Mathematics - Real Life | Maths 7th STd

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. We find that

1. Heena is two times taller than Amir.

Or

2. Amir’s height is 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 times the marbles that Amit has. Or

2. Amit has part of what Rita has.

Yet 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 of the speed of the Cheetah.

Speed of Man 20 km per hour

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.

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 = or 3 : 2.

This is same as the ratio for Rita’s sto 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.

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. To 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 .

We have, ;

We find that , which means that .

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:

Year Wins Losses

Last year 8 2 In which year was the record better?

This year 4 2 How can you say so?

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, )

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

= .

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

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?

Solution

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 =

Hence, cost of 10 bowls = × 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 km.

Hence, with 30 litres of petrol it would go 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.

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?

8.3 Percentage – Another Way of Comparing Quantities

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% = = 0.01

To understand this, let us consider the following example.

Rina made a table top of 100 different coloured tiles. She counted yellow, green, red and blue tiles separately and filled the table below. Can you help her complete the table?

Try These

Percentages when total is not hundred

In all these examples, the total number of items add up to 100. For example, Rina had 100 tiles in all, there were 100 children and 100 shoe pairs. How do we calculate Percentage of an item if the total number of items do not add up to 100? In such cases, we need to convert the fraction to an equivalent fraction with denominator 100. Consider the following example. You have a necklace with twenty beads in two colours.

We see that these three methods can be used to find the Percentage when the total does not add to give 100. In the method shown in the table, we multiply the fraction by . This does not change the value of the fraction. Subsequently, only 100 remains in the denominator.

Anwar has used the unitary method. Asha has multiplied by to get 100 in the denominator. You can use whichever method you find suitable. May be, you can make your own method too.

The method used by Anwar can work for all ratios. Can the method used by Asha also work for all ratios? Anwar says Asha’s method can be used only if you can find a natural number which on multiplication with the denominator gives 100. Since denominator was 20, she could multiply it by 5 to get 100. If the denominator was 6, she would not have been able to use this method. Do you agree?

2. Mala has a collection of bangles. She has 20 gold bangles and 10 silver bangles. What is the percentage of bangles of each type? Can you put it in the tabular form as done in the above example?

Think, Discuss and Write

8.3.2 Converting Fractional Numbers to Percentage

Fractional numbers can have different denominator. To compare fractional numbers, we need a common denominator and we have seen that it is more convenient to compare if our denominator is 100. That is, we are converting the fractions to Percentages. Let us try converting different fractional numbers to Percentages.

Example 7 Write as per cent.

Solution We have,

=

Example 8 Out of 25 children in a class, 15 are girls. What is the percentage of girls?

Solution Out of 25 children, there are 15 girls.

Therefore, percentage of girls = . There are 60% girls in the class.

Example 9 Convert to per cent.

Solution We have,

From these examples, we find that the percentages related to proper fractions are less than 100 whereas percentages related to improper fractions are more than 100.

Think, Discuss and Write

(i) Can you eat 50% of a cake? Can you eat 100% of a cake?

Can you eat 150% of a cake?

(ii) Can a price of an item go up by 50%? Can a price of an item go up by 100%?

Can a price of an item go up by 150%?

8.3.3 Converting Decimals to Percentage

We have seen how fractions can be converted to per cents. Let us now find how decimals can be converted to per cents.

Example 10 Convert the given decimals to per cents:

(a) 0.75 (b) 0.09 (c) 0.2

Solution (a) 0.75 = 0.75 × 100 % (b) 0.09 = = 9 %

= × 100 % = 75%

(c) 0.2 = × 100% = 20 %

8.3.4 Converting Percentages to Fractions or Decimals

We have so far converted fractions and decimals to percentages. We can also do the reverse. That is, given per cents, we can convert them to decimals or fractions. Look at the table, observe and complete it:

Parts always add to give a whole

In the examples for coloured tiles, for the heights of children and for gases in the air, we find that when we add the Percentages we get 100. All the parts that form the whole when added together gives the whole or 100%. So, if we are given one part, we can always find out the other part. Suppose, 30% of a given number of students are boys.

This means that if there were 100 students, 30 out of them would be boys and the remaining would be girls.

Then girls would obviously be (100 – 30)% = 70%.

Try These

Think, Discuss and Write

Consider the expenditure made on a dress

20% on embroidery, 50% on cloth, 30% on stitching.

Can you think of more such examples?

8.3.5 Fun with Estimation

Percentages help us to estimate the parts of an area.

Example 11 What per cent of the adjoining figure is shaded?

Solution We first find the fraction of the figure that is shaded. From this fraction, the percentage of the shaded part can be found.

You will find that half of the figure is shaded. And,

Thus, 50 % of the figure is shaded.

8.4 Use of Percentages

8.4.1 Interpreting Percentages

We saw how percentages were helpful in comparison. We have also learnt to convert fractional numbers and decimals to percentages. Now, we shall learn how percentages can be used in real life. For this, we start with interpreting the following statements:

— 5% of the income is saved by Ravi. — 20% of Meera’s dresses are blue in colour.

— Rekha gets 10% on every book sold by her.

What can you infer from each of these statements?

By 5% we mean 5 parts out of 100 or we write it as . It means Ravi is saving

5 out of every 100 that he earns. In the same way, interpret the rest of the statements given above.

8.4.2 Converting Percentages to “How Many”

Consider the following examples:

Example 12 A survey of 40 children showed that 25% liked playing football. How many children liked playing football?

Solution Here, the total number of children are 40. Out of these, 25% like playing football. Meena and Arun used the following methods to find the number. You can choose either method.

Try These

Example 13 Rahul bought a sweater and saved 20 when a discount of 25% was given. What was the price of the sweater before the discount?

Solution Rahul has saved 20 when price of sweater is reduced by 25%. This means that 25% reduction in price is the amount saved by Rahul. Let us see how Mohan and Abdul have found the original cost of the sweater.

Try These

8.4.3 Ratios to Percents

Sometimes, parts are given to us in the form of ratios and we need to convert those to percentages. Consider the following example:

Example 14 Reena’s mother said, to make idlis, you must take two parts rice and one part urad dal. What percentage of such a mixture would be rice and what percentage would be urad dal?

Solution In terms of ratio we would write this as Rice : Urad dal = 2 : 1.

Now, 2 + 1=3 is the total of all parts. This means part is rice and part is urad dal.

Then, percentage of rice would be .

Percentage of urad dal would be .

Example 15 If ` 250 is to be divided amongst Ravi, Raju and Roy, so that Ravi gets two parts, Raju three parts and Roy five parts. How much money will each get? What will it be in percentages?

Solution The parts which the three boys are getting can be written in terms of ratios as 2 : 3 : 5. Total of the parts is 2 + 3 + 5 = 10.

Amounts received by each Percentages of money for each

= 50 Ravi gets

= 75 Raju gets

Roy gets

Try These

8.4.4 Increase or Decrease as Per cent

There are times when we need to know the increase or decrease in a certain quantity as percentage. For example, if the population of a state increased from 5,50,000 to 6,05,000. Then the increase in population can be understood better if we say, the population increased by 10 %.

How do we convert the increase or decrease in a quantity as a percentage of the initial amount? Consider the following example.

Example 16 A school team won 6 games this year against 4 games won last year. What is the per cent increase?

Solution The increase in the number of wins (or amount of change) = 6 – 4 = 2.

Percentage increase =

= = = 50

Example 17 The number of illiterate persons in a country decreased from 150 lakhs to 100 lakhs in 10 years. What is the percentage of decrease?

Solution Original amount = the number of illiterate persons initially = 150 lakhs.

Amount of change = decrease in the number of illiterate persons = 150 – 100 = 50 lakhs

Therefore, the percentage of decrease

= =

Try These

8.5 Prices Related to an Item or Buying and Selling

I bought it for 600

The buying price of any item is known as its cost price. It is written in short as CP.

The price at which you sell is known as the selling price or in short SP.

What would you say is better, to you sell the item at a lower price, same price or higher price than your buying price? You can decide whether the sale was profitable or not depending on the CP and SP. If CP < SP then you made a profit = SP – CP.

If CP = SP then you are in a no profit no loss situation.

If CP > SP then you have a loss = CP – SP.

Let us try to interpret the statements related to prices of items.

• A toy bought for 72 is sold at 80.

• A T-shirt bought for 120 is sold at 100.

• A cycle bought for 800 is sold for 940.

Let us consider the first statement.

The buying price (or CP) is 72 and the selling price (or SP) is 80. This means SP is more than CP. Hence profit made = SP – CP = 80 – 72 = 8

Now try interpreting the remaining statements in a similar way.

8.5.1 Profit or Loss as a Percentage

The profit or loss can be converted to a percentage. It is always calculated on the CP.

For the above examples, we can find the profit % or loss %.

Let us consider the example related to the toy. We have CP = 72, SP = 80,
Profit = 8. To find the percentage of profit, Neha and Shekhar have used the following methods.

Neha does it this way

Profit per cent = × 100 = On

= =

On 100, profit

= . Thus, profit per cent =

Similarly you can find the loss per cent in the second situation. Here,

CP = 120, SP = 100.

Therefore, Loss = 120 – 100 = 20

Try the last case.

Now we see that given any two out of the three quantities related to prices that is, CP, SP, amount of Profit or Loss or their percentage, we can find the rest.

Example 18 The cost of a flower vase is 120. If the shopkeeper sells it at a loss of 10%, find the price at which it is sold.

Solution We are given that CP = 120 and Loss per cent = 10. We have to find the SP.

Thus, by both methods we get the SP as
108.