Introduction to Percentages

Percentages are all around us and are an important component for representing figures and statistics. However, just because they’re a new way to represent numbers, doesn’t mean that they can’t understood intuitively.

The Big Idea: What is Percentage?

Percent is a combination of the words per and cent meaning per hundred. A percentage is thus a number that represents a fraction of a given quantity out of 100. 

Represented usually by the symbol %, percentages are an important part in relating figures to a common scale and also, bringing disproportionate figures to common denominations so that they can be understood better.

Percentages are after all, an extended concept of fractions and are used to represent numbers from a lot of hundred denominations. Let’s explain this concept further by taking the example of a class with students-boys and girls numbering to 20 and 30 respectively. The number of boys can be represented as a fraction which would be \({20 \over 50}\) or \({2 \over 5}\) on taking out common factors.

But wait, look at the denominator-it’s five. If you need to bring this proportion to a base 100, you would multiply it with the same number, thus giving \(\begin {align}40-(2\times\frac{100}5)\end {align}\).

This is another way of saying that there are 40 boys for every 100 students in the class and 60 girls for every 100 students. Notice how the sum comes up to 100.

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Percentages are also a central tool for mensuration, accounting, rate studies and a wide number of mathematical fields that require specific numeral denominations.  

Why Percentages? Why Are Percentages Important?

How well do you know if something has dropped or increased since last year? Can you tell by what amount has a nation’s population increased over five years?

What these questions require is some good old-fashioned percentage calculations.

Imagine for example the case of a company tracking the sales that occur annually from 2010 to 2015. How would the company know how well it’s performing?

You guessed it right. Simply subtract the annual values by the initial value and multiply it by 100. This shows an important property of percentages- they help gauge changes.

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\(\begin{align}{\text{%  }}\frac{{{\text{Increase}}}}{{{\text{Decrease}}}}{\text{ =}}\left( {\frac{{\left( {{\text{Current Value - Old Value}}} \right)}}{{{\text{Old Value}}}}} \right) \times {\text{100}}\end{align}\)

Now let’s look at another example of two friends who’ve scored different marks in different courses. Say one got \({30 \over 50}\) and the other got \({47 \over 60}\).

Instead of using the conventional method of bringing both the denominators to be same to compare both the values, simply represent both in terms of percentages.

This would make both the marks 60% and 78.33%. Now doesn’t that let you compare the values easier?

Common Doubts with Regards To Percentages

Percentages that use 100 as a denomination might seem simple to understand by know but you’ll likely encounter percentages that far exceed 100 or may even go below zero.
Not to worry though, these are simple abstractions of the very same concept used to show changes, especially those of greater or more serious magnitudes.

The best thing to do is to treat such percentages as improper fractions where the numerator is larger than the denominator. If you divide the numerator by the denominator in such a case, you’ll end up with a quotient and a remainder that can be represented further as a mixed fraction.

For example, think of a huge company that doubles their net profits in a year, or imagine a school whose enrollment triples due to more people moving into the vicinity. Both these cases would result in a percentage change of greater than 100!

Take another example in the other way, suppose there’s a profit produced of Rs 1000 by a shop while the profit in the previous month was Rs. 700. Since the newer value is larger than the first, it’s obvious that the percentage increase will be larger than 100% i.e., \(\begin{align}\left( {\frac{{1000}}{{700}}} \right) \times 100\end{align}\) or 142.85%. Be warned though, this only shows you how high 1000 is compared to 700. To properly know the amount at which this has occurred, you would have to subtract both values.

Useful Tips and Tricks to Solve Problems on Percentages

When dealing with problems that say that the number is “50%” higher or lower than the principal amount, simply add or subtract the percent value from one and multiply it with the principal number.

For example, 50% higher than Rs.100 would be 1.5*100 = Rs.150.

If you want to convert a percentage back to the original value, retrace the steps and simply reciprocate the percent to 100.

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For example, 30% of 200 is 60. If you didn’t know the value 200, just multiply 60 by \({100 \over 30}\).

For more sophisticated problems in interest rates, you would have to deal with percentages that tend to change over years, months, days, weeks and even years. People who are proficient in simple and compound interest cases would know about this. The trick is to simply divide the percentage value with its respective number of chronological denominations.

For example, 2% interest compounded year on year on a weekly basis would return 2% divided by 52 since there are 52 weeks in a conventional year.

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