PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

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.

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% = \(1 \over 100\) = 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?

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 \(100\over 100\) . 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 \(5 \over 5\) 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?

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

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: Write \(1\over 3\) as per cent.

SOLUTION :We have,

\(\begin{align}   & \frac{1}{3}=\frac{1}{3}\times \frac{100}{100}=\frac{1}{3}\times 100% \\  & =\frac{100}{3}%=33\frac{1}{3}% \\ \end{align}\)

EXAMPLE : 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 = \(\frac{15}{25}\times 100\)= 60. There are 60% girls in the class.

EXAMPLE : Convert \(\frac{5}{4}\) to per cent.

SOLUTION :We have, \(5 \over 4\)= \(5 \over 4\)×  100 %=125 %

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

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

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

 Convert the given decimals to per cents:

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

SOLUTION

(a) 0.75 = 0.75 × 100 % = \(75 \over 100\) × 100 % = 75%

(b) 0.09 = \(9 \over 100\) = 9 %

(c) 0.2 = \(2\over 10\)× 100% = 20 %

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%

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

Fun with Estimation

Percentages help us to estimate the parts of an area.

 EXAMPLE : 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, \(1 \over 2\) =\(1 \over 2\)×100 % =50 %

Thus, 50 % of the figure is shaded.

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

USE OF PERCENTAGES

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 \(5 \over 100\) . 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

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

Converting Percentages to “How Many”

 Consider the following examples:

EXAMPLE : 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.

EXAMPLE : Rahul bought a sweater and saved ₹ 200 when a discount of 25% was given. What was the price of the sweater before the discount?

SOLUTION :Rahul has saved ₹200 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.

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

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 : 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 \(2 \over 3\) part is rice and \(1 \over 3\) part is urad dal

Then, percentage of rice would be \(2 \over 3\) × 100%=\(200 \over 3\)  = \(66 {2 \over 3}\) %.

Percentage of urad dal would be \(1 \over 3\) ×100%= \(100\over 3\)= 33\(1 \over 3\) %

EXAMPLE : 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

\(2 \over 10\) × ₹250 = ₹ 50                               Ravi gets \(2\over 10\) ×100 %=20 % 

\(3 \over 10\) × ₹250= ₹75                                 Raju gets \(3 \over 10\) ×100 % =30 % 

\(5 \over 10\)  ×₹250 = ₹ 125                              Roy gets \(5 \over 10\) ×100 %= 50 %

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

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 :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.

\(\begin{align}   & \text{Percentage increase }=\text{ }\frac{\text{amount of change}}{\text{original amount or base}}\text{  }\times \text{ 1}00\text{ } \\  & =\text{ }\frac{\text{increase in the number of wins  }}{\text{original number of wins}}\text{ }\times \text{1}00=\frac{2}{4}\text{ }\times \text{ 1}00\text{ }=\text{ 5}0 \\ \end{align}\)

EXAMPLE :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

\(=\text{ }\frac{\text{amount of change  }}{\text{original amount}}\times \text{ 1}00=\text{ }\frac{\text{5}0}{150}\text{ }\times \text{ 1}00=\text{33}\frac{1}{3}\)

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

PRICES RELATED TO AN ITEM OR BUYING AND SELLING

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.

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

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.

Thus, the profit is ₹8 and profit Per cent is 1\(11\over9\).

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 : 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.

EXAMPLE : Selling price of a toy car is ₹540. If the profit made by shopkeeper is 20%, what is the cost price of this toy?

SOLUTION We are given that SP = ₹540 and the Profit = 20%. We need to find the CP.

Thus, by both methods, the cost price is ₹450

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

CHARGE GIVEN ON BORROWED MONEY OR SIMPLE INTEREST

Sohini said that they were going to buy a new scooter. Mohan asked her whether they had the money to buy it. Sohini said her father was going to take a loan from a bank. The money you borrow is known as sum borrowed or principal.

This money would be used by the borrower for some time before it is returned. For keeping this money for some time the borrower has to pay some extra money to the bank. This is known as Interest. You can find the amount you have to pay at the end of the year by adding the sum borrowed and the interest. That is, Amount = Principal + Interest. Interest is generally given in per cent for a period of one year. It is written as say 10% per year or per annum or in short as 10% p.a. (per annum).

10% p.a. means on every ₹100 borrowed, ₹10 is the interest you have to pay for one year. Let us take an example and see how this works

EXAMPLE : Anita takes a loan of ₹5,000 at 15% per year as rate of interest. Find the interest she has to pay at the end of one year.

SOLUTION The sum borrowed = ₹5,000, Rate of interest = 15% per year. This means if ` 100 is borrowed, she has to pay ₹15 as interest for one year. If she has borrowed ₹5,000, then the interest she has to pay for one year

\(\frac{\text{15}}{100}\text{ }\times \text{ 50}00=\)₹750

So, at the end of the year she has to give an amount of ₹5,000 + ₹ 750 = ₹5,750. We can write a general relation to find interest for one year. Take P as the principal or sum and R % as Rate per cent per annum. Now on every ₹100 borrowed, the interest paid is ₹R

Therefore, on ₹P borrowed, the interest paid for one year would be

\(=\text{ }\frac{R\times P}{100}=\frac{P\times R}{100}\)

PERCENTAGE – ANOTHER WAY OF COMPARING QUANTITIES

Interest for Multiple Years

If the amount is borrowed for more than one year the interest is calculated for the period the money is kept for. For example, if Anita returns the money at the end of two years and the rate of interest is the same then she would have to pay twice the interest i.e.,₹ 750 for the first year and ₹ 750 for the second. This way of calculating interest where principal is not changed is known as simple interest. As the number of years increase the interest also increases. For ₹100 borrowed for 3 years at 18%, the interest to be paid at the end of 3 years is 18 + 18 + 18 = 3 × 18 = ₹ 54. We can find the general form for simple interest for more than one year. We know that on a principal of ₹P at R% rate of interest per year, the interest paid for one year is \(\frac{R\times P}{100}\). Therefore, interest I paid for T years would be

\(=\text{ }\frac{T\times R\times P}{100}=\frac{P\times R\times T}{100}(OR)\frac{PRT}{100}\)

And amount you have to pay at the end of T years is A = P + I

EXAMPLE : If Manohar pays an interest of ₹750 for 2 years on a sum of  ₹ 4,500, find the rate of interest.