Percentages are fundamental to daily life, whether to calculate discounts in shopping, analysis of data, or exam preparations for competitive tests. Simply put, a percentage is a proportion of 100 and is shown by the % symbol. Thirty percent, for instance, would mean 30 out of 100. Percentages facilitate an easy comparison of values and gauge changes easily. This article will walk you through the basic principles of percentages, such as formulas, percentage changes, and shortcuts for quick problem-solving. We have also added solved examples so that you practice and build confidence, especially if you're a competitive exam aspirant like the SAT, ACT, or other common tests. Go through this article thoroughly to reinforce your knowledge and enhance your problem-solving skills.

What is the Percentage of a Number?

In mathematics calculations, a percentage is a numeral or ratio that can be defined as a fraction of 100. In other words, we can say that the percentage is specified as a given fraction or part in every hundred. This implies that it is a fraction with 100 as the denominator and is commonly symbolized by the symbol “%” symbol.
For example, if we have to estimate the percent of a number, then divide the number by total and multiply it by 100. In a live test, Savita scored 45% marks, which means that she scored 45 marks out of 100.

Percentage Formula

To hold a better command on percentage calculation we need to know all percentage formulas. The basic formula used to calculate the percentage is equivalent to the ratio of actual value to the complete value multiplied by 100. The formula of the percentages is expressed as:

\(\text{Percentage formula}=\left(\frac{\text{Actual value}}{\text{Total value}}\right)\times100\)
For Example: \(\frac{2}{4}\times100=0.5\times100=50\%\)

Percentage Difference Formula

The percentage difference can be understood as the change in the value of an amount over some time in terms of percentage. If there are two values and we need to determine the percentage difference between the given two values, then this can be calculated by the below steps:
Step 1: Compute the difference (i.e subtract one value from the other) skip any negative sign if obtained.
Step 2: Estimate the average of the two values (add the values, then divide by 2).
Step 3: Finally divide the difference by the average obtained.
Step 4: Transform the obtained answer to a percentage for the result to be in percentages.

Percentage Difference Formula=\(\left|\frac{\text{First Value}−\text{Second Value}}{\left(\frac{\text{First Value}+\text{Second Value}}{2}\right)}\right|\times100\%\)

The modulus symbols represent absolute value so that any negative outcome becomes positive.

Percentage Increase and Decrease Formula

Two cases might appear while computing percentage difference namely:
Percentage increase.
Percentage decrease.
Let us learn how to calculate both through the formula:
The percentage increase is equivalent to subtracting the original number from the new number and dividing the obtained answer by the original number. Multiply the final answer by 100 for the answer to be in percentage.

Percentage Increase=\(\frac{\text{Rise in the Number}}{\text{Original Number}}\times100\%\)
Rise in the Value= New number – Original number
Likewise, percentage decrease is comparable to subtracting the new number from the original numeral and dividing the obtained answer by the original number. Multiply the final answer by 100 for the answer to be in percentage.

Percentage Decrease=\(\frac{\text{Decrease in the Number}}{\text{Original Number}}\times100\%\)

Decrease in the Number=Original number – New number
We should remember that when the new value/number is greater than the old number/value, it is a percentage increase, otherwise, it is a decreasing percentage.

Percentage Formula in terms of a Fraction

To transform a fraction into a percentage divide the top/numerator number by the bottom/denominator number and lastly multiply the result by 100%.

\(\frac{\text{Numerator Value}}{\text{Denominator Value}}\times100\%\)

Percentage Change Formula

Sometimes when it is required to get the increase or decrease in any quantity as percentages, which is also directed to as percentage change is given by the formula:

Percentage Change=\(\frac{\text{New value – Original value}}{\text{Original value}}\times100\)

How to Find Percentage?

To estimate the percentage of any value/ data/ number, we can apply the various formulas as discussed above as per the condition applied. Let us learn the basic method to find the percentage.
A% of a data = B
Here B is the necessary percentage.
If we wish to remove the % sign, then the formula is expressed as:
A/100 * given data = B
For example:
How to calculate 20% of 60.
Let 20% of 60 = Y
20/100 * 60 = Y
Y = 12

Similarly; 8% means 8 out of every 100, or in fraction we write 8/100.
In the same way, 50% can be composed as a fraction, 1/2, or a decimal, 0.5.

Percentage Table

Some common fraction and their percentages equivalents are given below.

How to Convert Fractions to Percentages?

A fraction can be represented by; \(\frac{x}{y}\) and can be converted into percentage by the below formula:

\(\frac{x}{y}\times 100\)

Therefore by the formula, it is clear that we can convert fraction to percentage merely by multiplying the given fraction by 100.

Note:

To convert percentages into fraction, divide it by 100.

Example: 25% = 25/100 = ¼

To convert a fraction into percentage, multiply in by 100.

Example: ⅕ = ⅕ x 100 = 20%

Difference between Percentage and Percent

The words percentage and percent are nearly related to one another. The tradition for operating percent and percentage is as specified. The word percent (or the symbol %) accompanies a specific number, on the other hand, the word percentage is used without a number.

An example of Percent:
More than 65% of the country’s population have been vaccinated with the first dose of Covid-19.
An example of Percentage:
A very large percentage of the world’s population has been exposed to Covid-19 pandemic.

Important Definitions related to Percentages

Let us know some of the important definitions related to the percentage.

Marks Percentage

Marks obtained by students in various exams during school and colleges are mostly out of 100. These marks are calculated in terms of percent. For example, consider if a student has scored X marks out of total marks. And, if we have to decide the percentage score; then we divide the scored mark from total marks and multiply the result by 100.

Tips and Tricks to solve Percentage based Questions Faster

Candidates can find different tips and tricks from below for solving the questions related to percentage.

Tip # 1:  Candidates need to make sure that they know all the important formulas of percentage which are mentioned below.

• Profit % = profit x 100 / cost price
• Loss % = loss x 100 / cost price
• Markup % = (markup / cost price) x 100
• Discount % = (discount / market price) x 100

Tip # 2: Successive Percentage Change: We can use successive percentage change formulas to solve percentage related problems where the product of two quantities equal the third quantity. For example,

⇒Length x Breadth = Area

⇒Price x Quantity purchased = Expenditure

⇒If any quantity is increased by x%, then y% and later on z%, the overall or effective percentage increase is:

⇒[(100 + x) / 100) (100 + y) / 100) (100 + z / 100) -1] x 100

Percentage Solved Example for Competitive Exams

Question 1: 20 gram is what percentage of 1 kg?

Solution 1: Here, quantity 1 = 20 grams and quantity 2 = 1kg = 1000 grams

⇒Hence, required percentage = 20/1000 × 100 = 2%

Question 2: If the price of sugar is increased by 10%, then by how much percent consumption should be reduced so that the expenditure remains the same?

Solution 2: Let the price be Rs. x /kg Consumption be y kg

⇒Hence, expenditure = price × consumption ⇒ Expenditure = xy

⇒Price of sugar is increased by 10% Hence, new price of sugar = 1.1x per kg

⇒Let new consumption be z kg

⇒Hence, new expenditure = (1.1x) × z Now, new expenditure = old expenditure

⇒ (1.1x) × z = x × y ⇒ z = y/1.1

Reduction in consumption = (y – z) = y – (y/1.1) = y/11

∴ Percentage reduction in consumption = [(y/11)/y] × 100 = 100/11 = 9.09%

Question 3: The population of a town 2 years ago was 245000. It increased by 15% in the first year and then increased by 20% in the second year. What is the current population of the town?

Solution 3: The population of a town 2 years ago was 245000 It increased by 15% in the first year

∴ The population after first year will be = (115 / 100) x 245000 = 281750

⇒The population then increased by 20% in the second year.

∴ The population after second year will be = (120 / 100) x 281750 = 338100

Question 4: An electric bully was bought at Rs. 4100. Its value depreciates at the rate of 7% per annum. Its value after one year will be:

Solution: Actual price of the electric bully = Rs. 4100

⇒ Depreciation rate = 7%

∴ Value after 1 year = 4100 – 7% of 4100 = 4100 – 4100 × (7/100) = Rs. 3813

Question 5: If A’s income is 40% more than the income of B, then what percentage of B’s income is less than income of A?

Solution: Let the income of B be 100

∴ Income of A = 140

⇒B’s income is less than income of A by (140 – 100) = 40

⇒Required percentage = (40 / 140) x 100 = 200 / 7 = 28 (4/7) %

Question 6: If A is 40% less than B, then B is how much percentage more than A?

Solution: Given, A is 40% less than B Let B be 100

⇒A = B – 40% of B = 100 – 40% of 100 = 100 – 40 = 60

∴ Required % = {(100 – 60)/60} × 100 = (40/60) × 100 = 66.66%

Conclusion

Mastering the concept of percentages is the key to achieving success in day-to-day life and, above all, competitive examinations such as the SAT, ACT, and other U.S. standardized exams. Familiarity with percentage formulas, percentage changes, and conversions between fractions and percentages will surely improve speed and accuracy in problem-solving. By solving the example problems offered in this article, you can gain a firm understanding of the concepts and apply them confidently to real-life situations and test problems. Whether figuring discounts, interpreting data, or answering tricky exam questions, a firm grasp of percentages can make all the difference. Keep practicing, remain regular, and you will be adequately equipped to face any percentage question thrown your way!