Page 1 COMPARING QUANTITIES 117 8.1 Recalling Ratios and Percentages W e know , ratio means comparing two quantities. A basket has two types of fruits, say, 20 apples and 5 oranges. Then, the ratio of the number of oranges to the number of apples = 5 : 20. The comparison can be done by using fractions as, 5 20 = 1 4 The number of oranges are 1 4 th the number of apples. In terms of ratio, this is 1 : 4, read as, “1 is to 4” Number of apples to number of oranges = 20 4 51 = which means, the number of apples are 4 times the number of oranges. This comparison can also be done using percentages. There are 5 oranges out of 25 fruits. So percentage of oranges is 54 20 20% 25 4 100 ×= = OR [Denominator made 100]. Since contains only apples and oranges, So, percentage of apples + percentage of oranges = 100 or percentage of apples + 20 = 100 or percentage of apples = 100 – 20 = 80 Thus the basket has 20% oranges and 80% apples. Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the total number of students and are 18 in number. The picnic site is 55 km from the school and the transport company is charging at the rate of ` 12 per km. The total cost of refreshments will be ` 4280. Comparing Quantities CHAPTER 8 By unitary method: Out of 25 fruits, number of oranges are 5. So out of 100 fruits, number of oranges = 5 100 25 × = 20. OR Page 2 COMPARING QUANTITIES 117 8.1 Recalling Ratios and Percentages W e know , ratio means comparing two quantities. A basket has two types of fruits, say, 20 apples and 5 oranges. Then, the ratio of the number of oranges to the number of apples = 5 : 20. The comparison can be done by using fractions as, 5 20 = 1 4 The number of oranges are 1 4 th the number of apples. In terms of ratio, this is 1 : 4, read as, “1 is to 4” Number of apples to number of oranges = 20 4 51 = which means, the number of apples are 4 times the number of oranges. This comparison can also be done using percentages. There are 5 oranges out of 25 fruits. So percentage of oranges is 54 20 20% 25 4 100 ×= = OR [Denominator made 100]. Since contains only apples and oranges, So, percentage of apples + percentage of oranges = 100 or percentage of apples + 20 = 100 or percentage of apples = 100 – 20 = 80 Thus the basket has 20% oranges and 80% apples. Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the total number of students and are 18 in number. The picnic site is 55 km from the school and the transport company is charging at the rate of ` 12 per km. The total cost of refreshments will be ` 4280. Comparing Quantities CHAPTER 8 By unitary method: Out of 25 fruits, number of oranges are 5. So out of 100 fruits, number of oranges = 5 100 25 × = 20. OR 118 MATHEMATICS Can you tell. 1. The ratio of the number of girls to the number of boys in the class? 2. The cost per head if two teachers are also going with the class? 3. If their first stop is at a place 22 km from the school, what per cent of the total distance of 55 km is this? What per cent of the distance is left to be covered? Solution: 1. T o find the ratio of girls to boys. Ashima and John came up with the following answers. They needed to know the number of boys and also the total number of students. Ashima did this John used the unitary method Let the total number of students There are 60 girls out of 100 students. be x. 60% of x is girls. There is one girl out of 100 60 students. Therefore, 60% of x = 18 So, 18 girls are out of how many students? 60 100 x × = 18 OR Number of students = 100 18 60 × or, x = 18 100 60 × = 30 = 30 Number of students = 30. So, the number of boys = 30 – 18 = 12. Hence, ratio of the number of girls to the number of boys is 18 : 12 or 18 12 = 3 2 . 3 2 is written as 3 : 2 and read as 3 is to 2. 2. To find the cost per person. Transportation charge = Distance both ways × Rate = ` (55 × 2) × 12 = ` 110 × 12 = ` 1320 Total expenses = Refreshment charge + Transportation charge = ` 4280 + ` 1320 = ` 5600 Total number of persons =18 girls + 12 boys + 2 teachers = 32 persons Ashima and John then used unitary method to find the cost per head. For 32 persons, amount spent would be ` 5600. The amount spent for 1 person = ` 5600 32 = ` 175. 3. The distance of the place where first stop was made = 22 km. Page 3 COMPARING QUANTITIES 117 8.1 Recalling Ratios and Percentages W e know , ratio means comparing two quantities. A basket has two types of fruits, say, 20 apples and 5 oranges. Then, the ratio of the number of oranges to the number of apples = 5 : 20. The comparison can be done by using fractions as, 5 20 = 1 4 The number of oranges are 1 4 th the number of apples. In terms of ratio, this is 1 : 4, read as, “1 is to 4” Number of apples to number of oranges = 20 4 51 = which means, the number of apples are 4 times the number of oranges. This comparison can also be done using percentages. There are 5 oranges out of 25 fruits. So percentage of oranges is 54 20 20% 25 4 100 ×= = OR [Denominator made 100]. Since contains only apples and oranges, So, percentage of apples + percentage of oranges = 100 or percentage of apples + 20 = 100 or percentage of apples = 100 – 20 = 80 Thus the basket has 20% oranges and 80% apples. Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the total number of students and are 18 in number. The picnic site is 55 km from the school and the transport company is charging at the rate of ` 12 per km. The total cost of refreshments will be ` 4280. Comparing Quantities CHAPTER 8 By unitary method: Out of 25 fruits, number of oranges are 5. So out of 100 fruits, number of oranges = 5 100 25 × = 20. OR 118 MATHEMATICS Can you tell. 1. The ratio of the number of girls to the number of boys in the class? 2. The cost per head if two teachers are also going with the class? 3. If their first stop is at a place 22 km from the school, what per cent of the total distance of 55 km is this? What per cent of the distance is left to be covered? Solution: 1. T o find the ratio of girls to boys. Ashima and John came up with the following answers. They needed to know the number of boys and also the total number of students. Ashima did this John used the unitary method Let the total number of students There are 60 girls out of 100 students. be x. 60% of x is girls. There is one girl out of 100 60 students. Therefore, 60% of x = 18 So, 18 girls are out of how many students? 60 100 x × = 18 OR Number of students = 100 18 60 × or, x = 18 100 60 × = 30 = 30 Number of students = 30. So, the number of boys = 30 – 18 = 12. Hence, ratio of the number of girls to the number of boys is 18 : 12 or 18 12 = 3 2 . 3 2 is written as 3 : 2 and read as 3 is to 2. 2. To find the cost per person. Transportation charge = Distance both ways × Rate = ` (55 × 2) × 12 = ` 110 × 12 = ` 1320 Total expenses = Refreshment charge + Transportation charge = ` 4280 + ` 1320 = ` 5600 Total number of persons =18 girls + 12 boys + 2 teachers = 32 persons Ashima and John then used unitary method to find the cost per head. For 32 persons, amount spent would be ` 5600. The amount spent for 1 person = ` 5600 32 = ` 175. 3. The distance of the place where first stop was made = 22 km. COMPARING QUANTITIES 119 T o find the percentage of distance: Ashima used this method: John used the unitary method: 22 22 100 40% 55 55 100 =× = Out of 55 km, 22 km are travelled. OR Out of 1 km, 22 55 km are travelled. Out of 100 km, 22 55 × 100 km are travelled. That is 40% of the total distance is travelled. She is multiplying 100 the ratio by =1 100 and converting to percentage. ?? ?? ?? ?? ?? ?? ?? ?? TRY THESE Both came out with the same answer that the distance from their school of the place where they stopped at was 40% of the total distance they had to travel. Therefore, the percent distance left to be travelled = 100% – 40% = 60%. In a primary school, the parents were asked about the number of hours they spend per day in helping their children to do homework. There were 90 parents who helped for 1 2 hour to 1 1 2 hours. The distribution of parents according to the time for which, they said they helped is given in the adjoining figure ; 20% helped for more than 1 1 2 hours per day; 30% helped for 1 2 hour to 1 1 2 hours; 50% did not help at all. Using this, answer the following: (i) How many parents were surveyed? (ii) How many said that they did not help? (iii) How many said that they helped for more than 1 1 2 hours? EXERCISE 8.1 1. Find the ratio of the following. (a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour. (b) 5 m to 10 km (c) 50 paise to ` 5 2. Convert the following ratios to percentages. (a) 3 : 4 (b) 2 : 3 3. 72% of 25 students are good in mathematics. How many are not good in mathematics? 4. A football team won 10 matches out of the total number of matches they played. If their win percentage was 40, then how many matches did they play in all? 5. If Chameli had ` 600 left after spending 75% of her money , how much did she have in the beginning? Page 4 COMPARING QUANTITIES 117 8.1 Recalling Ratios and Percentages W e know , ratio means comparing two quantities. A basket has two types of fruits, say, 20 apples and 5 oranges. Then, the ratio of the number of oranges to the number of apples = 5 : 20. The comparison can be done by using fractions as, 5 20 = 1 4 The number of oranges are 1 4 th the number of apples. In terms of ratio, this is 1 : 4, read as, “1 is to 4” Number of apples to number of oranges = 20 4 51 = which means, the number of apples are 4 times the number of oranges. This comparison can also be done using percentages. There are 5 oranges out of 25 fruits. So percentage of oranges is 54 20 20% 25 4 100 ×= = OR [Denominator made 100]. Since contains only apples and oranges, So, percentage of apples + percentage of oranges = 100 or percentage of apples + 20 = 100 or percentage of apples = 100 – 20 = 80 Thus the basket has 20% oranges and 80% apples. Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the total number of students and are 18 in number. The picnic site is 55 km from the school and the transport company is charging at the rate of ` 12 per km. The total cost of refreshments will be ` 4280. Comparing Quantities CHAPTER 8 By unitary method: Out of 25 fruits, number of oranges are 5. So out of 100 fruits, number of oranges = 5 100 25 × = 20. OR 118 MATHEMATICS Can you tell. 1. The ratio of the number of girls to the number of boys in the class? 2. The cost per head if two teachers are also going with the class? 3. If their first stop is at a place 22 km from the school, what per cent of the total distance of 55 km is this? What per cent of the distance is left to be covered? Solution: 1. T o find the ratio of girls to boys. Ashima and John came up with the following answers. They needed to know the number of boys and also the total number of students. Ashima did this John used the unitary method Let the total number of students There are 60 girls out of 100 students. be x. 60% of x is girls. There is one girl out of 100 60 students. Therefore, 60% of x = 18 So, 18 girls are out of how many students? 60 100 x × = 18 OR Number of students = 100 18 60 × or, x = 18 100 60 × = 30 = 30 Number of students = 30. So, the number of boys = 30 – 18 = 12. Hence, ratio of the number of girls to the number of boys is 18 : 12 or 18 12 = 3 2 . 3 2 is written as 3 : 2 and read as 3 is to 2. 2. To find the cost per person. Transportation charge = Distance both ways × Rate = ` (55 × 2) × 12 = ` 110 × 12 = ` 1320 Total expenses = Refreshment charge + Transportation charge = ` 4280 + ` 1320 = ` 5600 Total number of persons =18 girls + 12 boys + 2 teachers = 32 persons Ashima and John then used unitary method to find the cost per head. For 32 persons, amount spent would be ` 5600. The amount spent for 1 person = ` 5600 32 = ` 175. 3. The distance of the place where first stop was made = 22 km. COMPARING QUANTITIES 119 T o find the percentage of distance: Ashima used this method: John used the unitary method: 22 22 100 40% 55 55 100 =× = Out of 55 km, 22 km are travelled. OR Out of 1 km, 22 55 km are travelled. Out of 100 km, 22 55 × 100 km are travelled. That is 40% of the total distance is travelled. She is multiplying 100 the ratio by =1 100 and converting to percentage. ?? ?? ?? ?? ?? ?? ?? ?? TRY THESE Both came out with the same answer that the distance from their school of the place where they stopped at was 40% of the total distance they had to travel. Therefore, the percent distance left to be travelled = 100% – 40% = 60%. In a primary school, the parents were asked about the number of hours they spend per day in helping their children to do homework. There were 90 parents who helped for 1 2 hour to 1 1 2 hours. The distribution of parents according to the time for which, they said they helped is given in the adjoining figure ; 20% helped for more than 1 1 2 hours per day; 30% helped for 1 2 hour to 1 1 2 hours; 50% did not help at all. Using this, answer the following: (i) How many parents were surveyed? (ii) How many said that they did not help? (iii) How many said that they helped for more than 1 1 2 hours? EXERCISE 8.1 1. Find the ratio of the following. (a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour. (b) 5 m to 10 km (c) 50 paise to ` 5 2. Convert the following ratios to percentages. (a) 3 : 4 (b) 2 : 3 3. 72% of 25 students are good in mathematics. How many are not good in mathematics? 4. A football team won 10 matches out of the total number of matches they played. If their win percentage was 40, then how many matches did they play in all? 5. If Chameli had ` 600 left after spending 75% of her money , how much did she have in the beginning? 120 MATHEMATICS 6. If 60% people in a city like cricket, 30% like football and the remaining like other games, then what per cent of the people like other games? If the total number of people are 50 lakh, find the exact number who like each type of game. 8.2 Finding the Increase or Decrease Per cent W e often come across such information in our daily life as. (i) 25% off on marked prices (ii) 10% hike in the price of petrol Let us consider a few such examples. Example 2: The price of a scooter was ` 34,000 last year. It has increased by 20% this year. What is the price now? Solution: OR Amita said that she would first find the increase in the price, which is 20% of ` 34,000, and then find the new price. 20% of ` 34000 = ` 20 34000 100 × = ` 6800 New price = Old price + Increase = ` 34,000 + ` 6,800 = ` 40,800 Similarly, a percentage decrease in price would imply finding the actual decrease followed by its subtraction the from original price. Suppose in order to increase its sale, the price of scooter was decreased by 5%. Then let us find the price of scooter. Price of scooter = ` 34000 Reduction = 5% of ` 34000 = ` 5 34000 100 × = ` 1700 New price = Old price – Reduction = ` 34000 – ` 1700 = ` 32300 W e will also use this in the next section of the chapter. 8.3 Finding Discounts Discount is a reduction given on the Marked Price (MP) of the article. This is generally given to attract customers to buy goods or to promote sales of the goods. Y ou can find the discount by subtracting its sale price from its marked price. So, Discount = Marked price – Sale price Sunita used the unitary method. 20% increase means, ` 100 increased to ` 120. So, ` 34,000 will increase to? Increased price = ` 120 34000 100 × = ` 40,800 Page 5 COMPARING QUANTITIES 117 8.1 Recalling Ratios and Percentages W e know , ratio means comparing two quantities. A basket has two types of fruits, say, 20 apples and 5 oranges. Then, the ratio of the number of oranges to the number of apples = 5 : 20. The comparison can be done by using fractions as, 5 20 = 1 4 The number of oranges are 1 4 th the number of apples. In terms of ratio, this is 1 : 4, read as, “1 is to 4” Number of apples to number of oranges = 20 4 51 = which means, the number of apples are 4 times the number of oranges. This comparison can also be done using percentages. There are 5 oranges out of 25 fruits. So percentage of oranges is 54 20 20% 25 4 100 ×= = OR [Denominator made 100]. Since contains only apples and oranges, So, percentage of apples + percentage of oranges = 100 or percentage of apples + 20 = 100 or percentage of apples = 100 – 20 = 80 Thus the basket has 20% oranges and 80% apples. Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the total number of students and are 18 in number. The picnic site is 55 km from the school and the transport company is charging at the rate of ` 12 per km. The total cost of refreshments will be ` 4280. Comparing Quantities CHAPTER 8 By unitary method: Out of 25 fruits, number of oranges are 5. So out of 100 fruits, number of oranges = 5 100 25 × = 20. OR 118 MATHEMATICS Can you tell. 1. The ratio of the number of girls to the number of boys in the class? 2. The cost per head if two teachers are also going with the class? 3. If their first stop is at a place 22 km from the school, what per cent of the total distance of 55 km is this? What per cent of the distance is left to be covered? Solution: 1. T o find the ratio of girls to boys. Ashima and John came up with the following answers. They needed to know the number of boys and also the total number of students. Ashima did this John used the unitary method Let the total number of students There are 60 girls out of 100 students. be x. 60% of x is girls. There is one girl out of 100 60 students. Therefore, 60% of x = 18 So, 18 girls are out of how many students? 60 100 x × = 18 OR Number of students = 100 18 60 × or, x = 18 100 60 × = 30 = 30 Number of students = 30. So, the number of boys = 30 – 18 = 12. Hence, ratio of the number of girls to the number of boys is 18 : 12 or 18 12 = 3 2 . 3 2 is written as 3 : 2 and read as 3 is to 2. 2. To find the cost per person. Transportation charge = Distance both ways × Rate = ` (55 × 2) × 12 = ` 110 × 12 = ` 1320 Total expenses = Refreshment charge + Transportation charge = ` 4280 + ` 1320 = ` 5600 Total number of persons =18 girls + 12 boys + 2 teachers = 32 persons Ashima and John then used unitary method to find the cost per head. For 32 persons, amount spent would be ` 5600. The amount spent for 1 person = ` 5600 32 = ` 175. 3. The distance of the place where first stop was made = 22 km. COMPARING QUANTITIES 119 T o find the percentage of distance: Ashima used this method: John used the unitary method: 22 22 100 40% 55 55 100 =× = Out of 55 km, 22 km are travelled. OR Out of 1 km, 22 55 km are travelled. Out of 100 km, 22 55 × 100 km are travelled. That is 40% of the total distance is travelled. She is multiplying 100 the ratio by =1 100 and converting to percentage. ?? ?? ?? ?? ?? ?? ?? ?? TRY THESE Both came out with the same answer that the distance from their school of the place where they stopped at was 40% of the total distance they had to travel. Therefore, the percent distance left to be travelled = 100% – 40% = 60%. In a primary school, the parents were asked about the number of hours they spend per day in helping their children to do homework. There were 90 parents who helped for 1 2 hour to 1 1 2 hours. The distribution of parents according to the time for which, they said they helped is given in the adjoining figure ; 20% helped for more than 1 1 2 hours per day; 30% helped for 1 2 hour to 1 1 2 hours; 50% did not help at all. Using this, answer the following: (i) How many parents were surveyed? (ii) How many said that they did not help? (iii) How many said that they helped for more than 1 1 2 hours? EXERCISE 8.1 1. Find the ratio of the following. (a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour. (b) 5 m to 10 km (c) 50 paise to ` 5 2. Convert the following ratios to percentages. (a) 3 : 4 (b) 2 : 3 3. 72% of 25 students are good in mathematics. How many are not good in mathematics? 4. A football team won 10 matches out of the total number of matches they played. If their win percentage was 40, then how many matches did they play in all? 5. If Chameli had ` 600 left after spending 75% of her money , how much did she have in the beginning? 120 MATHEMATICS 6. If 60% people in a city like cricket, 30% like football and the remaining like other games, then what per cent of the people like other games? If the total number of people are 50 lakh, find the exact number who like each type of game. 8.2 Finding the Increase or Decrease Per cent W e often come across such information in our daily life as. (i) 25% off on marked prices (ii) 10% hike in the price of petrol Let us consider a few such examples. Example 2: The price of a scooter was ` 34,000 last year. It has increased by 20% this year. What is the price now? Solution: OR Amita said that she would first find the increase in the price, which is 20% of ` 34,000, and then find the new price. 20% of ` 34000 = ` 20 34000 100 × = ` 6800 New price = Old price + Increase = ` 34,000 + ` 6,800 = ` 40,800 Similarly, a percentage decrease in price would imply finding the actual decrease followed by its subtraction the from original price. Suppose in order to increase its sale, the price of scooter was decreased by 5%. Then let us find the price of scooter. Price of scooter = ` 34000 Reduction = 5% of ` 34000 = ` 5 34000 100 × = ` 1700 New price = Old price – Reduction = ` 34000 – ` 1700 = ` 32300 W e will also use this in the next section of the chapter. 8.3 Finding Discounts Discount is a reduction given on the Marked Price (MP) of the article. This is generally given to attract customers to buy goods or to promote sales of the goods. Y ou can find the discount by subtracting its sale price from its marked price. So, Discount = Marked price – Sale price Sunita used the unitary method. 20% increase means, ` 100 increased to ` 120. So, ` 34,000 will increase to? Increased price = ` 120 34000 100 × = ` 40,800 COMPARING QUANTITIES 121 TRY THESE Example 3: An item marked at ` 840 is sold for ` 714. What is the discount and discount %? Solution: Discount = Marked Price – Sale Price = ` 840 – ` 714 = ` 126 Since discount is on marked price, we will have to use marked price as the base. On marked price of ` 840, the discount is ` 126. On MP of ` 100, how much will the discount be? Discount = 126 100 840 × = 15% Y ou can also find discount when discount % is given. Example 4: The list price of a frock is ` 220. A discount of 20% is announced on sales. What is the amount of discount on it and its sale price. Solution: Marked price is same as the list price. 20% discount means that on ` 100 (MP), the discount is ` 20. By unitary method, on `1 the discount will be ` 20 100 . On ` 220, discount = ` 20 220 100 × = ` 44 The sale price = (` 220 – ` 44) or ` 176 Rehana found the sale price like this — A discount of 20% means for a MP of ` 100, discount is ` 20. Hence the sale price is ` 80. Using unitary method, when MP is ` 100, sale price is ` 80; When MP is ` 1, sale price is ` 80 100 . Hence when MP is ` 220, sale price = ` 80 220 100 × = ` 176. 1. A shop gives 20% discount. What would the sale price of each of these be? (a) A dress marked at ` 120 (b) A pair of shoes marked at ` 750 (c) A bag marked at ` 250 2. A table marked at ` 15,000 is available for ` 14,400. Find the discount given and the discount per cent. 3. An almirah is sold at ` 5,225 after allowing a discount of 5%. Find its marked price. Even though the discount was not found, I could find the sale price directly.Read More

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### Points to Remember- Comparing Quantities

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