RATIO AND PROPORTION

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Ross Charles
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1 8 RATIO AND PROPORTION. Find the ratio of each of the following in simplest form : (i) Rs 3.2 to Rs 2.7 (ii) hours to day (iii) 4 months to 3 years (iv) 3 paise to Rs 7.80 (v) 7 kg to 2 quintals (vi) g to 2 kg Ans. (i) Rs 3.2 to Rs 2.7 = (ii) hours to day = Rs = = 3 Rs hours day = = 3 : hours = = : 8 24 hours 8 (iii) 4 months to 3 years = 4 months to 3 2 months = 4 = = : 36 (iv) 6 paise to Rs 7.80 = 6 p aise Rs 7.80 = 6 paise p aise = 6 = = : (v) 7 kg to 2 quintals = 7 kg 2 quintals = 7 kg 3 = 2 00 kg 8 = 3 : 8 (vi) 0 g to 2 kg = 0 g 2 kg = 0 g 3 = = 3 : g Express each of the following ratios in simplest form : (i) 4 : 3 (ii) 46 : 36 (iii) 3 4 : : 4 6 (vi) 2 : 3 : Ans. (i) 4 : 3 = 4 = 7 = 7 : (ii) 46 : 36 = 46 = 2 = 2 : 3 36 (iii) (iv) 3 4 : : = : 3 : : 60 : 60 = 4 : 48 : = : : LCM of 3, 4 and 6 = : : = : 2 : 2 = 28 : 3 :

2 3. In a school, there are 420 boys and 80 girls. Find the ratio of: (i) girls to total number of students (ii) boys to total number of students Ans. Number of boys = 420 and number of girls = 80 Total number of students = = 600 (i) Ratio of girls to total number of students = 80 : 600 = 80 = 3 = 3 : (ii) Ratio of boys to total number of students = 420 : 600 = 420 = 7 = 7 : The population of a village is 340. If the number of males is 206, find the ratio of males to females. Ans. Total population of a village = 340; Number of males = 206 Number of females = = 47 Thus, ratio of males to females = 206 : 47 = 206 = 7 = 7 : 47. A line segment 32 cm long is divided into two parts in the ratio 3 : Find the length of each part. Ans. Total ratio = 3 + = 8, Length of one part = 3 32 = 2 cm, Length of another 8 part = cm 8 = 6. The ratio of tin and zinc in an alloy is 3 : 4. How much tin is there in 0. g of the alloy? Ans. Sum of ratio = = 7. Quantity of tin in alloy = 3 0. g = 3. g = 4. g 7 7. Divide Rs 66 between Amit and Kunal in the ratio 7 : 4. Ans. Sum of terms of ratio = = Amit s share = Rs 7 66 = Rs 7 60 = Rs 423 Kunal s share = Rs 4 66 = Rs 4 60 = Rs Divide Rs 002 among A, B, C in the ratio 3 : 4 : 7. Ans. Sum of ratio = = 4, A s share = Rs = = Rs 2 4 B s share = Rs = = Rs C s share = Rs = = Rs

3 . Divide 82 into three parts in the ratio : : Ans. LCM of 0, and 20 = 60 : : 0 20 = 60 : 60 : = 6 : 4 : 3 Total ratio = = 3, First s part = = 6 4 = 84 Second s part = = 4 4 = 6, Third s part = 3 82 = 3 4 = Divide Rs 300 among A, B, C in such way that A gets double of what B and B gets double of what C gets. Ans. Let C gets = Rs, B gets = Rs 2 = Rs 2, A gets = Rs 2 2 = Rs = 4 A : B : C = 4 : 2 :, sum of terms of ratio = = 7 A s share = Rs Rs = =, B s share = Rs = = Rs 860, C s share = Rs 300 = Rs The sides of a triangle are in the ratio 2 : 3 : 4. If its perimeter is 4 cm, find the lengths of sides of the triangle. Ans. Sum of the terms of ratio = =, Sum of side i.e. perimeter = 4 cm st side = 2 4 = 2 cm; 2nd side = 3 4 = 8 cm; 3rd side = 4 4 = 4 6 = 24 cm. 2. The angles of a triangle are in the ratio 7 : : 3. Find the measure of each angle of the triangle. Ans. Sum of the terms of ratio = = We know that the sum of angles of triangle = 80 Ist angle = 7 3rd angle = 3 80 = 7 2 = 84, 2nd angle = 80 = 3 2 = = 2 = 60, 3. Find the ratio of the price of coffee to that of tea, when coffee costs Rs 24 per 00 gm and tea costs Rs 80 per kilogram. 24 Ans. The ratio of the price of coffee to tea = = = = 3 : 000 3

4 4. An office opens at.30 A.M. and closes at.30 P.M. Official lunch time is from 2.30 P.M. to.00 P.M. What is the ratio of (i) lunch interval to total office hours? (ii) lunch interval to working hours? Ans. Opening time =.30 A.M. Closing time =.30 P.M. or 7.30 Total office hours = ( ) = 8.00 hours Lunch interval =.00 P.M P.M. = ( ) = 0.30 hours = 30 = hr (i) Lunch Interval : Total office hour. = : = = = : (ii) Working hours = 8 = = Lunch interval: working hours = : = 2 = = = : India has a fleet of 28 naval ships, Pakistan has 4 naval ships and Bangladesh has 2 naval ships. Find the ratio of the sizes of three fleets, in lowest terms. Ans. India has naval ship = 28, Pakistan has naval ship = 4, Bangladesh has naval ship = 2 Required ratio = 28 : 4 : 2 = 4 : 7 : 6. A pole of length m 80 cm is divided into two parts such that their lengths are in the ratio : 7, find the length of each part of the pole. Ans. Total length of pole = m 80 cm or 80 cm ( metre = 00 cm) Ratio of length = : 7; Sum of terms of ratio = + 7 = 2 7 Length of Ist part = 80 = 7 cm; Length of IInd part = 80 = 0 cm Divide Rs. 60 between Ramu and Munni in the ratio 3 : 2. Ans. Total amount to be divided = Rs 60; Sum of the terms of ratio = = Fraction of Ramu = 3 ; Fraction of Munni = 2 ; Part of Ramu = 3 60 = Rs 336; Part of Munni = 2 60 = Rs Arnav and Prerna are aged 8 years and 0 years respectively. Their mother divides Rs 0 in the ratio of their ages. How much does each get? 4

5 Ans. Ratio of ages = 8 : 0 = 4:. Sum of terms of ratio = 4 + = Total money = Rs 0, Part of Arnav = Part of Prerna = 0 = Rs = Rs 40. An amount of hundred rupees is divided among two persons in the ratio :. 0 How much money does each get? Ans. Ratio = : = = = 3 : 2; Sum of ratio = = Total amount = Rs 00; Part of first person = 00 = Rs 60 2 Part of second person = 00 = Rs The lengths of the sides of a triangle are in the ratio 2 : 3 : 4. If the perimeter of the triangle is 63 cm, find the lengths of the sides of the triangle. Ans. Ratio of sides of triangle = 2 : 3 : 4; Sum of ratio of sides of triangle = = If the perimeter = 63 cm; First side = 3 63 = 7 3 = 2 cm; Third side = 2. An alloy of zinc and copper weighs copper is : 4, find the weight of copper in it = 7 2 = 4 cm; Second side = 4 63 = 7 4 = 28 cm. 2 2 kg. If in the alloy, the ratio of zinc to 2 Ans. Weight of alloy = 2 kg = kg. Given ratio = : 4; Sum of the terms of the 2 2 ratio = + 4 = ; Weight of copper = 4 2 kg ; = 2 = 0 kg Divide a sum of Rs,30 among A, B and C such that B gets equal to one - third of A and C gets equal to half of B. Ans. Let the share A = ; Share of B = 3 rd of A = = ; Share of C = of B = = 2 3 6

6 Given ratio = : : 3 6 = 6 : 6 : 6 = 6 : 2 : ( LCM of 3 and 6 = 6) 3 6 Sum of the terms of the ratio = = ; Total sum = Rs 30 A share = 6 Rs 30 = 6 Rs 0 = Rs 00; B share = 2 Rs 30 = Rs 300 C share = Rs 30 = Rs 0 = A mixture weighing kg contains three substances A, B and C in the ratio : :. 2 3 Find the quantity of each substance in the mixture. Ans. Weight of mixture = kg. Given ratio = : : 2 3 = 30 : 30 : = : 6 : 0 Sum of the terms of the ratio = = 3 Quantity of substance A in mixture = kg 3 = kg = 7 kg Quantity of substance B in mixture = 6 kg 3 = 6 kg = 30 kg Quantity of substance C in mixture = 0 kg 3 = 0 kg = 0 kg. 24. Mr.Gupta divides Rs 4,00 among his three children Ashok, Mohit and Geeta in such a way that Ashok gets equal to four times of what Mohit gets and Mohit gets equal to 2. times of what Geeta gets. Find the share of each. Ans. Let the share of Geeta =. Share of Mohit is (2. times of Geeta) = 2. Share of Ashok is (4 times of Mohit) = 4 2. = 0. Ratio = : 2. : 0 = 2 : 2. 2 : 0 2 = 2 : : 20 Sum of the terms of ratio = = 27 Share of Geeta = 2 Rs 400 = 2 Rs 0 = Rs Share of Mohit = Rs = Rs 0 = Rs 70 Share of Ashok = 20 Rs = 20 Rs 0 = Rs

7 2. In a proportion, the first, second and fourth terms are 32, 2 and 27 respectively. Find the third term. Ans. Let the third term of the proportion be x 32 : 2 : : x : x = x = = = 2 3 = The ratio of length to breadth of a rectangular playground is 3 : 0. If the length of the playground is 260 meters, find its breadth. Ans. Let the breadth be x. Length = 260 meters. Length : Breadth = 3 : : x = 3 : x = x = = Hence, breadth = 200 metre. 27. The ratio between weights of copper and zinc in an alloy is : 7. If the weight of zinc in the alloy is.8 kg. find : (i) the weight of copper in the alloy. (iii) the ratio of weight of copper to that of alloy. Ans. Weight of Zinc =.8 kg Copper : Zinc = : 7 (i) Let the weight of copper be x, then x :.8 = : 7 7 x =.8 = x = Hence, weight of copper = 2.6 kg = = (ii) Weight of copper : weight of alloy 2.6 : 22.4 = = 26 = = : Find the fourth term of the proportion whose first, second and third terms are 8, 27 and 32 respectively. Ans. Let the fourth term be x. 8 : 27 : : 32 : x 8 x = x = = 3 6 = 48 8 Thus fourth term = The ratio of the length to the width of a school ground is 2. Find its length, if width is 40 metres. Ans. Let the length = x m, width = 40 m; The ratio of length to width = x : 40 As per condition : 2 = x : 40 2 x = 20 x = 40 = 20 = 00 m 2 Length of school ground = 00 m. 7

8 30. On a map, a 4 cm tall building is drawn and in front of it a. cm tall tree is drawn. If the actual height of the building is 2 metre, find the actual height of the tree. Ans. Let the actual height of tree be x; In map height of building = 4 cm. Height of tree =. cm. Actual height of building = 2 m. 4 : = 2 : x 4 x =. 2 x = Actual height of tree is 4. metre = 4.0 metre. 8

In the ratio a : b, a and b are called the terms of ratio, `a' is the antecedent and `b' is the consequent.

Ratio: The number of times one quantity contains another quantity of the same kind is called ratio of the two quantities. The ratio of a to b is written as a : b a b a b In the ratio a : b, a and b are