Understanding Percentage

Let us take the following example of business done by 2 brothers Ashok & Bharat. Suppose they were in the business of buying vegetables from the wholesale market, and selling them in the city market.

On day 1, the transactions were as follows.

It is obvious that Bharat made more profit on Day 1. Hence on this day he was a better businessman.

On day 2, the transactions were as follows.

On Day 1, it was easier to see that Bharat was the better business man as he made more profit while the Sale Amount for both was the same at Rs 1,000.

But on Day 2 we have a situation where both the profit amounts and the same amounts are different. How do we now compare them?

The idea of ratios gives a way to do this comparison. We can see both the profits as a fraction of the sale amounts. The person who makes a larger fraction of his sale amount as profit is a better businessman! Let us see the figures.

So we can say that for every rupee of sale amount, Ashok made a profit of Rs 0.20 while Bharat made a profit of Rs 0.375. Hence Bharat is a better business man!

Percentages

Percentages are actually fractions or decimals in "commercial" disguise.

Businessmen are not really comfortable in working with fractions. So they evolved the idea of percentages. Here instead of comparing the profit on every rupee of the sale amount, they compare the profit on every Rs 100 of the same amount. This is possibly because all of us are familiar with 100 and can visualise it easily. Recasting the above table in terms of percentages we get the following table

So for every Rs 100 of Sale Amount, Ashok made a profit of Rs 20 while Bharat made a profit of Rs 37.50.

Businessmen feel more comfortable to see 0.2 as Rs 20 per Rs 200 & 0.375 as Rs 37.50 per Rs 100.

So a percentage is just another way of writing a fraction. It is a fraction where the denominator is always 100.

Let us see some fractions & the different ways

The notation '%' is used to indicate that the number before it is a percentage.

It gives the quantity "per cent" or per hundred.

Converting Fraction To Percentage - Multiply the fraction by 100

Converting Percentage to Fraction - Divide by 100

Since percentages are decimals in disguise, they can multiplied & divided just as decimals. Hence they obey the law of commutativity.

Other Areas

There are other aspects of life where percentages are used.

In any assessment, the marks obtained are always indicated in percentages i.e they are given as if the total marks was 100. So if you get 40 out of a test where the total marks was 80, you have scored 50%.

Any data concerning various countries is always given in percentages, because the size and population could be widely different.

Other Understandings

Percentage amounts themselves can be expressed as decimals or fractions. For example the profit percentage could be 12.50%. The businessman can imagine this as a profit of Rs 12.50 on every sale of Rs 100!

Though we can say 12 1/2% it is generally not done as the entire purpose of percentages was to avoid fractions!

They can be less than 1 or greater than 100. A businessman could make a profit of Rs 110 on a sale of Rs 100! His profit would be 110%.

The same number can be a different percentage of different numbers. 5 is 5% of 110, 25% of 20 & 50% of 10.

Caution While Comparing Percentages

In understanding percentages, the context is important. Comparing percentages without understanding the context may give a wrong picture. A bigger percentage by itself may not be significant. It could be that of a small number!

In real terms, a profit of 20% may not be bigger than a profit of 10%. 20% could be out of 100 and 10% could be out of 500!

Percentages obey the Laws of Arithmetic

Because percentages look different, it is many times forgotten that they are just fractions.

Take this problem: what is 54% of 10? This may look confusing.

It looks simpler if written as - what is 10% of 54? The answer obviously is 5.4.

This is because a percentage is a fraction and fractions are commutative in multiplication.

54% of 10 is actually 54/100 X 10 which can be rewritten as 10/100 X 54 which is 10% of 54.

Percentages, Fractions & Decimals

All the above indicate the same idea expressed in different formats. Much like the "teacher" in class is "Leela" in her house and "Beti Leelu" to her mother.

Percentage is a fraction where the denominator is always 100! 12% is the fraction 12/100.

Percentages can easily be expressed in decimal form because they are fractions with a denominator 100.

12% is 12/100 and hence is 0.12. Converting from decimal format to percentage format just needs a multiplication by 100.

Operations with Percentages

Normally students need to be familiar with 3 types of life situations. They are given in increasing order of difficulty from the point of view of students.

1. Find the part given the whole & the percentage - What is 30% of 150?

2. Find the percentage given the part & the whole - What percentage of 250 is 50?

3. Given the part & the percentage, find the whole - If 20 students form 40% of the strength of a class, find the class strength?

The basic understanding is Part/Whole *100 = Percentage