9 Ratio and Proportion IN T H I S CH A P T E R, Y O U W I L L LEARN: Ratio Dividing a quantity in a given ratio Proportion and cross product rule Solving real life problems using unitary method introduction In our daily life, many a times we come across situation where we have to compare two quantities of the same kind in terms of their measurements/magnitudes. For example: (i) Suppose Raja s age is 12 years and his sister Rani s age is 9 years. For comparing their ages, we can find the difference of their age. Raja s age Rani s age = 12 years 9 years = 3 years. We can say that Raja s age is 3 years more than that of Rani s age. Thus, we have compared the ages of Raja and Rani by finding difference between their ages. Such a comparison is called comparison by difference. (ii) Suppose the cost of a Ritz s car is 40000 and the cost of a Bajaj s scooter is 0000. Cost of Ritz s car cost of Bajaj s scooter = 40000 0000 = 400000. We can say that the cost of Ritz s car is 400000 more than that of the cost of Bajaj s scooter. In this case, we note that the comparison by difference does not make much sense. For a better comparison between their costs, we use division. Cost of Ritz s car Cost of Bajaj s scooter `40000 = `0000 We can say that the cost of Ritz s car is nine times the cost of Bajaj s scooter. Thus, in certain situations, comparison by division makes much better sense than comparison by difference. The comparison by division is called ratio. ratio Ratio is a comparison of the sizes (or measures) of two or more quantities of same kind by division. = 1 9. If a and b are two quantities of the same kind (in same units), then the fraction a b is called the ratio of a to b. It is written as a : b (read as a is to b ). Thus, the ratio of a to b = a or a : b. b The quantities a and b are called the terms of the ratio; a is called the first term (or antecedent) and b is called the second term (or consequent). For example: (i) The ratio of 8 to 13 = 8 or 8 : 13. 13
ratio and ProPortion 179 (ii) The ratio of 19 kg to 2 kg = 19 or 19 : 2. 2 (iii) The ratio of hours to 11 hours = or : 11. 11 remarks Since ratio is a fraction, both of its terms (numerator and denominator) can be multiplied or divided by the same (non-zero) number. For example, 4 : 6 = 4 6 = 4 3 6 3 = 12 18 4 : 6 = 12 : 18. Also, 4 : 6 = 4 6 = 4 2 = 2 6 2 3 4 : 6 = 2 : 3. Usually, the ratio is expressed in its simplest form i.e. lowest terms. The order of terms in a ratio is very important. The ratio of 8 to 13 = 8 or 8 : 13, 13 the ratio of 13 to 8 = 13 or 13 : 8. 8 The ratio 8 : 13 is different from 13 : 8. Ratio exists only between two quantities of the same kind. For example, (i) there is no ratio between 30 and 12 m. (ii) it is meaningless to compare the weight of one child with the height of another child. Sometimes the quantities may be of the same kind but may be expressed in different units. You must convert them into same units. Usually, the bigger unit is converted into smaller unit. For example, if the height of Ramu is 1 metre and the height of Munni is 80 centimetre, then fi rst convert 1 metre to centimetre i.e. 1 metre = 100 centimetre. The ratio of their heights = 100 80 = or : 4. 4 Ratio has no units. Ratio being a fraction has no units (because the units in the numerator and denominator cancel out). Equivalent ratios Two ratios are called equivalent if the fractions corresponding to them are equivalent. Thus, the ratio 3 : 2 is equivalent to 6 : 4 or 1 : 10. Example 1. Number of boys in a school is 1200 and the number of girls is 800. Find the ratio of the number of boys to that of girls in the simplest form. Solution. Number of boys = 1200, number of girls = 800. Ratio of number of boys to number of girls = 1200 800 = 3 2. Hence, the required ratio in simplest form is 3 : 2. Example 2. Find the ratio of each of the following in simplest form: (i) 2.0 to 7 paise (iii) 300 g to 2 kg (ii) 60 cm to 2.4 m (iv) 8 months to 2 years. Divide by H.C.F. of 1200 and 800 i.e. by 400
180 Learning mathematics Vi Solution. (i) 2.0 = (2.0 100) paise = 20 paise. Ratio of 2.0 to 7 paise = 20 7 = 10 3 or 10 : 3. (ii) 2.4 m = (2.4 100) cm = 240 cm. Ratio of 60 cm to 2.4 m = (iii) 2 kg = (2 1000) g = 2000 g. Ratio of 300 g to 2 kg = 60 240 = 1 4 300 2000 = 3 20 (iv) 2 years = (2 12) months = 24 months. Ratio of 8 months to 2 years = 8 24 = 1 3 or 1 : 4. or 3 : 20. or 1 : 3. Example 3. A labourer earns 1200 a month and spends 800. Find the ratio of his (i) expenditure to income (ii) saving to income (iii) saving to expenditure. Solution. Here, income = 1200, expenditure = 800, saving = 1200 800 = 400. (i) Ratio of expenditure to income = 800 1200 = 3 or 2 : 3. 2 (ii) Ratio of saving to income = 400 1200 = 1 or 1 : 3. 3 (iii) Ratio of saving to expenditure = 400 800 = 1 or 1 : 2. 2 Example 4. Out of 1800 students in a school, 70 opted basketball, 800 opted cricket and the remaining opted table tennis. If a student can opt only one game, find the ratio of: (i) number of students who opted basketball to the number of students opting table tennis. (ii) number of students who opted cricket to the number of students opting basketball. (iii) number of students who opted basketball to the total number of students. Solution. Total number of students (opting for a game) = 1800, number of students opting basketball = 70 and number of students opting cricket = 800. Since a student can opt only for one game, the number of students opting table tennis = 1800 70 800 = 20. (i) Ratio of number of students opting basketball to the number of students opting table tennis 70 = 3 or 3 : 1 1 = 20 (ii) Ratio of number of students opting cricket to the number of students opting basket ball 16 or 16 : 1. = 800 = 70 1 (iii) Ratio of number of students opting basketball to the total number of students 70 = or : 12. 12 = 1800 Example. Present age of father is 42 years and that of his son is 14 years. Find the ratio of: (i) Present age of father to the present age of son. (ii) Age of father after 10 years to the age of son after 10 years. (iii) Age of father to the age of son when son was 12 years old. (iv) Age of father to the age of son when father was 30 years old. Convert the given quantities in same units
ratio and ProPortion 181 Solution. Present age of father = 42 years, present age of son = 14 years. (i) Ratio of the present age of father to the present age of the son = 42 = 3 or 3 : 1. 14 1 (ii) Age of a father after 10 years = 42 years + 10 years = 2 years, age of son after 10 years = 14 years + 10 years = 24 years. Ratio of age of father after 10 years to age of son after 10 years = 24 (iii) When son was 12 years old i.e. 2 years before, 2 = 6 age of father = 42 years 2 years = 40 years. Ratio of age of father to the age of son when son was 12 years old = 12 (iv) When father was 30 years old i.e. 12 years before, age of son = 14 years 12 years = 2 years. Ratio of age of father to the age of son when father was 30 years old = 2 to divide a quantity in a given ratio Example 6. Divide 300 among Ravi and Isha in the ratio 2 : 3. Solution. Here, the two terms of the ratio 2 : 3 are 2 and 3. 13 or 13 : 6. 40 = 3 30 = 1 10 or 10 : 3. 1 or 1 : 1. Sum of these terms = 2 + 3 =. This means that if the money is divided into equal parts then Ravi should get 2 parts and Isha should get 3 parts i.e. Ravi should get 2 of the total money and Isha should get 3 of the total money. Since the total money to be divided is 300, Ravi gets 2 of 300 = 2 300 = 120. and Isha gets 3 of 300 = 3 300 = 180. Example 7. Two girls are aged 12 years and 1 years. They want to divide 180 in the ratio of their ages. How much money would each get? Solution. The ages of the girls are 12 years and 1 years. The ratio in which money is to be divided is 12 : 1 i.e. 4 :. Sum of terms of the ratio = 4 + = 9. Since total money to be divided is 180, girl aged 12 years get 4 of 180 = c 4 # 180m = 80 9 9 and girl aged 1 years get of 180 = c # 180m = 100. 9 9 Example 8. The ratio of the marks secured by Ayush and Amrita is : 6. If Amrita secured 78 marks, what did Ayush score? Solution. Given, ratio of marks secured by Ayush and Amrita is : 6 marks secured by Ayush = but marks secured by Amrita is 78 marks secured by Amrita 6 marks secured by Ayush = 78 6 marks secured by Ayush = 6 78 = 13 = 6. Hence, marks secured by Ayush = 6. Symbol stands for the word implies
182 Learning mathematics VI Exercise 9.1 1. Express the following ratios in simplest form: (i) 20 : 40 (ii) 40 : 20 (iii) 81 : 108 (iv) 98 : 63. 2. Fill in the missing numbers in the following equivalent ratios: (i) 14 =... = 6 (ii) 1 =... = 10 =.... 21 3... 18 6... 30 3. Find the ratio of each of the following in simplest form: (i) 2.1 m to 1.2 m (ii) 91 cm to 1.04 m (iii) 3. kg to 20 g (iv) 60 paise to 4 rupees (v) 1 minute to 1 seconds (vi) 1 mm to 2 cm (vii) 30 minutes to 1. hours (viii) 00 g to 2 kg. 4. The length and the breadth of a rectangular field are 0 m and 1 m respectively. Find the ratio of the length to the breadth of the field.. There are 102 teachers in a school of 3300 students. Find the ratio of the number of teachers to the number of students. 6. Distances travelled by Hamid and Ravinder in an hour are 12 km and 9 km respectively. Find the ratio of speed of Hamid to the speed of Ravinder. 7. In a class, there are 30 boys and 2 girls. Find the ratio of the number of (i) boys to that of girls. (ii) girls to that of total number of students. (iii) boys to that of total number of students. 8. The population of a village is 4800. If the number of females is 2160, find the ratio of number of males to that of females. 9. In a school, out of 4320 students, 2300 are girls. Find the ratio of (i) number of girls to the total number of students. (ii) number of boys to the number of girls. 10. In a year, Seema earns 1,0,000 and saves 0,000. Find the ratio of (i) money she earns to the money she saves. (ii) money that she saves to the money she spends. 11. The monthly expenses of a student have increased from 30 to 00. Find the ratio of (i) increase in expenses to original expenses. (ii) original expenses to increased expenses. (iii) increased expenses to increase in expenses. 12. Ratio of distance of the school from John s home to the distance of the school from Mary s home is 2 : 1. (i) Who lives nearer to the school? (ii) Complete the following table which shows some possible distances that John and Mary could live from the school. Distance from John s home to school (in km) 10 4 Distance from Mary s home to school (in km) 4 3 1 (iii) If the distance of John s home to distance of Meenu s home from school is 1 : 2, then who lives nearer to school? 13. Out of 30 students in a class, 6 like football, 12 like cricket and the remaining like tennis. Find the ratio of (i) number of students liking football to number of students liking tennis. (ii) number of students liking cricket to total number of students.