We are often presented with a requirement to compare numbers and outcomes to find what lies in our best interest, to choose what’s a better option, to analyze performances, and to keep note of our progress from time to time etc. In this module, we will learn how to calculate percentage in different scenarios and compare numbers using percentage formulas.

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For example, the comparison can be made in the following manners:

- By simply comparing two or more definite integer values. For example, 10 is greater than 8.
- By calculating the ratio. For example, 3/4 > 1/4
- By calculating the percentage. For example, Rohit scores 82% in an exam; whereas Mukesh scored 78%. Clearly, Rohit performed better than Mukesh as 82% > 78%.

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## What’s a Percentage?

The word percentage is derived from the Latin word *per centum*, where *centum* means hundred. Therefore, the word percent refers to nothing but per hundred which implies that the percentage value (X) is the numerator of a fraction whose denominator is 100. Simply put, the percentage value of X is X/100.

The percentage is denoted by the symbol %. For example, 2%, 7%, 15.5% and 30% etc. Therefore, 2% is nothing but 2 parts per 100 (2/100) or 30% is nothing but 30 out of 100 (30/100).

## Calculating Percentage Formulas

The percentage can be calculated following different steps depending on different scenarios as follows:

### 1. When the Base Value is 100

For instance, there are 100 balls in a bag out of which 25 balls are red, 40 balls are white, and 35 balls are black. To calculate what percentage of balls is white, we need to consider:

Number of white balls = 40

Total number of balls = 100

⇒ There are 40 white balls per 100 balls which can be written as \( \frac{40}{100} \)

And since percent means per hundred,

White balls (% in the bag) = 40%

### 2. When the Total or Base Value is Not 100

In real-life scenarios, there are many cases where a percentage value of a given condition needs to be calculated with the base either being less or more than 100. For instance, there are 200 balls in the bag out of which 25 balls are red, 40 balls are white, and 35 balls are black along with 50 balls of blue and yellow each.

If we need to find what percentage of balls is white, we need to do the following:

Here, Number of white balls = 40

Total number of balls = 200 (base value)

Therefore, there are 40 white balls per 200 balls.

Now, to convert this fraction into a percentage, we need to multiply and divide it by 100 thereby making it into an equivalent fraction with denominator 100 (*because percent means per hundred*).

⇒ Percentage (%) of white balls = \( \frac{40}{200} \times \frac{100}{100} \)

Here, the value of 100 in the denominator takes the form of % and the answer is calculated with the rest of the values.

White balls (%) = \( \frac{40}{200} \times \) 100 % = **20%**

### Alternate Method using Unitary

Another method to calculate percentage when the base value is not 100 lies in the **unitary method** which we have studied in our previous classes. Using the unitary method, we are trying to find that if there are 40 white balls for total 200 balls, how many white balls can there be for total 100 balls?

Mathematically denoting,

For every 200 total balls → 40 white balls

For every 1 → white ball(s)

Therefore, **Per 100** total balls → white balls = **20%**

## Comparing using Percentage Formula

There are a lot of instances where we try to compare our results or performances with others by not keeping the same base which is where comparison using percentage formula helps.

For instance, in two different exams, Mukesh scored 420/500 and Amit scored 400/450. Considering absolute values, it looks like Amit has scored less than Mukesh; however, if we compare the performances by calculating with percentage formula, the picture becomes crystal clear.

Calculating the percentage:

- For Mukesh, marks percentage = \( \frac{420}{500} \) × 100 = 84%
- For Amit, marks percentage = \( \frac{400}{450} \) × 100 = 88.88%

Clearly, the performance of Amit is better than that of Mukesh** **by** **(88.88% – 84%) = **4****.88%.**

## Solved Examples for You

**Question 1**: **Ankit is 5 feet (5′) tall. This year he grows by 3 inches (3″) which makes him 5′3″ tall. Next year, he again grows by 3″ making him 5′6″. Find out and compare the growth rates of Ankit’s height for both the years.**

**Answer :** We know that 5′ = 60″, analyzing the growth rate:

For this year, Growth = 3″ Baseline = 60″

∴ Growth (%) = \( \frac{3}{60} \) × 100 = 5%

For next year, Growth = 3” Baseline = 63”

∴ Growth (%) = \( \frac{3}{63} \) × 100 = 4.76%

Clearly, the growth rate of the current year is greater than that of the next year by (5 – 4.76) %** = 0.24%.**

**Question 2:** **There are two bags, A and B. Bag A contains 90 red marbles out 450 total marbles; whereas Bag B contains 70 red marbles out of 280 total marbles. Which bag contains the higher percentage of red marbles?**

**Answer :** Considering Bag A:

No. of red marbles = 90

Total no. of marbles = 450

∴ Red marble (%) in Bag A= \( \frac{90}{450} \) × 100 = 20%

Considering Bag B:

No. of red marbles = 70

Total no. of marbles = 280

∴ Red marble (%) in Bag B = \( \frac{70}{280} \) × 100 = 25%

Clearly, Bag B contains higher percentage of red marbles than Bag A by (25 − 20)% = **5%**

**Question 3: What is the per cent equation?**

**Answer**: In per cent equation, the base is the number of which we will be taking a percentage and the amount will be the value that we get when we take the per cent of the base. Thus, it means that in any per cent problem, there will be three basic values which should concern us: the per cent, the base, and finally the resulting amount.

**Question 4: How do I calculate a percentage of a percentage?**

**Answer**: In order to calculate the percentage of a percentage, we need to start by a division of each percentage by 100 so we convert them to a decimal form. After that, we multiply the decimals collectively to get our result. Finally, multiply the result you get by 100 to obtain the final percentage.

**Question 5: What’s a Percentage?**

**Answer**: This term is taken from the Latin language. The Latin word is per centum and it translates to a hundred. Thus, it means per hundred that implies the percentage value x is the fraction’s numerator with a denominator of 100. In other words, the percentage value of x will be x/100. We use the symbol of% to denote it.

**Question 6: What per cent is 3% of 5%?**

**Answer**: 60 % is 3% of 5%. We get this by turning the denominator of 5 to 100. Thus, 5×20=100, so we now multiply the top and bottom of the fraction by 20. Thus, it gives us our answer that is 60 %.

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