Page 1 COMP ARING QUANTITIES 153 153 153 153 153 8.1 INTRODUCTION In our daily life, there are many occasions when we compare two quantities. Suppose we are comparing heights of Heena and Amir. W e find that 1. Heena is two times taller than Amir. Or 2. Amir’ s height is 1 2 of Heena’s height. Consider another example, where 20 marbles are divided between Rita and Amit such that Rita has 12 marbles and Amit has 8 marbles. We say, 1. Rita has 3 2 times the marbles that Amit has. Or 2. Amit has 2 3 part of what Rita has. Y et another example is where we compare speeds of a Cheetah and a Man. The speed of a Cheetah is 6 times the speed of a Man. Or The speed of a Man is 1 6 of the speed of the Cheetah. Do you remember comparisons like this? In Class VI, we have learnt to make comparisons by saying how many times one quantity is of the other. Here, we see that it can also be inverted and written as what part one quantity is of the other. Chapter 8 Comparing Quantities 150 cm 75 cm Heena Amir 150 75 Speed of Cheetah Speed of Man 120 km per hour 20 km per hour 202021 © NCERT not to be republished Page 2 COMP ARING QUANTITIES 153 153 153 153 153 8.1 INTRODUCTION In our daily life, there are many occasions when we compare two quantities. Suppose we are comparing heights of Heena and Amir. W e find that 1. Heena is two times taller than Amir. Or 2. Amir’ s height is 1 2 of Heena’s height. Consider another example, where 20 marbles are divided between Rita and Amit such that Rita has 12 marbles and Amit has 8 marbles. We say, 1. Rita has 3 2 times the marbles that Amit has. Or 2. Amit has 2 3 part of what Rita has. Y et another example is where we compare speeds of a Cheetah and a Man. The speed of a Cheetah is 6 times the speed of a Man. Or The speed of a Man is 1 6 of the speed of the Cheetah. Do you remember comparisons like this? In Class VI, we have learnt to make comparisons by saying how many times one quantity is of the other. Here, we see that it can also be inverted and written as what part one quantity is of the other. Chapter 8 Comparing Quantities 150 cm 75 cm Heena Amir 150 75 Speed of Cheetah Speed of Man 120 km per hour 20 km per hour 202021 © NCERT not to be republished MATHEMATICS 154 154 154 154 154 In the given cases, we write the ratio of the heights as : Heena’s height : Amir’s height is 150 : 75 or 2 : 1. Can you now write the ratios for the other comparisons? These are relative comparisons and could be same for two different situations. If Heena’ s height was 150 cm and Amir’ s was 100 cm, then the ratio of their heights would be, Heena’s height : Amir’s height = 150 : 100 = 150 100 3 2 = or 3 : 2. This is same as the ratio for Rita’s to Amit’s share of marbles. Thus, we see that the ratio for two different comparisons may be the same. Remember that to compare two quantities, the units must be the same. A ratio has no units. EXAMPLE 1 Find the ratio of 3 km to 300 m. SOLUTION First convert both the distances to the same unit. So, 3 km = 3 × 1000 m = 3000 m. Thus, the required ratio, 3 km : 300 m is 3000 : 300 = 10 : 1. 8.2 EQUIVALENT RATIOS Different ratios can also be compared with each other to know whether they are equivalent or not. T o do this, we need to write the ratios in the form of fractions and then compare them by converting them to like fractions. If these like fractions are equal, we say the given ratios are equivalent. EXAMPLE 2 Are the ratios 1:2 and 2:3 equivalent? SOLUTION To check this, we need to know whether 1 2 2 3 = . We have, 1 1 3 3 2 2 3 6 × = = × ; 2 3 2 2 3 2 4 6 = × × = W e find that 3 6 4 6 < , which means that 1 2 2 3 < . Therefore, the ratio 1:2 is not equivalent to the ratio 2:3. Use of such comparisons can be seen by the following example. EXAMPLE 3 Following is the performance of a cricket team in the matches it played: Y ear Wins Losses Last year 8 2 In which year was the record better? This year 4 2 How can you say so? 202021 © NCERT not to be republished Page 3 COMP ARING QUANTITIES 153 153 153 153 153 8.1 INTRODUCTION In our daily life, there are many occasions when we compare two quantities. Suppose we are comparing heights of Heena and Amir. W e find that 1. Heena is two times taller than Amir. Or 2. Amir’ s height is 1 2 of Heena’s height. Consider another example, where 20 marbles are divided between Rita and Amit such that Rita has 12 marbles and Amit has 8 marbles. We say, 1. Rita has 3 2 times the marbles that Amit has. Or 2. Amit has 2 3 part of what Rita has. Y et another example is where we compare speeds of a Cheetah and a Man. The speed of a Cheetah is 6 times the speed of a Man. Or The speed of a Man is 1 6 of the speed of the Cheetah. Do you remember comparisons like this? In Class VI, we have learnt to make comparisons by saying how many times one quantity is of the other. Here, we see that it can also be inverted and written as what part one quantity is of the other. Chapter 8 Comparing Quantities 150 cm 75 cm Heena Amir 150 75 Speed of Cheetah Speed of Man 120 km per hour 20 km per hour 202021 © NCERT not to be republished MATHEMATICS 154 154 154 154 154 In the given cases, we write the ratio of the heights as : Heena’s height : Amir’s height is 150 : 75 or 2 : 1. Can you now write the ratios for the other comparisons? These are relative comparisons and could be same for two different situations. If Heena’ s height was 150 cm and Amir’ s was 100 cm, then the ratio of their heights would be, Heena’s height : Amir’s height = 150 : 100 = 150 100 3 2 = or 3 : 2. This is same as the ratio for Rita’s to Amit’s share of marbles. Thus, we see that the ratio for two different comparisons may be the same. Remember that to compare two quantities, the units must be the same. A ratio has no units. EXAMPLE 1 Find the ratio of 3 km to 300 m. SOLUTION First convert both the distances to the same unit. So, 3 km = 3 × 1000 m = 3000 m. Thus, the required ratio, 3 km : 300 m is 3000 : 300 = 10 : 1. 8.2 EQUIVALENT RATIOS Different ratios can also be compared with each other to know whether they are equivalent or not. T o do this, we need to write the ratios in the form of fractions and then compare them by converting them to like fractions. If these like fractions are equal, we say the given ratios are equivalent. EXAMPLE 2 Are the ratios 1:2 and 2:3 equivalent? SOLUTION To check this, we need to know whether 1 2 2 3 = . We have, 1 1 3 3 2 2 3 6 × = = × ; 2 3 2 2 3 2 4 6 = × × = W e find that 3 6 4 6 < , which means that 1 2 2 3 < . Therefore, the ratio 1:2 is not equivalent to the ratio 2:3. Use of such comparisons can be seen by the following example. EXAMPLE 3 Following is the performance of a cricket team in the matches it played: Y ear Wins Losses Last year 8 2 In which year was the record better? This year 4 2 How can you say so? 202021 © NCERT not to be republished COMP ARING QUANTITIES 155 155 155 155 155 SOLUTION Last year, Wins: Losses = 8 : 2 = 4 : 1 This year, Wins: Losses = 4 : 2 = 2 : 1 Obviously, 4 : 1 > 2 : 1 (In fractional form, 4 1 2 1 > ) Hence, we can say that the team performed better last year. In Class VI, we have also seen the importance of equivalent ratios. The ratios which are equivalent are said to be in proportion. Let us recall the use of proportions. Keeping things in proportion and getting solutions Aruna made a sketch of the building she lives in and drew sketch of her mother standing beside the building. Mona said, “There seems to be something wrong with the drawing” Can you say what is wrong? How can you say this? In this case, the ratio of heights in the drawing should be the same as the ratio of actual heights. That is Actual height of building Actual height of mother = Height of building in drawing Height of mother in the drawin ng . Only then would these be in proportion. Often when proportions are maintained, the drawing seems pleasing to the eye. Another example where proportions are used is in the making of national flags. Do you know that the flags are always made in a fixed ratio of length to its breadth? These may be different for different countries but are mostly around 1.5 : 1 or 1.7 : 1. We can take an approximate value of this ratio as 3 : 2. Even the Indian post card is around the same ratio. Now, can you say whether a card with length 4.5 cm and breadth 3.0 cm is near to this ratio. That is we need to ask, is 4.5 : 3.0 equivalent to 3 : 2? We note that 4 5 3 0 4 5 3 0 45 30 3 2 . : . . . = = = Hence, we see that 4.5 : 3.0 is equivalent to 3 : 2. We see a wide use of such proportions in real life. Can you think of some more situations? We have also learnt a method in the earlier classes known as Unitary Method in which we first find the value of one unit and then the value of the required number of units. Let us see how both the above methods help us to achieve the same thing. EXAMPLE 4 A map is given with a scale of 2 cm = 1000 km. What is the actual distance between the two places in kms, if the distance in the map is 2.5 cm? 202021 © NCERT not to be republished Page 4 COMP ARING QUANTITIES 153 153 153 153 153 8.1 INTRODUCTION In our daily life, there are many occasions when we compare two quantities. Suppose we are comparing heights of Heena and Amir. W e find that 1. Heena is two times taller than Amir. Or 2. Amir’ s height is 1 2 of Heena’s height. Consider another example, where 20 marbles are divided between Rita and Amit such that Rita has 12 marbles and Amit has 8 marbles. We say, 1. Rita has 3 2 times the marbles that Amit has. Or 2. Amit has 2 3 part of what Rita has. Y et another example is where we compare speeds of a Cheetah and a Man. The speed of a Cheetah is 6 times the speed of a Man. Or The speed of a Man is 1 6 of the speed of the Cheetah. Do you remember comparisons like this? In Class VI, we have learnt to make comparisons by saying how many times one quantity is of the other. Here, we see that it can also be inverted and written as what part one quantity is of the other. Chapter 8 Comparing Quantities 150 cm 75 cm Heena Amir 150 75 Speed of Cheetah Speed of Man 120 km per hour 20 km per hour 202021 © NCERT not to be republished MATHEMATICS 154 154 154 154 154 In the given cases, we write the ratio of the heights as : Heena’s height : Amir’s height is 150 : 75 or 2 : 1. Can you now write the ratios for the other comparisons? These are relative comparisons and could be same for two different situations. If Heena’ s height was 150 cm and Amir’ s was 100 cm, then the ratio of their heights would be, Heena’s height : Amir’s height = 150 : 100 = 150 100 3 2 = or 3 : 2. This is same as the ratio for Rita’s to Amit’s share of marbles. Thus, we see that the ratio for two different comparisons may be the same. Remember that to compare two quantities, the units must be the same. A ratio has no units. EXAMPLE 1 Find the ratio of 3 km to 300 m. SOLUTION First convert both the distances to the same unit. So, 3 km = 3 × 1000 m = 3000 m. Thus, the required ratio, 3 km : 300 m is 3000 : 300 = 10 : 1. 8.2 EQUIVALENT RATIOS Different ratios can also be compared with each other to know whether they are equivalent or not. T o do this, we need to write the ratios in the form of fractions and then compare them by converting them to like fractions. If these like fractions are equal, we say the given ratios are equivalent. EXAMPLE 2 Are the ratios 1:2 and 2:3 equivalent? SOLUTION To check this, we need to know whether 1 2 2 3 = . We have, 1 1 3 3 2 2 3 6 × = = × ; 2 3 2 2 3 2 4 6 = × × = W e find that 3 6 4 6 < , which means that 1 2 2 3 < . Therefore, the ratio 1:2 is not equivalent to the ratio 2:3. Use of such comparisons can be seen by the following example. EXAMPLE 3 Following is the performance of a cricket team in the matches it played: Y ear Wins Losses Last year 8 2 In which year was the record better? This year 4 2 How can you say so? 202021 © NCERT not to be republished COMP ARING QUANTITIES 155 155 155 155 155 SOLUTION Last year, Wins: Losses = 8 : 2 = 4 : 1 This year, Wins: Losses = 4 : 2 = 2 : 1 Obviously, 4 : 1 > 2 : 1 (In fractional form, 4 1 2 1 > ) Hence, we can say that the team performed better last year. In Class VI, we have also seen the importance of equivalent ratios. The ratios which are equivalent are said to be in proportion. Let us recall the use of proportions. Keeping things in proportion and getting solutions Aruna made a sketch of the building she lives in and drew sketch of her mother standing beside the building. Mona said, “There seems to be something wrong with the drawing” Can you say what is wrong? How can you say this? In this case, the ratio of heights in the drawing should be the same as the ratio of actual heights. That is Actual height of building Actual height of mother = Height of building in drawing Height of mother in the drawin ng . Only then would these be in proportion. Often when proportions are maintained, the drawing seems pleasing to the eye. Another example where proportions are used is in the making of national flags. Do you know that the flags are always made in a fixed ratio of length to its breadth? These may be different for different countries but are mostly around 1.5 : 1 or 1.7 : 1. We can take an approximate value of this ratio as 3 : 2. Even the Indian post card is around the same ratio. Now, can you say whether a card with length 4.5 cm and breadth 3.0 cm is near to this ratio. That is we need to ask, is 4.5 : 3.0 equivalent to 3 : 2? We note that 4 5 3 0 4 5 3 0 45 30 3 2 . : . . . = = = Hence, we see that 4.5 : 3.0 is equivalent to 3 : 2. We see a wide use of such proportions in real life. Can you think of some more situations? We have also learnt a method in the earlier classes known as Unitary Method in which we first find the value of one unit and then the value of the required number of units. Let us see how both the above methods help us to achieve the same thing. EXAMPLE 4 A map is given with a scale of 2 cm = 1000 km. What is the actual distance between the two places in kms, if the distance in the map is 2.5 cm? 202021 © NCERT not to be republished MATHEMATICS 156 156 156 156 156 SOLUTION Arun does it like this Meera does it like this Let distance = x km 2 cm means 1000 km. then, 1000 : x = 2 : 2.5 So, 1 cm means 1000 km 2 1000 2 2 5 x = . Hence, 2.5 cm means 1000 2.5 km 2 × 1000× ×2.5 2 2.5 2.5 x x x = × × = 1250 km 1000 × 2.5 = x × 2 x = 1250 Arun has solved it by equating ratios to make proportions and then by solving the equation. Meera has first found the distance that corresponds to 1 cm and then used that to find what 2.5 cm would correspond to. She used the unitary method. Let us solve some more examples using the unitary method. EXAMPLE 5 6 bowls cost ` 90. What would be the cost of 10 such bowls? SOLUTION Cost of 6 bowls is ` 90. Therefore, cost of 1 bowl = ` 90 6 Hence, cost of 10 bowls = ` 90 6 × 10 = ` 150 EXAMPLE 6 The car that I own can go 150 km with 25 litres of petrol. How far can it go with 30 litres of petrol? SOLUTION With 25 litres of petrol, the car goes 150 km. With 1 litre the car will go 150 25 km. Hence, with 30 litres of petrol it would go 150 30 25 × km = 180 km In this method, we first found the value for one unit or the unit rate. This is done by the comparison of two different properties. For example, when you compare total cost to number of items, we get cost per item or if you take distance travelled to time taken, we get distance per unit time. Thus, you can see that we often use per to mean for each. For example, km per hour, children per teacher etc., denote unit rates. 202021 © NCERT not to be republished Page 5 COMP ARING QUANTITIES 153 153 153 153 153 8.1 INTRODUCTION In our daily life, there are many occasions when we compare two quantities. Suppose we are comparing heights of Heena and Amir. W e find that 1. Heena is two times taller than Amir. Or 2. Amir’ s height is 1 2 of Heena’s height. Consider another example, where 20 marbles are divided between Rita and Amit such that Rita has 12 marbles and Amit has 8 marbles. We say, 1. Rita has 3 2 times the marbles that Amit has. Or 2. Amit has 2 3 part of what Rita has. Y et another example is where we compare speeds of a Cheetah and a Man. The speed of a Cheetah is 6 times the speed of a Man. Or The speed of a Man is 1 6 of the speed of the Cheetah. Do you remember comparisons like this? In Class VI, we have learnt to make comparisons by saying how many times one quantity is of the other. Here, we see that it can also be inverted and written as what part one quantity is of the other. Chapter 8 Comparing Quantities 150 cm 75 cm Heena Amir 150 75 Speed of Cheetah Speed of Man 120 km per hour 20 km per hour 202021 © NCERT not to be republished MATHEMATICS 154 154 154 154 154 In the given cases, we write the ratio of the heights as : Heena’s height : Amir’s height is 150 : 75 or 2 : 1. Can you now write the ratios for the other comparisons? These are relative comparisons and could be same for two different situations. If Heena’ s height was 150 cm and Amir’ s was 100 cm, then the ratio of their heights would be, Heena’s height : Amir’s height = 150 : 100 = 150 100 3 2 = or 3 : 2. This is same as the ratio for Rita’s to Amit’s share of marbles. Thus, we see that the ratio for two different comparisons may be the same. Remember that to compare two quantities, the units must be the same. A ratio has no units. EXAMPLE 1 Find the ratio of 3 km to 300 m. SOLUTION First convert both the distances to the same unit. So, 3 km = 3 × 1000 m = 3000 m. Thus, the required ratio, 3 km : 300 m is 3000 : 300 = 10 : 1. 8.2 EQUIVALENT RATIOS Different ratios can also be compared with each other to know whether they are equivalent or not. T o do this, we need to write the ratios in the form of fractions and then compare them by converting them to like fractions. If these like fractions are equal, we say the given ratios are equivalent. EXAMPLE 2 Are the ratios 1:2 and 2:3 equivalent? SOLUTION To check this, we need to know whether 1 2 2 3 = . We have, 1 1 3 3 2 2 3 6 × = = × ; 2 3 2 2 3 2 4 6 = × × = W e find that 3 6 4 6 < , which means that 1 2 2 3 < . Therefore, the ratio 1:2 is not equivalent to the ratio 2:3. Use of such comparisons can be seen by the following example. EXAMPLE 3 Following is the performance of a cricket team in the matches it played: Y ear Wins Losses Last year 8 2 In which year was the record better? This year 4 2 How can you say so? 202021 © NCERT not to be republished COMP ARING QUANTITIES 155 155 155 155 155 SOLUTION Last year, Wins: Losses = 8 : 2 = 4 : 1 This year, Wins: Losses = 4 : 2 = 2 : 1 Obviously, 4 : 1 > 2 : 1 (In fractional form, 4 1 2 1 > ) Hence, we can say that the team performed better last year. In Class VI, we have also seen the importance of equivalent ratios. The ratios which are equivalent are said to be in proportion. Let us recall the use of proportions. Keeping things in proportion and getting solutions Aruna made a sketch of the building she lives in and drew sketch of her mother standing beside the building. Mona said, “There seems to be something wrong with the drawing” Can you say what is wrong? How can you say this? In this case, the ratio of heights in the drawing should be the same as the ratio of actual heights. That is Actual height of building Actual height of mother = Height of building in drawing Height of mother in the drawin ng . Only then would these be in proportion. Often when proportions are maintained, the drawing seems pleasing to the eye. Another example where proportions are used is in the making of national flags. Do you know that the flags are always made in a fixed ratio of length to its breadth? These may be different for different countries but are mostly around 1.5 : 1 or 1.7 : 1. We can take an approximate value of this ratio as 3 : 2. Even the Indian post card is around the same ratio. Now, can you say whether a card with length 4.5 cm and breadth 3.0 cm is near to this ratio. That is we need to ask, is 4.5 : 3.0 equivalent to 3 : 2? We note that 4 5 3 0 4 5 3 0 45 30 3 2 . : . . . = = = Hence, we see that 4.5 : 3.0 is equivalent to 3 : 2. We see a wide use of such proportions in real life. Can you think of some more situations? We have also learnt a method in the earlier classes known as Unitary Method in which we first find the value of one unit and then the value of the required number of units. Let us see how both the above methods help us to achieve the same thing. EXAMPLE 4 A map is given with a scale of 2 cm = 1000 km. What is the actual distance between the two places in kms, if the distance in the map is 2.5 cm? 202021 © NCERT not to be republished MATHEMATICS 156 156 156 156 156 SOLUTION Arun does it like this Meera does it like this Let distance = x km 2 cm means 1000 km. then, 1000 : x = 2 : 2.5 So, 1 cm means 1000 km 2 1000 2 2 5 x = . Hence, 2.5 cm means 1000 2.5 km 2 × 1000× ×2.5 2 2.5 2.5 x x x = × × = 1250 km 1000 × 2.5 = x × 2 x = 1250 Arun has solved it by equating ratios to make proportions and then by solving the equation. Meera has first found the distance that corresponds to 1 cm and then used that to find what 2.5 cm would correspond to. She used the unitary method. Let us solve some more examples using the unitary method. EXAMPLE 5 6 bowls cost ` 90. What would be the cost of 10 such bowls? SOLUTION Cost of 6 bowls is ` 90. Therefore, cost of 1 bowl = ` 90 6 Hence, cost of 10 bowls = ` 90 6 × 10 = ` 150 EXAMPLE 6 The car that I own can go 150 km with 25 litres of petrol. How far can it go with 30 litres of petrol? SOLUTION With 25 litres of petrol, the car goes 150 km. With 1 litre the car will go 150 25 km. Hence, with 30 litres of petrol it would go 150 30 25 × km = 180 km In this method, we first found the value for one unit or the unit rate. This is done by the comparison of two different properties. For example, when you compare total cost to number of items, we get cost per item or if you take distance travelled to time taken, we get distance per unit time. Thus, you can see that we often use per to mean for each. For example, km per hour, children per teacher etc., denote unit rates. 202021 © NCERT not to be republished COMP ARING QUANTITIES 157 157 157 157 157 THINK, DISCUSS AND WRITE An ant can carry 50 times its weight. If a person can do the same, how much would you be able to carry? EXERCISE 8.1 1. Find the ratio of: (a) ` 5 to 50 paise (b) 15 kg to 210 g (c) 9 m to 27 cm (d) 30 days to 36 hours 2. In a computer lab, there are 3 computers for every 6 students. How many computers will be needed for 24 students? 3. Population of Rajasthan = 570 lakhs and population of UP = 1660 lakhs. Area of Rajasthan = 3 lakh km 2 and area of UP = 2 lakh km 2 . (i) How many people are there per km 2 in both these States? (ii) Which State is less populated? 8.3 PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES Anita’s Report Rita’s Report T otal 320/400 T otal 300/360 Percentage: 80 Percentage: 83.3 Anita said that she has done better as she got 320 marks whereas Rita got only 300. Do you agree with her? Who do you think has done better? Mansi told them that they cannot decide who has done better by just comparing the total marks obtained because the maximum marks out of which they got the marks are not the same. She said why don’t you see the Percentages given in your report cards? Anita’s Percentage was 80 and Rita’s was 83.3. So, this shows Rita has done better. Do you agree? Percentages are numerators of fractions with denominator 100 and have been used in comparing results. Let us try to understand in detail about it. 8.3.1 Meaning of Percentage Per cent is derived from Latin word ‘per centum’ meaning ‘per hundred’. Per cent is represented by the symbol % and means hundredths too. That is 1% means 1 out of hundred or one hundredth. It can be written as: 1% = 1 100 = 0.01 202021 © NCERT not to be republishedRead More
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