NCERT Solutions for Class 6 Maths Chapter - 12: Ratio and ProportionExercise 12.11. There are 20 girls and 15 boys in a class. (a) What is the ratio of number of girls to the number of boys? Answer: 4: 3 Explanation: Number of girls = 20 Number of boys = 15 Ratio of number of girls to the number of boys = Number of girls/Number of boys = 20/15 = 4/3 (5 is the common factor of both the numbers 20 and 15) Ratio of number of girls to the number of boys = 4: 3 (b) What is the ratio of number of girls to the total number of students in the class? Answer: 4: 7 Explanation: Number of girls = 20 Number of boys = 15 Total number of students = Number of girls + Number of boys = 15 + 20 = 35 Ratio of number of girls to the total number of students = Number of girls/Total number of students = 20/35 = 4/7 (5 is the common factor of both the numbers 20 and 35) Ratio of number of girls to the number of students = 4: 7 2. Out of 30 students in a class, 6 like football, 12 like cricket and remaining like tennis. Explanation: Total students in a class = 30 Students liking football = 6 Students liking cricket = 12 Remaining students = Total students - (student liking football + cricket) = 30 - (6 + 12) = 30 - 18 = 12 Thus, students liking tennis = 12 Find the ratio of: (a) Number of students liking football to number of students liking tennis. Answer: 1:2 Explanation: Total students in a class = 30 Students liking football = 6 Students liking tennis = 12 Ratio of the students liking football to the students liking tennis = Number of students like football/ Number of students liking tennis = 6/12 = 1/2 (6 is the common factor of both the numbers 6 and 12) Thus, Ratio of the students liking football to the students liking tennis = 1: 2 (b) Number of students liking cricket to total number of students. Answer: Explanation: Total students in a class = 30 Students liking cricket = 12 Ratio of the students liking cricket to the total students = Number of students liking cricket / Total number of students = 12/30 = 2/5 (6 is the common factor of both the numbers 12 and 30) Thus, Ratio of the students liking cricket to the total students = 2: 5 3. See the figure and find the ratio of (a) Number of triangles to the number of circles inside the rectangle. Answer: 3: 2 Explanation: Total number of figures inside the rectangle = 7 Number of triangles = 3 Number of circles = 2 Number of squares = 2 Ratio of the number of triangles to the number of circles = Number of triangles/Number of circles = 3/2 Thus, Ratio = 3: 2 (b) Number of squares to all the figures inside the rectangle. Answer: 2: 7 Explanation: Total number of figures inside the rectangle = 7 Number of triangles = 3 Number of circles = 2 Number of squares = 2 Ratio of the number of squares to the number of figures inside the rectangle = Number of squares /Number of figures = 2/7 Thus, Ratio = 2: 7 (c) Number of circles to all the figures inside the rectangle. Answer: 2:7 Explanation: Total number of figures inside the rectangle = 7 Number of triangles = 3 Number of circles = 2 Number of squares = 2 Ratio of the number of circles to the number of figures inside the rectangle = Number of circles /Number of figures = 2/7 Thus, Ratio = 2: 7 4. Distances travelled by Hamid and Akhtar in an hour are 9 km and 12 km. Find the ratio of speed of Hamid to the speed of Akhtar. Answer: 3: 4 Explanation: Distance travelled by Hamid = 9km Speed = Distance/Time Speed = 9/Time Distance travelled by Akhtar = 12km Speed = Distance/Time Speed = 12/Time Ratio of speed of Hamid to the speed of Akhtar = Speed of Hamir/Speed of Akhtar = (9/Time)/ (12/Time) Since, the time are the same, it will cancel out. Ratio = 9/12 Ratio = 3/4 (3 is the common factor of both the numbers 9 and 12) Ratio = 3: 4 5. Fill in the following blanks: [Are these equivalent ratios?] Answer: 5, 12, 25, Yes Explanation: Here, we will consider two ratios at a time to find the missing value. Step 1: Let's compare the denominator of both the ratios. Dividing 18 by 6, we get: 18/6 = 3 It means that the numerator of the given ratio can also be divided by 3. 15/3 = 5 (3 × 5)/ (3 × 6) Thus, the missing number is 5. Step 2: Here, we need to find the denominator. So, we will compare the numerators of the next two ratios. 5 × 2 = 10 Similarly, multiplying 2 by 6, we get: 6 × 2 = 12 Thus, the denominator of the other missing ratio is 12. Step 3: Let's compare the denominator of both the ratios. 30/12 = 2.5 Or 12 × 2.5 = 30 Multiplying the numerator by 2.5, we get: 10 × 2.5 = 25 The numerator of the third missing ratio is 25. Thus, the given question with all the missing numbers can be written as: Whenever two ratios are specified in an equivalent equation, they are always tends to be equal. Hence, all the four ratios specified in the given question are equal. 6. Find the ratio of the following: Explanation: To find the ratio, we need to convert the fraction to its lowest form. It means that the numerator and the denominator of the fraction have only 1 as the common factor. So, we will divide the ratio by its common factors, i.e., the common factors of both the numerator and the denominator. (a) 81 to 108 Answer: 3: 4 Explanation: Factors of 81: 3 × 3 × 3 × 3 Factors of 108: 3 × 3 × 3 × 4 81/108 can be written as: 81/108 = 3 × 3 × 3 × 3/ 3 × 3 × 3 × 4 Cancelling the common numbers, we get: 81/108 = 3/4 Thus, the ratio of 81 to 108 is 3: 4. (b) 98 to 63 Answer: 14: 9 Explanation: Factors of 98: 2 × 7 × 7 Factors of 63: 3 × 3 × 7 98/63 can be written as: 98/63 = (2 × 7 × 7)/ (3 × 3 × 7) Cancelling the common numbers, we get: = 14/9 Ratio = 14: 9 (c) 33 km to 121 km Answer: 3: 11 Explanation: Factors of 33: 3 × 11 Factors of 121: 11 × 11 33/121 can be written as: 33/121 = (3 × 11) / (11 × 11) Cancelling the common numbers, we get: 33/121 = 3/11 Ratio = 3: 11 (d) 30 minutes to 45 minutes Answer: 2: 3 Explanation: Factors of 30: 2 × 3 × 5 Factors of 45: 3 × 3 × 5 30/45 can be written as: 30/45 = (2 × 3 × 5) / (3 × 3 × 5) Cancelling the common numbers, we get: 30/45 = 2/3 Thus, Ratio = 2: 3 7. Find the ratio of the following: (a) 30 minutes to 1.5 hours Answer: 1: 3 Explanation: To find the ratio, the units of both the numerator and the denominator should be the same. 1 hour = 60 minutes 1.5 hours = 1.5 × 60 = 90 minutes Factors of 30: 3 × 2 × 5 Factors of 90: 3 × 3 × 2 × 5 30/90 can be expressed as: 30/90 = (3 × 2 × 5)/ (3 × 3 × 2 × 5) Cancelling the common numbers, we get: 30/90 = 1/3 Thus, Ratio of 30 minutes to 1.5 hours= 1: 3 (b) 40 cm to 1.5 m Answer: 4: 15 Explanation: To find the ratio, the units of both the numerator and the denominator should be the same. 1 m = 100 cm 1.5 m = 1.5 × 100 = 150 cm Factors of 40: 2 × 2 × 2 × 5 Factors of 150: 2 × 3 × 5 × 5 40/150 can be expressed as: 40/150 = (2 × 2 × 2 × 5)/ (2 × 3 × 5 × 5) Cancelling the common numbers, we get: 40/150 = 4/15 Thus, Ratio of 40 cm to 1.5 m= 4: 15 (c) 55 paise to ₹ 1 Answer: 11: 20 Explanation: To find the ratio, the units of both the numerator and the denominator should be the same. 1 rupee = 100 paisa Factors of 55: 11 × 5 Factors of 100: 2 × 2 × 5 × 5 55/100 can be expressed as: 55/100 = (11 × 5) / (2 × 2 × 5 × 5) Cancelling the common numbers, we get: 55/100 = 11/20 Ratio = 11: 20 Ratio of 55 paise to ₹ 1 = 11: 20 (d) 500 mL to 2 litres Answer: 1:4 Explanation: To find the ratio, the units of both the numerator and the denominator should be the same. 1 litre = 1000 ml 2 litres = 2 × 1000 = 2000 ml Factor of 2000: 4 × 500 500/2000 can be written as: 500/2000 = (500)/ (4 × 500) Cancelling the common numbers, we get: = 1/4 Ratio = 1:4 Thus, ratio of 500 mL to 2 litres is 1: 4. 8. In a year, Seema earns ₹ 1,50,000 and saves ₹ 50,000. Find the ratio of: (a) Money that Seema earns to the money she saves. Answer: 3: 1 Explanation: Money earned by Seema = 150000 Money saved by Seema = 50,000 Ratio of money earned to the money saved = Earned money/ Saved money = 150000/ 50000 = (3 × 50000) /50000 = 3/1 Thus, the ratio of money earned to the money saved is 3: 1. (b) Money that she saves to the money she spends. Answer: 1: 2 Explanation: Money earned by Seema = 150000 Money saved by Seema = 50,000 Money spends by Seema = Money earned - Money saved = 150000 - 50000 = 1, 0 0, 000 Ratio of money saved to the money spend = Saved money/ Spent money = 50000/ 100000 = (50000) / (2 × 50000) = 1/2 Thus, the ratios of money saved to the money spend 1: 2. 9. There are 102 teachers in a school of 3300 students. Find the ratio of the number of teachers to the number of students. Answer: 17: 550 Explanation: Total teachers in a school = 102 Total number of students in a school = 3300 Ratio of the number of teachers to the number of students = Number of teachers/ Number of students = 102/3300 Factors of 102: 2× 3 × 17 Factors of 3300: 3 × 11 × 2 × 50 102/3300 = (2× 3 × 17) / (3 × 11 × 2 × 50) 102/3300 = 17/550 Ratio of the number of teachers to the number of students = 17: 550 10. In a college, out of 4320 students, 2300 are girls. Find the ratio of (a) Number of girls to the total number of students. Answer: 115: 216 Explanation: Total students = 4320 Number of girls = 2300 Number of boys = Total students - Number of girls = 4320 - 2300 = 2020 Ratio of the number of girls to the total number of students = Number of girls/ Total students = 2300/4320 Factors of 2300: 23 × 2 × 2 × 5 × 5 Factors of 4320: 2 × 2 × 2 × 2 × 2 × 5 × 3 × 3 × 3 2300/4320 = (23 × 2 × 2 × 5 × 5) / (2 × 2 × 2 × 2 × 2 × 5 × 3 × 3 × 3) Cancelling out the common factors, we get: 2300/4320 = (23 × 5)/ (2 × 2 × 2 × 3 × 3 × 3) 2300/4320 = 115/216 Thus, Ratio of the number of girls to the total number of students = 115: 216 (b) Number of boys to the number of girls. Answer: 101: 115 Explanation: Total students = 4320 Number of girls = 2300 Number of boys = Total students - Number of girls = 4320 - 2300 = 2020 Ratio of the number of boys to the number of girls = Number of boys/ Number of girls = 2020/2300 Factors of 2300: 23 × 2 × 2 × 5 × 5 Factors of 2020: 2 × 2 × 5 × 101 2020/2300 = (2 × 2 × 5 × 101) / (23 × 2 × 2 × 5 × 5) Cancelling out the common factors, we get: 2300/4320 = (101)/ (23 × 5) 2300/4320 = 101/115 Thus, Ratio of the number of boys to the number of girls = 101:115 (c) Number of boys to the total number of students. Answer: 101: 216 Explanation: Total students = 4320 Number of girls = 2300 Number of boys = Total students - Number of girls = 4320 - 2300 = 2020 Ratio of the number of boys to the total number of students = Number of boys/ Number of students = 2020/4320 Factors of 2020: 2 × 2 × 5 × 101 Factors of 4320: 2 × 2 × 2 × 2 × 2 × 5 × 3 × 3 × 3 2020/4320 = (2 × 2 × 5 × 101/ (2 × 2 × 2 × 2 × 2 × 5 × 3 × 3 × 3) Cancelling out the common factors, we get: 2020/4320 = (101)/ (2 × 2 × 2 × 3 × 3 × 3) 2020/4320 = 101/216 Thus, Ratio of the number of boys to the total number of students = 101: 216 11. Out of 1800 students in a school, 750 opted basketball, 800 opted cricket and remaining opted table tennis. Explanation: Total number of students = 1800 Number of students opting basketball = 750 Number of students opting cricket = 800 Number of students opting basketball tennis = Total students - (Number of students opting basketball + Number of students opting cricket) = 1800 - (750 + 800) = 1800 - 1550 = 250 Number of students opting basketball tennis = 250 If a student can opt only one game, find the ratio of: (a) Number of students who opted basketball to the number of students who opted table tennis. Answer: 3: 1 Explanation: Total number of students = 1800 Number of students opting basketball = 750 Number of students opting cricket = 800 Number of students opting basketball tennis = 250 Ratio of number of students who opted basketball to the number of students who opted table tennis = Students opting basketball/ Students opting tennis = 750/250 Factors of 750: 3 × 250 750/250 = (3 × 250)/ 250 = 3/1 Thus, Ratio = 3: 1 (b) Number of students who opted cricket to the number of students opting basketball. Answer: 16: 15s Explanation: Total number of students = 1800 Number of students opting basketball = 750 Number of students opting cricket = 800 Number of students opting basketball tennis = 250 Ratio of number of students who opted cricket to the number of students who opted table basketball = Students opting cricket / Students opting basketball = 800/750 Factors of 750: 3 × 2 × 5 × 5 × 5 Factors of 800: 2 × 2 × 2 × 2 × 2 × 5 × 5 800/750 can be written as 800/750 = (2 × 2 × 2 × 2 × 2 × 5 × 5)/ (3 × 2 × 5 × 5 × 5) Cancelling out the common terms, we get: 800/750 = (2 × 2 × 2 × 2)/ (3 × 5) 800/750 = 16/15 Thus, Ratio = 16: 15 750/250 = (3 × 250)/ 250 = 3/1 Thus, Ratio = 3: 1 (c) Number of students who opted basketball to the total number of students. Answer: 5: 12 Explanation: Total number of students = 1800 Number of students opting basketball = 750 Number of students opting cricket = 800 Number of students opting basketball tennis = 250 Ratio of number of students who opted basketball to the number of students = Students opting basketball/ Total students = 750/1800 Factors of 750: 3 × 2 × 5 × 5 × 5 Factors of 1800: 3 × 3 × 2 × 2 × 2 × 5 × 5 750/1800 can be written as: 750/1800 = (3 × 2 × 5 × 5 × 5) / (3 × 3 × 2 × 2 × 2 × 5 × 5) Cancelling out the common terms, we get: 750/1800 = 5/ (3 × 2 × 2) 750/1800 = 5 / 12 Thus, Ratio = 5: 12 12. Cost of a dozen pens is ₹ 180 and cost of 8 ball pens is ₹ 56. Find the ratio of the cost of a pen to the cost of a ball pen. Answer: 3: 2 Explanation: 1 dozen = 12 pens Cost of a dozen pens = ₹ 180 Cost of 1 pen = 180/12 = 15 Cost of 1 pen = ₹ 15 Cost of 8 ball pens = ₹ 56 Cost of 1 ball pen = 56/8 = 7 Cost of 1 ball pen = ₹ 7 Ratio of the cost of a pen to the cost of a ball pen = Cost of 1 pen/Cost of 1 ball pen = 15/7 Thus, Ratio = 15: 7 13. Consider the statement: Ratio of breadth and length of a hall is 2: 5. Complete the following table that shows some possible breadths and lengths of the hall.
Answer: Explanation: 20, 100 Here, we will consider two ratios at a time to find the missing value. Step 1:
To find the numerator, we will find the relation between their denominators. 25 × 2 = 50 Similarly, 10 × 2 = 20 Thus, the numerator of the second ratio is 20.
Step 2:
To find the denominator, we will find the relation between their numerators. 20 × 2 = 40 Similarly, 50 × 2 = 100 The numerator of the third ratio is 100. Thus, The given ratios with their missing value can be represented as:
14. Divide 20 pens between Sheela and Sangeeta in the ratio of 3: 2. Answer: 12 and 8 Explanation: Sheela has the ratio of 3 and Sangeeta has the ratio of 2. So, ratio of pens can be represented as: Ratio/ (Sum of ratio) × Total pens Share of Sheela = 3/ (3 + 2) × 20 = 3/5 × 20 = 12 Share of Sangeeta = 2/ (3 + 2) × 20 = 2/5 × 20 = 8 15. Mother wants to divide ₹ 36 between her daughters Shreya and Bhoomika in the ratio of their ages. If age of Shreya is 15 years and age of Bhoomika is 12 years, find how much Shreya and Bhoomika will get. Answer: ₹20 and ₹16 Explanation: To find the division of rupees 36, we will first find the ratio of the ages of Shreya and Bhoomika. Ratio = Age of Shreya/ Age of Bhoomika Ratio = 15/12 Ratio = (3 × 5)/ (3 × 4) = 5/4 Ratio = 5: 4 So, ratio of money can be represented as: Ratio/ (Sum of ratio) × Total money Share of Shreya = 5/ (5 + 4) × 36 = 5/9 × 36 = 20 Thus, the share of Shreya is rupees 20. Share of Bhoomika = 4/ (5 + 4) × 36 = 4/9 × 36 = 16 Thus, the share of Bhoomika is rupees 16. 16. Present age of father is 42 years and that of his son is 14 years. Find the ratio of (a) Present age of father to the present age of son. Answer: 3: 1 Explanation: Ratio = Present age of father/ Present age of son Present age of father = 42 years Present age of son = 14 years Ratio = 42/14 = (2 × 3 × 7) / (2 × 7) = 3/1 Ratio = 3: 1 Thus, the ratio of the present age of father to the present age of son is 3: 1. (b) Age of the father to the age of son, when son was 12 years old. Answer: 10: 3 Explanation: Age of son = 12 years It means 2 years younger than the present age. Age of father = present age - 2 Age of father = 42 - 2 = 40 years Ratio = 40/12 = (2 × 2 × 2 × 5) / (2 × 3 × 2) Cancelling out the common factors, we get: Ratio = 10/3 Ratio = 10: 3 (c) Age of father after 10 years to the age of son after 10 years. Answer: 13: 6 Explanation: Present age of father = 42 years Present age of son = 14 years Age of father after 10 years = 42 + 10 = 52 years Age of son after 10 years = 14 + 10 = 24 years Ratio = Age of father after 10 years/ Age of son after 10 years = 52/24 = (2 × 2 × 13)/ (2 × 2 × 2 × 3) Cancelling out the common factors, we get: = 13/6 Ratio = 13: 6 (d) Age of father to the age of son when father was 30 years old. Answer: 15: 1 Explanation: Present age of father = 42 years Present age of son = 14 years Age of father = 30 years It means 12 years younger than the present age. Age of son = Present age - 12 = 14 - 12 = 2 years Ratio = 30/2 = 15/1 Ratio = 15: 1 Thus, the ratio of the age of the father to the age of son is 15: 1. Exercise 12.21. Determine if the following are in proportion. (a) 15, 45, 40, 120 Answer: Yes Explanation: When the two ratios are equal, they are said to be in proportion. We can write the above two ratios as: 15: 45:: 40: 120 '::' symbol divides the two ratios. We can also use an equal (=) sign to compare the two ratios. We need to convert the ratios to their lowest form. It makes the comparison easier. 15:45 can be written as: 15/45 15/(15 × 3) Cancelling out the common factors, we get: 15/45 = 1/3 So, Ratio = 1:3 40:120 can be written as, 40/120 = 40/ (40 × 3) Cancelling out the common factors, we get: 40/120 = 1/3 So, Ratio = 1:3 15:45 = 40:120 1: 3 = 1: 3 Both the ratios are equal. Hence, we can say that the given numbers are in proportion. (b) 33, 121, 9, 96 Answer: No Explanation: When the two ratios are equal, they are said to be in proportion. We can write the above two ratios as: 33: 121:: 9: 96 '::' symbol divides the two ratios. We can also use an equal (=) sign to compare the two ratios. We need to convert the ratios to their lowest form. It makes the comparison easier. 33:121 can be written as: 33/121 (11 × 3)/(11 × 11) Cancelling out the common factors, we get: 33/121 = 3/11 So, Ratio = 3: 11 9: 96 can be written as, 9/96 = (3 × 3)/ (3 × 32) Cancelling out the common factors, we get: 9/96 = 3/32 So, Ratio = 3:32 3: 11 is not equal to 3: 32 33: 121 is not equal to 9: 96. Hence, we can say that the given numbers are not in proportion. (c) 24, 28, 36, 48 Answer: No Explanation: When the two ratios are equal, they are said to be in proportion. We can write the above two ratios as: 24: 28:: 36: 48 '::' symbol divides the two ratios. We can also use an equal (=) sign to compare the two ratios. We need to convert the ratios to their lowest form. It makes the comparison easier. 24: 28 can be written as: 24/28 (4 × 6)/(4 × 7) Cancelling out the common factors, we get: 24/28 = 6/7 So, Ratio = 6: 7 36: 48 can be written as, 36/48 = (12 × 3)/ (12 × 4) Cancelling out the common factors, we get: 36/48 = 3/4 So, Ratio = 3:4 6: 7 is not equal to 3: 4 24: 28 is not equal to 36: 48 Hence, we can say that the given numbers are not in proportion. (d) 32, 48, 70, 210 Answer: No Explanation: When the two ratios are equal, they are said to be in proportion. We can write the above two ratios as: 32: 48:: 70: 210 '::' symbol divides the two ratios. We can also use an equal (=) sign to compare the two ratios. We need to convert the ratios to their lowest form. It makes the comparison easier. 32: 48 can be written as: 32/48 (16 × 2)/(16 × 3) Cancelling out the common factors, we get: 32/48 = 2/3 So, Ratio = 2: 3 70: 210 can be written as, 70/210 = (70 × 1)/ (70 × 3) Cancelling out the common factors, we get: 70/120 = 1/3 So, Ratio = 1: 3 2: 3 is not equal to 1: 3 32: 48 is not equal to 70: 120 Hence, we can say that the given numbers are not in proportion. (e) 4, 6, 8, 12 Answer: Yes Explanation: When the two ratios are equal, they are said to be in proportion. We can write the above two ratios as: 4: 6:: 8: 12 '::' symbol divides the two ratios. We can also use an equal (=) sign to compare the two ratios. We need to convert the ratios to their lowest form. It makes the comparison easier. 4: 6 can be written as: 4/6 (2 × 2)/(2 × 3) Cancelling out the common factors, we get: 4/6 = 2/3 So, Ratio = 2: 3 8: 12 can be written as, 8/12 = (4 × 2)/ (4 × 3) Cancelling out the common factors, we get: 8/12 = 2/3 So, Ratio = 2: 3 2: 3 = 2: 3 4: 6 = 8: 12 Hence, we can say that the given numbers are in proportion. (f) 33, 44, 75, 100 Answer: Yes Explanation: When the two ratios are equal, they are said to be in proportion. We can write the above two ratios as: 33: 44:: 75: 100 '::' symbol divides the two ratios. We can also use an equal (=) sign to compare the two ratios. We need to convert the ratios to their lowest form. It makes the comparison easier. 33: 44 can be written as: 33/44 (11 × 3)/(11 × 4) Cancelling out the common factors, we get: 33/44 = 3/4 So, Ratio = 3: 4 75: 100 can be written as, 75/100 = (25 × 3)/ (25 × 4) Cancelling out the common factors, we get: 75/100 = 3/4 So, Ratio = 3: 4 3: 4 = 3: 4 33: 44 = 75: 100 Hence, we can say that the given numbers are in proportion. 2. Write True (T) or False (F) against each of the following statements: (a) 16 : 24 :: 20 : 30 Answer: True Explanation: '::' can also be written as '='. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. 16: 24 can be written as: 16/24 = (8 × 2)/ (8 × 3) Cancelling out the common factors, we get: = 2/3 Ratio = 2: 3 20:30 can be written as: 20/30 = (10 × 2)/ (10 × 3) Cancelling out the common factors, we get: = 2/3 Ratio = 2: 3 The given ratios are equal. Hence, the given two ratios are in proportion. (b) 21: 6:: 35: 10 Answer: True Explanation: '::' can also be written as '='. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. 21: 6 can be written as: 21/6 = (3 × 7)/ (3 × 2) Cancelling out the common factors, we get: = 7/2 Ratio = 7: 2 35: 10 can be written as: 35/10 = (5 × 7)/ (5 × 2) Cancelling out the common factors, we get: = 7/2 Ratio = 7: 2 The given ratios are equal. Hence, the given two ratios are in proportion. (c) 12: 18:: 28: 12 Answer: False Explanation: '::' can also be written as '='. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. 12: 18 can be written as: 12/18 = (6 × 2)/ (6 × 3) Cancelling out the common factors, we get: = 2/3 Ratio = 2: 3 28: 12 can be written as: 28/12 = (4 × 7)/ (4 × 3) Cancelling out the common factors, we get: = 7/3 Ratio = 7: 3 The ratios are not equal. Hence, the given two ratios are not in proportion. (d) 8: 9:: 24: 27 Answer: True Explanation: '::' can also be written as '='. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. 8: 9 can be written as: 8/9 It is already present in its lowest form, i.e., both the numerator and the denominator have the common factor 1. 24: 27 can be written as: 24/27 = (3 × 8)/ (3 × 9) Cancelling out the common factors, we get: = 8/9 Ratio = 8: 9 The given ratios are equal. Hence, the given two ratios are in proportion. (e) 5.2 : 3.9 :: 3 : 4 Answer: False Explanation: '::' can also be written as '='. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. 5.2: 3.9 can be written as: 5.2/3.9 = (1.3 × 4)/ (1.3 × 3) Cancelling out the common factors, we get: = 4/3 Ratio = 4: 3 3: 4 is already present in its lowest form. The ratios are not equal. Hence, the given two ratios are not in proportion. (f) 0.9 : 0.36 :: 10 : 4 Answer: True Explanation: '::' can also be written as '='. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. 0.9: 0.36 can be written as: 0.9/0.36 = (0.09 × 10)/ (0.09 × 4) Cancelling out the common factors, we get: = 10/4 Ratio = 10: 4 10: 4 = 10: 4 The given ratios are equal. Hence, the given two ratios are in proportion. 3. Are the following statements true? (a) 40 persons: 200 persons = ₹ 15: ₹ 75 Answer: True Explanation: '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. The above statement can be represented as: 40: 200 = 15: 75 40: 200 can be written as: 40/200 = (40 × 1)/ (40 × 5) = 1/5 Ratio = 1: 5 15: 75 can be written as: 15/75 = (15 × 1)/ (15 × 5) Cancelling out the common factors, we get: = 1/5 Ratio = 1: 5 The given ratios are equal. Hence, the given two ratios are in proportion. The answer is true. (b) 7.5 litres: 15 litres = 5 kg: 10 kg Answer: True Explanation: '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. So, we will ignore the units. The above statement can be represented as: 7.5: 15 = 5: 10 7.5: 15 can be written as: 7.5/ 15 = (7.5 × 1)/ (7.5 × 2) = 1/2 Ratio = 1: 2 5: 10 can be written as: 5/10 = (5 × 1)/ (5 × 2) Cancelling out the common factors, we get: = 1/2 Ratio = 1: 5 The given ratios are equal. Hence, the given two ratios are in proportion. The answer is true. (c) 99 kg: 45 kg = ₹ 44: ₹ 20 Answer: True Explanation: '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. So, we will ignore the units. The above statement can be represented as: 99: 45 = 44: 20 99: 45 can be written as: 99/45 = (9 × 11)/ (9 × 5) = 11/5 Ratio = 11: 5 44: 20 can be written as: 44/20 = (4 × 11)/ (4 × 5) Cancelling out the common factors, we get: = 11/5 Ratio = 11: 5 The given ratios are equal. Hence, the given two ratios are in proportion. The answer is true. (d) 32 m: 64 m = 6 sec: 12 sec Answer: True Explanation: '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. The above statement can be represented as: 32: 64 = 6: 12 32: 64 can be written as: 32/64 = (32 × 1)/ (32 × 2) = 1/2 Ratio = 1: 2 6: 12 can be written as: 6/12 = (6 × 1)/ (6 × 2) Cancelling out the common factors, we get: = 1/2 Ratio = 1: 2 The given ratios are equal. Hence, the given two ratios are in proportion. The answer is true. (e) 45 km: 60 km = 12 hours: 15 hours Answer: False Explanation: '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. So, we will ignore the units. The above statement can be represented as: 45: 60 = 12: 15 45: 60 can be written as: 45/60 = (15 × 3)/ (15 × 4) = 3/4 Ratio = 3: 4 12: 15 can be written as: 12/15 = (3 × 4)/ (3 × 5) Cancelling out the common factors, we get: = 4/5 Ratio = 4: 5 The given ratios are not equal. Hence, the given two ratios are not in proportion. The answer is false. 4. Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion. (a) 25 cm: 1 m and ₹ 40: ₹ 160 Answer: Yes Middle terms: 1 m and ₹ 40 Extreme terms: 25 cm and ₹ 160 Explanation: 'and', '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. But, the units in a ratio should be the same. If not, we need to first convert the units of both the numerator and the denominator. The above statement can be represented as: 25: 100 = 40: 160 1m = 100 cm We have converted the unit meter into centimetre. We can also convert the unit centimetre to meter. The answer at the end will be the same in both cases. 25: 100 can be written as: 25/100 = (25 × 1)/ (25 × 4) Cancelling out the common factors, we get: = 1/4 Ratio = 1: 4 40: 160 can be written as: 40/160 = (40 × 1)/ (40 × 4) Cancelling out the common factors, we get: = 1/4 Ratio = 1: 4 The given ratios are equal. Hence, the given two ratios are in proportion. The answer is yes. (b) 39 litres: 65 litres and 6 bottles: 10 bottles Answer: Yes Middle terms: 65 litres and 6 bottles Extreme terms: 39 litres and 10 bottles Explanation: 'and', '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. But, the units in a ratio should be the same. If not, we need to first convert the units of both the numerator and the denominator. The above statement can be represented as: 39: 65 = 6: 10 39: 65 can be written as: 39/65 = (13 × 3)/ (13 × 5) Cancelling out the common factors, we get: = 3/5 Ratio = 3: 5 6: 10 can be written as: 6/10 = (2 × 3)/ (2 × 5) Cancelling out the common factors, we get: = 3/5 Ratio = 3: 5 The given ratios are equal. Hence, the given two ratios are in proportion. The answer is yes. (c) 2 kg: 80 kg and 25 g: 625 g Answer: No Explanation: 'and', '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. But, the units in a ratio should be the same. The above statement can be represented as: 2: 80 = 25: 625 2: 80 can be written as: 2/80 = (2 × 1)/ (2 × 40) Cancelling out the common factors, we get: = 1/40 Ratio = 1: 40 25: 625 can be written as: 25/625 = (25 × 1)/ (25 × 25) Cancelling out the common factors, we get: = 1/25 Ratio = 1: 25 The given ratios are not equal. Hence, the given two ratios are not in proportion. The answer is no. (d) 200 mL: 2.5 litre and ₹ 4: ₹ 50 Answer: Yes Middle terms: 2.5 litres and ₹ 4 Extreme terms: 200 ml and ₹ 50 Explanation: 'and', '=' can also be written as '::'. We need to check that the given numbers are in proportion or not. For the comparison, the two ratios are converted to their lowest form. The units in the ratio does not matter while comparison. But, the units in a ratio should be the same. If not, we need to first convert the units of both the numerator and the denominator. The above statement can be represented as: 200: 2500 = 4: 50 1 Litre = 1000 Ml 2.5 litres = 2.5 × 1000 = 2500 millilitres Here, we have converted the litres into millilitres. We can also convert millilitres to litres. The answer in both the cases would be the same. 200: 2500 can be written as: 200/2500 = (100 × 2)/ (100 × 25) Cancelling out the common factors, we get: = 2/25 Ratio = 2: 25 4: 50 can be written as: 4/50 = (2 × 2)/ (2 × 25) Cancelling out the common factors, we get: = 2/25 Ratio = 2: 25 The given ratios are equal. Hence, the given two ratios are in proportion. The answer is yes. Exercise 12.31. If the cost of 7m of cloth is ₹ 1470, find the cost of 5 m of cloth. Answer: ₹ 1050 Explanation: Cost of 7 m of cloth = ₹ 1470 Cost of 1 m of cloth = 1470/7 Cost of 1 m of cloth = ₹ 210 Cost of 5 m of cloth = 210 × 5 = ₹ 1050 Thus, the cost of 5m of cloth is rupees 1050. 2. Ekta earns ₹ 3000 in 10 days. How much will she earn in 30 days? Answer: ₹ 9000 Explanation: Amount earned by Ekta in 10 days = ₹ 3000 Amount earned by Ekta in 1 day = ₹ 3000/10 = ₹ 300 Amount earned by Ekta in 30 days = Amount earned 1 day × 30 = 300 × 30 = 9000 Thus, Ekta earns rupees 9000 is 30 days. 3. If it has rained 276 mm in the last 3 days, how many cm of rain will fall in one full week (7 days)? Assume that the rain continues to fall at the same rate. Answer: 64.4 cm Explanation: The amount rained in the last three days = 276 mm The amount rained in 1 da = 276/3 = 92 mm The amount rained in 1 day = 92 mm The amount rained in 7 days = 92 × 7 = 644 mm The unit is in millimetres. But, according to the question, the unit should be in centimetres. So, we will convert mm to cm. 1 cm = 10 mm 1 mm = 1/10 cm 644 mm = 644/10 cm = 64.4 cm Thus, the amount rained in the 7 days is 64.4 cm. 4. Cost of 5 kg of wheat is ₹ 91.50. (a) What will be the cost of 8 kg of wheat? Answer: ₹ 146.40 Explanation: Cost of 5 kg wheat = ₹ 91.50 Cost of 1 kg wheat = ₹ 91.50/5 = ₹ 18.30 Cost of 8 kg wheat = cost of 1 kg × 8 = ₹ 18.30 × 8 = ₹ 146.40 (b) What quantity of wheat can be purchased in ₹ 183? Answer: 10 kg Explanation: Cost of 1 kg wheat = ₹ 18.30 Total cost of wheat given = ₹ 183 The quantity of wheat purchased = Total quantity / Rate per kg = ₹ 183 /₹ 18.30 = (₹ 18.30 × 10)/ (₹ 18.30 × 1) = 10 Thus, 10 kg of wheat can be purchased in ₹ 183. 5. The temperature dropped 15 degree Celsius in the last 30 days. If the rate of temperature drop remains the same, how many degrees will the temperature drop in the next ten days? Answer: 5 degree Celsius Explanation: The temperature drop in the last 30 days = 15 degree Celsius The temperature drop in 1 day = 15/30 = 1/2 = 0.5 degree Celsius The temperature drop in the next 10 days = 0.5 degree Celsius × 10 = 5 degree Celsius 6. Shaina pays ₹ 15000 as rent for 3 months. How much does she has to pay for a whole year, if the rent per month remains same? Answer: ₹ 60000 Explanation: 1 year = 12 months Rent paid by Shaina for 3 months = ₹ 15000 Rent paid for 1 month = ₹ 15000/3 = ₹ 5000 Rent paid by Shaina for the whole year = ₹ 5000 × 12 = ₹ 60000 Or Rent paid by Shaina for 3 months = ₹ 15000 1 year = 12 months 3 × 4 = 12 Rent paid by Shaina for the whole year = Rent for 3 months × 4 = ₹ 15000 × 4 = ₹ 60000 7. Cost of 4 dozen bananas is ₹ 180. How many bananas can be purchased for ₹ 90? Answer: 24 Explanation: Cost of 4 dozen bananas = ₹ 180 Cost of 1 dozen bananas = ₹ 180/4 = ₹ 45 The bananas purchased in ₹ 90 = 90/Cost of 1 dozen bananas = 90/45 = 2 Thus, 2 dozen bananas can be purchased for ₹ 90. 1 dozen = 12 2 dozens = 12 × 2 2 dozens = 24 bananas 8. The weight of 72 books is 9 kg. What is the weight of 40 such books? Answer: 5 kg Explanation: Weight of 72 books = 9kg Weight of 1 book = 9/72 Weight of 1 book = 1/8 Weight of 40 books = 40 × weight of 1 book = 40 × 1/8 = 5 kg Thus, the weight of 40 books is 5 kg. 9. A truck requires 108 litres of diesel for covering a distance of 594 km. How much diesel will be required by the truck to cover a distance of 1650 km? Answer: 300 litres Explanation: Diesel required for covering a distance of 594 km = 108 litres Diesel required for covering a distance of 1 km = 108/594 Factorizing the above ratio, = (54 × 2)/ (54 × 11) Cancelling out the common terms, we get: 2/11 Diesel required for covering a distance of 1650 km = 2/11 × 1650 = 300 litres Thus, 300 litres of diesel will be required by the truck to cover a distance of 1650 km. 10. Raju purchases 10 pens for ₹ 150 and Manish buys 7 pens for ₹ 84. Can you say who got the pens cheaper? Answer: Manish Explanation: Cost of 10 pens purchased by Raju = ₹ 150 Cost of 1 pen = 150/10 = ₹ 15 Cost of 7 pens purchased by Manish = ₹ 84 Cost of 1 pen = 84/7 = ₹ 12 Thus, Manish got the pens cheaper. 11. Anish made 42 runs in 6 overs and Anup made 63 runs in 7 overs. Who made more runs per over? Answer: Anup Explanation: Runs made by Anish in 6 overs = 42 Runs made by Anish in 1 over = 42/6 = 7 runs Runs made by Anup in 7 overs = 63 Runs made by Anup in 1 over = 63/7 = 9 runs Thus, Anup made more runs per over. Next Topicclass 7 Maths |