine capacities of P and Q are 1500 cc and 1200 cc, respectively. To cover 800 km, P takes 16 hours. Find the 37. If a, b and c are in continued proportion, then which of time taken by Q to cover 600 km. (in hours) the following is equal to a : c? (A) 15 (B) 17 (A) a 2 : b 2 (B) (a 2 + b 2):(b 2 + c 2) 2 (C) b : c 2 1 (D) All of these (C) 20 (D) 9 2 38. The ratio of the present ages of a man and his wife is 5 : 4. 30. The volume of a solid figure is proportional to the square Which of the following cannot be a possible ratio of their of its radius when its height is constant and to its height ages 20 years ago? when its radius is constant. The solid figure A has a radius (A) 7 : 5 (B) 3 : 2 of 7 units, a height of 9 units and a volume of 1386 cubic (C) 13 : 10 (D) 6 : 5 units. Find the volume of a solid figure (in cubic units) 39. The monthly tariff of a DTH provider consists of two whose radius is 14 units and height is 3 units _______. Difficulty Level-2: Moderate parts – a fixed part for providing the service and a 31. The ratio of the number of sparrows on the left and right variable part which varies with the number of channels branches of a tree is 7 : 13. If 12 sparrows shift to the left opted for. The monthly tariffs of two customers who branch from the right, then the number of sparrows on opted for 30 channels and 50 channels are `350 and the left branch is equal to that on the right branch. How `450, respectively. Find the monthly tariff of a customer many sparrows were there on the left branch, before the who opted for 60 channels (in `) _______. shift? (Assume that the branches of the tree are classified 40. There are five vessels, with equal capacities, each as either left branches or right branches.) containing some milk. The quantities of milk in the 5 (A) 84 (B) 14 vessels are in the ratio 4 : 5 : 6 : 7 : 8. The total quantity (C) 28 (D) 40 of milk in the five vessels is equal to 75% of the total x y z x y z capacities of the 5 vessels. How many of the vessels are at 32. If = = = 8 and = = =k 2a + b 2b + c 2c + a 2a 2b 2c least 64% full of milk? where a + b + c ≠ 0, then k = _______. (A) 2 (B) 1 33. The annual incomes of Varun and Vikram are in the ratio (C) 4 (D) 3 8 : 3 and their annual expenditures are in the ratio 4 : 1. 41. 3x + y – 5z = 0 and 4x + 5y – 14z = 0. Find x : y : z. If each saves `2000 per annum, then what is the annual (A) 1 : 1 : 1 (B) 2 : 1 : 1 income of Varun? (in `) _______ (C) 1 : 2 : 1 (D) 1 : 1 : 2 ka kb kc 34. If = = = l and k ≠ 0, a + b + c ≠ 0, then 42. A garrison of 2,000 men has provisions for 20 weeks at b +c c +a a +b the rate of 2.5 kg per day per man. After 4 weeks, 500 what is the value of ‘l ’? more men join the garrison. For how many more weeks, k (A) k (B) will the remaining provisions last at the rate of 2 kg per 3 day per man? k k (A) 12 (B) 15 (C) (D) 2 4 (C) 16 (D) 20 Directions for questions 35 and 36: These questions are based 43. A sum of money that was supposed to be divided between on the information given below. A and B in the ratio of 3 : 5 was divided among A, B and C Chapter 1.indd 14 7/13/2018 8:33:31 PM  RATIO, PROPORTION AND VARIATION 3.15 in the ratio of 6 : 5 : 4. Due to this, A gained `3,000. What charged at a certain fixed rate per call. The monthly was the loss incurred by B? bills of Ramesh and Suresh who made 98 outgoing calls (in `) _______ and 218 outgoing calls, respectively, were `300 and `450, respectively. Find the monthly bill of a person who has 44. The volume of a sphere varies directly as the cube of its made 160 outgoing calls (in `) _______. radius. If three cubes of radii 3 cm, 4 cm and 5 cm are melted and recast into one sphere, then find the radius 48. A stone is dropped from a height of one km. The distance of the sphere _______. it falls through varies directly with the square of the time taken to fall through that distance. If it travels 64 m in 4 45. Arjun buys 5 erasers for every 3 pencils bought by him. seconds, find the distance it covers in the 5th second. For every 24 articles of the combined lot of erasers and pencils bought by him, he buys 2 geometry boxes. If the (A) 36 m (B) 24 m total number of all three types of articles bought by him (C) 28 m (D) 44 m is 104, what is the number of pencils bought by him? 49. A and B are two numbers. If 10 is the mean proportional (A) 65 (B) 36 of A and B and 10000 is the third proportional of A (C) 8 (D) 60 and B, what will be the value of the larger of A and B? 46. The ratio of incomes of A and B is 3 : 4. Each of them _______ spends a part of his income and saves the rest. The ratio of 50. The ratio of the incomes of A and B is 5 : 4 and ratio their expenditures is 2 : 3. Whose savings, as a proportion of their expenditures is 3 : 2. If B saves one-third of his of his income, is higher? _______ income, find the ratio of their savings. 47. The monthly telephone bill has a fixed tariff for up to (A) 7 : 8 (B) 11 : 12 50 outgoing calls. Outgoing calls in excess of 50 are (C) 5 : 6 (D) 3 : 4 EXERCISE-3 Directions for questions 1 to 50: For the multiple choice ques- 5. A varies directly as the sum of the two quantities B and Difficulty Level-3: Advanced tions, select the correct alternative from the given choices. C. B in turn varies directly as x and C varies inversely as x. For the non-multiple choice questions, write your answer in When x = 2, A = 6 and when x = 4, A = 9. Find the value the space provided. of A when x = 16. (A) 2½ (B) 1 1. If a : b = 2 : 3, b : c = 6 : 5, c : d = 10 : 13 and e : d = 2 : 1, then (C) 8½ (D) 32¼ 480 find abc : ed2. 6. There are three unequal quantities x, y and z in continued 2. p, q, r, s and t are five integers satisfying p = 3q = 4r and proportion. Which of the following equals z : x? 2q = 5s = 12t. Which of the following pairs contains a z 2 − y2 number that can never be an integer? (A) y 2 : x 2 (B) y2 − x 2 (A) (2p/15, q/t) (B) (p/t, 4r/t) (C) (p/4, rs/180) (D) (p/8, s/r) (C) z 2 : y 2 (D) All of the above 3. A quantity Q is obtained by adding three quantities. The 7. A certain sum is divided among A, B and C in a manner first is a constant, the second varies directly as the square that for every rupee that A gets, B gets 75 paise and for root of y, and the third varies directly as the cube root of every rupee that B gets, C gets 50 paise. If C ’s share in y. When y = 1, Q = 90, when y = 64, Q = 450, and when y = the total sum is `420, then find the share of A (in `) 729, Q = 1270. Find the constant. _______. (A) 5 (B) 10 8. If b/a is a proper fraction satisfy the equation 15a 2 – 26ab (C) 15 (D) 20 + 8b 2 = 0, find b : a. (A) 2 : 5 (B) 1 : 5 4. A salesman for a company gets an incentive for every unit of product he sells apart from his fixed salary. He gets (C) 4 : 5 (D) 3 : 4 `8,000 and `9,000 for 150 units and 200 units he sold, 9. A number is divided into five parts. Twice the first part, respectively. If he sells 400 units, what is his income per thrice the second part and four times the fourth part are unit? equal. Twice the second part, five times the third part and (A) `32.50 (B) `30 six times the last part are equal. Which of the following is (C) `27.50 (D) `20 always true if all the parts are integers? Chapter 1.indd 15 7/13/2018 8:33:31 PM  3.16 UNIT 3 – CHAPTER 1 (A) The first part is a multiple of 72. (A) 46 (B) 56 (B) The second part is divisible by the fourth part. (C) 59 (D) 53 (C) The first part is a factor of the last part. 16. The volume of a gas is inversely proportional to the (D) The product of the first and fourth parts is divisible pressure acting on it when the temperature is constant by 30. and directly proportional to the temperature when the 10. Balloons A and B are fitted with special valves and placed pressure acting on it is constant. When the temperature in an observation chamber. Initially, the volume of is 40 and the pressure acting on it is 64, its volume is 200. helium in balloon A was seven times that in balloon B. Find the pressure when the temperature and its volume The helium in balloon A leaks at a constant rate and half are 50 and 400 respectively. of the helium which leaks from balloon A at any given (A) 32 (B) 40 time enters balloon B. After 3 hours, the ratio of the (C) 44 (D) 50 volumes of helium in balloons A and B is 13 : 7. After how 17. The earnings of A and B are in the ratio 3 : 7 and that of many hours from the start, will the helium in balloons A B and C is 4 : 9 and that of D and C is 7 : 6. If the sum of and B be in the ratio 7 : 37? the earnings of A, B, C and D is `52950, then what are the (A) 7 (B) 4 earnings of D? (C) 11 (D) 14 (A) `21,050 (B) `22,050 Directions for questions 11 to 14: These questions are based on (C) `23,050 (D) `24,050 the data given below. 18. Three friends, Aravind, Bharath and Chandu are about to have their breakfast. Aravind has 7 apples, Bharath has There are two software companies in a city – ‘Smart Softcom’ 5 apples and Chandu has no apples but has 12 coins. He and ‘Fast Softcom’. The ratio of the number male employ- offers to pay for some apples. They agree to share the 12 ees to the number of female employees in ‘Smart Softcom’ is apples equally among themselves and agree that Chandu 5 : 3 and that in ‘Fast Softcom’ is 7 : 5. The ratio of number of would pay 12 coins for his share. Bharath suggests that he employees in the age groups (in completed years) 21 to 30, 31 be paid 5 coins and Aravind be paid 7 coins. Aravind says to 40 and 41 to 50 in ‘Smart Softcom’ and ‘Fast Softcom’ are that he should get more than 7 coins. How much should 4 : 5 : 7 and 2 : 1 : 3 ,respectively. The number of employees in Aravind get? the age group 21 to 30 in both the companies is the same and (A) 11 coins (B) 10 coins Difficulty Level-3: Advanced Smart Softcom has 400 employees more than ‘Fast Softcom’. (C) 12 coins (D) 9 coins 11. How many employees are there in Fast Softcom? 19. The kinetic energy of a moving body varies directly with (A) 1600 (B) 1200 its mass when its velocity is constant and with the square (C) 800 (D) 600 of its velocity when its mass is constant. A body has a mass of 7.2 kg and a velocity of 0.2 m/sec and a kinetic energy 12. How many male employees are there in the two of 0.144 joules. Find the kinetic energy of a body having companies together? a mass of 3.6 kg and a velocity of 0.8 m/sec (in joules) (A) 1100 (B) 1300 _______. (C) 1500 (D) 1700 20. If a : b = 2 : 3 and p : q = 3 : 2, what is the value of (2a 2p3 + 13. What is the difference in the number of employees in the 3b 2q 3):(3abpq 2 + 4a 2p 2q)? age group 41 to 50 in Smart Softcom and Fast Softcom? (A) 1 : 1 (B) 2 : 3 (A) 300 (B) 200 (C) 6 : 7 (D) Cannot be determined (C) 100 (D) 160 a +b a −b 14. What is the ratio of the number of male employees in 21. If = , then which of the following is true? Fast Softcom to the number of female employees in c +d c −d Smart Softcom? (A) ab + cd = 0 (B) ac + bd = 0 (A) 7 : 6 (B) 2 : 1 (C) ad − bc = 0 (D) ab − cd = 0 (C) 1 : 2 (D) 7 : 10 22. The radii of cylinders of equal heights vary directly with 15. The electricity bill for a month varies directly as the the square root of their volumes. The radii of cylinders number of units consumed. The charge per unit is `1.35 of equal volumes vary inversely as the square root up to 50 units used. If the number of units consumed of their heights. The radius of a cylinder is 10 cm, its is more than 50, then the cost of each additional unit volume is 1500 cm3 and height is 5 cm. Find the radius is `2.70. If the consumption in the first month is 97 of the cylinder whose volume is 2400 cm3 and height is units, then what should the consumption in the second 2 cm. month be such that the average bill for the two months (A) 10 cm (B) 15 cm combined is `135 per month? (C) 20 cm (D) 25 cm Chapter 1.indd 16 7/13/2018 8:33:32 PM  RATIO, PROPORTION AND VARIATION 3.17 free fall for 10 seconds is 25 kilojoules. Find the kinetic a2 + b2 b2 + c 2 23. If a ≠ c and = = k , find k. energy of a body with mass 2.5 kg which is under free fall a +b b +c for 3 seconds (in kilojoules) (A) a + c (B) a – c (A) 102.5 (B) 1.175 ac (C) 125 (D) 1.125 (C) c – a (D) a +c 29. If (x + y) varies directly as (x − y), then (x 2 + y 2) will vary 24. A product can be manufactured using 4 different as processes – A, B, C, and D. In each process there is a fixed (A) x 2 − y 2 (B) xy component to the cost and another component that (C) Both (A) and (B) (D) None of these varies with the number of units produced. The following table gives the costs incurred in the four processes. 30. The total surface area of a cylinder having a certain height is the sum of two parts. One of the parts varies A B C D directly with the radius and the other parts varies directly Fixed cost 10,000 12,000 15,000 20,000 with the square of the radius. The total surface area is (in `) 7200 units when the radius is 30 units and 3600 units Cost per unit 20 15 10 8 when the radius is 20 units. Find the total surface area (in `) when the radius is 10 units. (A) 1040 (B) 1200 If 500 units have to be produced, which process will (C) 1260 (D) 1170 result in the least cost? (A) A (B) B 31. The lateral surface area of a right square pyramid varies (C) C (D) D directly with the edge of its base when the slant height (the shortest distance from the vertex to any one of its 25. The monthly telephone bill has a fixed tariff of `250 edges) is constant. Also, the lateral surface area of the for up to 50 outgoing calls. For over 50 calls, there is a pyramid varies directly with the slant height when the charge of `1.25 per call. The ratio of the bills paid by edge of its base is constant. The edge of the base of a Aravind and Prasad for a particular month is 2 : 3 and the right square pyramid is 7 cm, its slant height is 14 cm and number of outgoing calls made by Aravind is 90. What is its lateral surface area is 196 cm2. Find the lateral surface the number of outgoing calls made by Prasad? area of a right square pyramid which has the edge of its Difficulty Level-3: Advanced (A) 210 (B) 250 base as 12 cm and slant height as 20 cm. (C) 160 (D) 180 (A) 240 cm2 (B) 360 cm2 2 26. A machine’s output varies directly with its effective input (C) 480 cm (D) 640 cm2 in kilos when its efficiency is constant and it varies directly 32. The sum of the present ages of a woman and her daughter with its efficiency when its effective input is constant. The is 60 years. When the woman attains her husband’s effective input is the input minus the waste material. The present age, the ratio of the ages of her husband and her machine produces an output of 1008 kg when its input daughter will be 2 : 1. Find the present age (in years) of is 2400 kg, efficiency is 70% and waste material is 40% of her daughter. the input. Find the output of the machine if its input is (A) 10 (B) 15 1680 kg, efficiency is 80% and waste material is 30% of (C) 20 (D) 25 the input (in kg) _______. 27. The expenses for yoga classes in a colony are partly 33. The consumption of diesel per hour of a bus varies constant and partly varying with the number of members. directly as the square of its speed. When the bus is If there are 50 members, then each of the members travelling at 40 km/h its consumption is 1 l/h. If each has to bear `220 per month and if there are 10 more litre costs `40 and other expenses per hour cost `40, members, then the share of each of the members comes then what would be the minimum expenditure required down by `15 per month. How many members would be to cover a distance of 400 km? there if the share of each member is `160? (A) `600 (B) `700 (A) 150 (B) 130 (C) `800 (D) `900 (C) 90 (D) Cannot be determined ax 2 + by 2 77 xy 2 + ab 2 17 34. If 2 2 = and 2 = , what is the value 28. The kinetic energy that a body acquires when it falls ax − by 13 xy − ab 2 7 freely for a time t varies directly with the square of t for a given mass. For bodies of different masses for a given of x 2 : b 2 ? value of t, the kinetic energy varies directly as the mass. (A) 1 : 4 (B) 9 : 4 The kinetic energy of a body of mass 5 kg which is under (C) 2 : 3 (D) 49 : 64 Chapter 1.indd 17 7/13/2018 8:33:33 PM  3.18 UNIT 3 – CHAPTER 1 35. The cost of a bars of a precious metal varies directly as the 40. The mean proportional between two numbers is 12. The square of the weight of the bar. Metal bars of weights in third proportional of the same numbers is 96. Find the the ratio 4 : 5 : 6 were bought from three different places greater of the two numbers. and melted