Comparing Quantities with Percentages Concept and Applications

What is Comparing Quantities?

Comparing quantities is the method of determining the amount of comparing units with respect to another standard or reference unit. Two comparing units must have a common reference point; otherwise, they cannot be compared. Comparing is something which we have done since we were babies, for example, while sharing chocolate with your elder sibling, you always compare the two chocolate pieces for which one is larger. Comparison of quantities is a similar approach, but instead of visual analysis, we go for practical analysis using number. 

What is the Percentage?

To understand the concept of percentage, let us take 30 milk chocolates out of 50 chocolates in a box. Now suppose we have another box of chocolates with the same chocolates. This means that now we have 50+50 = 100 chocolates, and the number of milk chocolates goes to 30 + 30 = 60 chocolates. If we look at the numbers carefully, we see that we have 60 milk chocolates out of 100. This notation of representing quantities out of 100 is known as a percent.

Percent means per – Hundred (use cent to denote hundred), i.e., amount of quantity left if we have a total amount of 100 quantities. Another way of comparing quantities is by using proportions like ratios and fractions.

How to Find a Percentage?

The simple mathematical way of finding percentage is by multiplying the fraction with 100.

     Percentage = \[\frac{{Fractional Amount}}{Total Amount}\] X 100

To picturise this you can consider approaching through these steps

1. Take the number of parts you need to calculate, for example; you secured 80 marks 

2. Find the total quantity or the maximum possible value. Let the maximum marks for our previous example be 90

3. Denote the expression in fractional notation i.e.  \[\frac{{80}}{90}\].

4. Multiply the fraction to with the reference number, i.e. 100 for percentage. Hence our expression becomes  \[\frac{{80}}{90}\] x 100 = 88.89% (This is the required percentage).

Examples

Comparing quantities examples

1. Imagine you score 23.5 marks out of 25 marks in a Monday test, Then the percentage of your marks will be   \[\frac{{23.5}}{25}\] x 100 = 94%

2. If the minimum percentage required to pass a specific test is 95% and the total number of marks is 300 then the maximum marks to pass the test will be \[\frac{{x}}{300}\] x 100 = 95, if we solve for x then the value we get is 285, i.e. you need to score maximum marks of 285 to successfully pass the exam. 

Interesting facts

If you need to calculate x% of a given value y, then the same result will be given for y% of x.

i.e. if you need to calculate 4% of 75, then simply calculate 75% of 4, i.e. 3. 

Questions

Sample Question 1:  Calculate the percentage of marks scored by the student if the question paper had 45 questions of 2 marks each, and the student got 7 answers wrong.

Solution: Since the question paper had 45questions of 2 marks each hence

Total Marks = 45 x 2 = 90

Now, the student got 7 answers wrong, which means 

Marks lost for wrong answers = 7 x 2 = 14

If the total marks are 90 and marks for wrong answers are 14 then 

Marks awarded for correct answers = 90 - 14 = 76

Hence,

Percentage calculation = \[\frac{{Scored Marks}}{Total Marks}\] X 100

= \[\frac{{76}}{90}\] X 100 = 84.44%

Therefore the percentage of marks secured by the student is 84.44%

Sample Question 2: Calculate the percentage of marks scored by a team of four students if the minimum marks scored by the group is 42, Maximum marks are 50 which are the highest marks possible, and other two members have scored five less than the maximum and five more than the minimum.

Solution:

To find the total marks, 

We have four students, and maximum marks for everyone are 50

Maximum total marks of the group = 20 x 4 = 200

Since one student got five less than maximum, and other five more than minimum

Marks of Student 2 = 50 - 5 = 45

Marks of student 3 = 42 + 5 = 47

Total marks secured by the group = 42 + 45 + 47 + 50 = 184

Therefore, Percentage =  \[\frac{{184 }}{200}\] x 100 = 92%

Hence, the total percentage scored by the group is 92%.

Sample Question 3: If Hermione Granger secured a 99% by getting 49.5 marks and Ron secured 75% by securing 37.5 marks. Calculate the percentage scored by Harry if he secured 43 marks. 

Solution: 

Let the total marks be x

Hermione’s percentage = \[\frac{{49.5}}{x}\] X 100 = 99

On solving for x we get total marks = 50

Hence,

Percentage for Harry = \[\frac{{43}}{50}\] X 100 = 86%

Therefore we can see that the calculated percentage of marks for Harry is 86%.