How to Calculate the Percentage of Marks?

The word Percentage is derived from the Latin word “Percentum”, which means “per hundred”. A Percentage is a number that is used to represent a part of 100. 

A Percentage is more like a fraction. A Percentage is denoted by the sign %. Suppose out of 100 students 12 students were absent, then we say 12% of the students were absent. In other words, anything out of a hundred becomes a Percentage that means a fraction with denominator 100 can be called a Percentage. In this chapter we will learn how to find the Percentage of Marks. For example,

\[\frac{{70}}{{100}}\] = can be called as 70 percent. = 70%

\[\frac{{20}}{{100}}\] = can be called as 20 percent. = 20%

\[\frac{{53}}{{100}}\] = can be called as 53 percent. = 53%

\[\frac{{97}}{{100}}\] = can be called as 97 percent. = 97%

Formula to calculate Percentage

It is easier to find the Percentage if it is out of a hundred but how to calculate the Percentage if it is not out of 100? Suppose there are 60 students in a class and 3 of them were absent. So how to calculate the percentage of students absent?

Percentage=\[\frac{{Number\,of\,subject\,of\,which\,percentage\,is\,to\,be\,found\,(Part)}}{{Total\,Number(Base)}} \times 100\% \]

Here, we need to find the number of students absent. Part = 3, Base = 60.

Percentage = (3 students/60 students)*100%

=(3/60)*100%

=(1/20)*100%

=(10/2)%

 =   50 %

Therefore, 5% of the students were absent.

Use of Percentage as for comparing Quantities

Percentage being a convenient way of writing a fraction becomes an integral tool for comparing quantities. Let’s take an example to explain why Percentage is preferred more than a fraction. Suppose Yusra comes and tells her parents that she is better in Maths because she scored 80 Marks and Tasnim scored 70 Marks in Maths in their respective classes. Would you believe it?

No, because we only know the part and are not aware of the base (total) Marks. So let’s check the reality. Yusra scored 80 Marks out of 120 Marks whereas Tasnim scored 70 Marks out of 100 Marks. The fraction of Yusra’s Marks and Tasnim’s Marks are 80120 and 70100 respectively but still we are not sure who scored better. So now let’s calculate the Percentage,

Yusra’s percentage in maths \[ = \frac{{Part}}{{Base}} \times 100 = \frac{{80}}{{120}} = 66.6\% \]

Tasnim’s percentage in maths \[ = \frac{{Part}}{{Base}} \times 100 = \frac{{70}}{{100}} = 70\% \]

From the above example, we clearly noticed that choosing a bigger fraction from \[\frac{{80}}{{120}}\] and \[\frac{{70}}{{100}}\] is a little tricky but choosing a bigger Percentage from 66.6% and 70% is easier.  So now we can say that Tasnim’s score is better than Yusra’s score because 70% is more than 66.6%. Therefore, it is now clear that the Percentage is more helpful than fractions when it comes to comparing quantities.

How to calculate the Percentage of Marks?

How to find out the Percentage of Marks of a class having 30 students out of which 25 students passed in Maths and 5 students failed. The students who failed in Maths got 15, 30, 22, 7 and 35 out of 80 respectively. Calculate the Percentage of students who failed in Maths. And also show how to find the Percentage of Marks of students who failed in Maths.

Solution: a) Let the students who failed in Maths be A, B, C, D and E. Shown below is the method of how to calculate Percentage from Marks:

Percentage of marks of Student A \[ = \frac{{15}}{{80}} \times 100 = \frac{{75}}{4} = 18.75\% \].

Percentage of marks of Student B \[ = \frac{{30}}{{80}} \times 100 = \frac{{300}}{8} = 37.5\% \]

Percentage of marks of Student C \[ = \frac{{22}}{{80}} \times 100 = \frac{{110}}{4} = 27.5\% \]

Percentage of marks of Student D \[ = \frac{7}{{80}} \times 100 = \frac{{70}}{8} = 8.75\% \]

Percentage of marks of Student E \[ = \frac{{35}}{{80}} \times 100 = \frac{{175}}{4} = 43.75\% \]

Therefore, the students who got 18.75%, 37.5%, 27.5%, 8.75% and 43.75% failed in Maths.

 b) If 5 out of 30 students failed in Maths then the Percentage of students who failed in Maths are:  \[\frac{5}{{30}} \times 100 = \frac{{50}}{3} = 16.66\% \]

Therefore, 16.66% of the total students failed in Maths.

Important Conversions

Conclusion: To convert any form of number into Percentage we need to convert it into a fraction and then multiply by 100. Similarly, to convert a Percentage to any form of a number we need to first divide the Percentage by 100 and then convert the resultant fraction to the form of a number we want.

Examples: 

1) Convert the following into Percentage:

a) 1/5          b) 0.6             c) 2:8       d) 9

 Solution:

1. \[\frac{1}{5}\] x100 = 20%

2. 0.6 = \[\frac{6}{{10}}\] x100 = 60% 

3. 2: 8 = \[\frac{2}{8}\] x100 = \[\frac{{100}}{4}\] = 25%

4. 9 = \[\frac{9}{1}\] x100 = 900%

2) Convert 20% into:

a) Fraction       b) decimal        c) Ratio     

Solution:

1. 20 % = 20/100 = 1/5

2. 20 % = 20/100 = 1/5 = 0.2

3. 20 % = 20/100 = 1/5 = 1 : 5

3) What is 10% of 30?

Solution: 

According to the formula, Percentage = \[ = \frac{{(Part)}}{{(Base)}}\, \times \,100\]

Here, Percentage = 10, 

          Base = 30

          Part = ?

Let part be x.

\[10 = \frac{x}{30} \times 100\]

x = 3

Thus, 10% of 30 is 3.

4) What Percentage of 120 is 40?

Solution:

According to the formula, Percentage = \[ = \frac{{(Part)}}{{(Base)}}\, \times \,100\]

Let the Percentage be x

\[x = \frac{{40}}{{120}} \times 100 = 33.33\% \]

Thus, 33.33% of 120 is 40.

Percentage Increase and Decrease

Any increase or decrease in number or quantity can be defined in Percentage. For example, the price of the car has decreased by 10%, the price of the plot has increased by 30% in the last few years, etc. The Percentage increase or decrease is always calculated on the basis of initial value and not final value. Suppose there is a discount of 10% on a t-shirt worth Rs 1000. Discount, that is the amount reduced is 10% on Rs 1000. 10% of Rs 1000 is Rs 100. So if the discount is of Rs 100 then the final value will be Rs 1000 - Rs 100 = Rs 900.

Decrease Percentage

Initial Value > Final Value.

Final Value = Initial Value - Decreased Value.

Initial Value = Final Value + Decreased Value.

Decreased Value = Initial Value - Final Value

Where,

Decreased Value = x% of Initial value 

                            = \[\frac{x}{100}\]× initial value

Increase Percentage

Initial Value < Final Value.

Final Value = Initial Value + Increased Value.

Initial Value = Final Value - Increased Value.

Increased Value = Final Value - Initial Value.

Where,

Increased Value = x% of Initial value 

                            =\[\frac{x}{100}\]× initial value

x Initial Value

Examples of question on the concept of Percentage change:

Example 1: Last year the number of students in a school was 1000. This year the number of students in the school is 1500. 

Solution: Initial number of students = 1000

Final number of students = 1500

Final number = Initial number + Increased number 

or

Number of students increased = Final number - Initial number

                                                 = 1500 - 1000

                                                 = 500

Let the increased Percentage be x.

X% of initial number= increased number

\[(\frac{x}{100})\] of 1000 = 500

\[(\frac{x}{100})\]  * 1000 = 500

\[x=(\frac {500}{1000})\]*100

X = 50

Therefore, the increased Percentage is 50%. That means 50% of the students have increased in school since last year.

Example 2: If in November, Nupur worked a total of 45 hours, in December she worked 65.5 hours – by what Percentage did Nupur’s working hours increase in December? And if she worked for only 45 hours again in January, then by what Percentage did her work change in January? 

Solution: Given

Number of hours nupur worked in the month of November = 45 hours

Number of hours she worked in December = 65.5 hours

And we know, In order to find the Percentage of a changing value, we first need to calculate the value change and then divide this value change to the initial value. And at last multiple by a 100. Which will provide us with the Percentage change. Formula to is

Percentage change =\[\frac{change in the value}{Initial value}\] × 100

For this sum, The initial value will be the working hours of the month, November = 45 hours. And the change in value will be

Change in value (Increased) = Working hours in December - Working hours in December 

= 65.5 - 45 hours. = 20.5 hours.

Applying these values to the formula of Percentage given above.

Percentage = \[\frac{20.5}{45}\]× 100 = 0.455 × 100 = 45.5%

Hence, Nupur, therefore, worked 45.5% more hours in December than she did in November.

And In the month of January, Nupur worked for only 45 hours again – the same as she did in November. 

Now, Find the Percentage difference between Nupur’s December working hours (65.5) and her January working hours (45)? 

In this case the initial value will be the working hours in December.

First Finding the change in working hours (decrease, in this situation) that is: 

Change = 65.5 - 45 hours = 20.5 hours

Applying these values in the formula

Percentage change =\[\frac{20.5}{65.5}\] × 100 = 0.312 × 100 = 31.2%

Hence, Nupur’s working hours were 31.2% lower in January than in the month of December.