Percentages - Key Formulas and Examples - Part One

Question

Introduction to Percentages

Percentages are a fundamental concept in mathematics that we encounter frequently in our daily lives, from calculating discounts while shopping to understanding financial growth rates. Despite being so common, percentages can sometimes be tricky to handle. They involve conversions between fractions, decimals, and ratios and are deeply embedded in various fields, including finance, economics, and even everyday decisions like budgeting and saving.

This article is designed to break down percentages into simple, digestible concepts that will enable you to solve percentage problems confidently. In this first part of the series, we&rsquo;ll explore key formulas, shortcuts, and problem-solving techniques related to percentages.

At its core, a percentage is a way to express a number as a part of 100. For example, if you score 45 out of 50 in a test, you can convert that fraction into a percentage to see your score as a part of 100, making it easier to understand and compare. Percentages allow us to scale numbers up or down, making them useful for understanding proportions and rates. Whether you are figuring out how much of your salary to save, determining the interest on a loan, or analyzing statistical data, percentages are indispensable.

Applications of Percentages

Percentages are versatile and can be applied in several ways:

Converting fractions to percentages &ndash; It&rsquo;s often easier to understand values when expressed as a percentage rather than as a fraction or decimal. For instance, saying that 75% of a project is completed is often more intuitive than stating that three-quarters of the work is done.
	Converting percentages to fractions &ndash; Conversely, you may need to convert percentages back into fractions to simplify complex calculations or understand certain ratios better.
	Finding percentages of numbers &ndash; This is a critical skill, especially in real-life applications such as calculating taxes, discounts, interest rates, and more.
	Percentage increase or decrease &ndash; Knowing how to calculate percentage changes is essential for understanding inflation, profit margins, and other trends over time.

Now, let&rsquo;s dive into some key formulas and examples to clarify how percentages work.

Some Basic Formulae

A. To Convert a Fraction into a Percentage

To convert any fraction $\frac{n}{m}$ into a percentage, multiply it by 100 and then append the percentage sign (%) to the result. 
Multiplying a fraction by 100 scales the fraction to represent parts per hundred, which is what a percentage represents. This method can be applied to any fraction.
$\frac{n}{m} \times 100 = \text{{percentage}} $

Example 1
What percentage is equivalent to $\frac{3}{5}$?

We know that $\frac{3}{5}$ is a fraction.  
Multiply the fraction by 100 to convert it into a percentage:  
$\frac{3}{5} \times 100 = 60$  
So, $\frac{3}{5}$ is equivalent to 60%.

(adsbygoogle = window.adsbygoogle || []).push({});

To convert a percentage into a fraction, simply drop the percentage sign and divide the number by 100.
$text{{percentage}} \div 100 = \frac{n}{m}
The process of converting a percentage back into a fraction involves dividing the percentage by 100. This conversion is useful when we need to deal with parts of a whole.

Example 2
What fraction is \(16\frac{2}{3}$%?

Step-by-Step Solution:  
Drop the percentage sign: $16\frac{2}{3}$.  
Convert the mixed number into an improper fraction:  
$16\frac{2}{3} = \frac{50}{3}$  
Divide this fraction by 100 to convert it back into a simple fraction:  
$\frac{50}{3} \times \frac{1}{100} = \frac{50}{300} = \frac{1}{6}$  
So, $16\frac{2}{3}\%$ is equivalent to $\frac{1}{6}$.

B. To Find a Percentage of a Given Number

To find x% of a number (N), multiply the percentage by the number and divide by 100:

$x\% \text{{ of }} N = \frac{x}{100} \times N$

Example 3
What is 75% of 400?

Start by writing the percentage as a fraction:  
75%= 75 100  
Multiply by the given number (400):  
75 100 &times;400=300  
So, 75% of 400 is 300.

Example 4
Find a number whose 4% is 72.

Let the unknown number be N. We know that 4% of N is 72, so we write the equation:  
4 100 &times;N=72  
Solve for $N$:  
N= 72&times;100 4 =1800  
So, the number is 1800.

(adsbygoogle = window.adsbygoogle || []).push({});

C. To Find What Percentage One Number Is of Another

To find what percentage a number a is of another number b, use the following formula:

a b &times;100=percentage

Example 5
What percentage of 25 kg is 3.5 kg?

We want to know what percentage 3.5 kg is of 25 kg.  
Apply the formula:  
3.5 25 &times;100=14%  
So, 3.5 kg is 14% of 25 kg.

Some Useful Shortcut Methods

A. Understanding Percentage Increase and Decrease

If A is x% more than B, then B is less than A by:
	x 100+x &times;100 %
	If A is x% less than B, then B is more than A by:
	x 100&minus;x &times;100 %

Explanation
Consider the relationships between the two quantities. If A is more than B by x%, the percentage difference between A and B isn&rsquo;t simply x% of B; instead, it&rsquo;s X% relative to A. This relationship is captured by the formulas above.

Example 6  
If Mohan's salary is 10% more than Sohan's salary, how much percent is Sohan's salary less than Mohan's?

Mohan's salary is 10% more than Sohan's salary.  
Apply the formula:  
10 100+10 &times;100 = 10 110 &times;100=9.09%  
So, Sohan's salary is 9.09% less than Mohan's.

Example 7:  
If A's income is 40% less than B's income, how much percent is B's income more than A's?

(adsbygoogle = window.adsbygoogle || []).push({});

A's income is 40% less than B's.  
Apply the formula:  
40 100&minus;40 &times;100 = 40 60 &times;100=66.67%  
So, B's income is 66.67% more than A's.

B. Comparing Percentages of a Common Base

If A is x% of C and B is y% of C, then:
A= x y &times;100% of B
This formula derives from the fact that both A and B are percentages of a common base, C. By comparing the percentages x% and Y%), we can determine the relationship between A and B.

Example 8:
If A is 20% of C and B is 25% of C, then what percentage is A) of B?

A = 20% of C, so:
A = 0.20 * C
B = 25% of C, so:
B = 0.25 * C

Now, to find A as a percentage of B, we calculate:

A B &times;100= 0.20&times;C 0.25&times;C &times;100= 0.20 0.25 &times;100=0.8&times;100=80%

A is 80% of B.

C. Percentage Relationship Between Two Numbers

The formula is used to compare two numbers that are either more or less than a third reference number.
	If two numbers are respectively $x\%$ and $y\%$ more than a third number, the relationship between them is given by:
	First Number as a percentage of Second Number = 100+y 100+x &times;100
	
	Derivation
	Let the third number be C.
	The first number (say A is x% more than C, i.e., A=C+ x 100 C=C(1+ x 100 ) .
	The second number (say B is y% more than C, i.e., B=C+ y 100 C=C(1+ y 100 ) .
	
	To find A as a percentage of B:
	A B &times;100= C(1+ x 100 ) C(1+ y 100 ) &times;100 = 1+ x 100 1+ y 100 &times;100= 100+x 100+y &times;100
	This gives the percentage relationship between the two numbers when they are both greater than a third reference number.

Similarly, if two numbers are $x\%$ and $y\%$ less than a third number, the relationship is:
	First Number as a percentage of Second Number = 100&minus;x 100&minus;y &times;100

(adsbygoogle = window.adsbygoogle || []).push({});

Example 9: Comparing Two Numbers More than a Third Number
Two numbers are respectively 20% and 50% more than a third number. What percent is the first of the second?

Let the third number be C.

The first number is 20% more than C, so:
A = C + 0.20C = 1.20C
The second number is 50% more than C, so:
B = C + 0.50C = 1.50C

Now, find A as a percentage of B:

A B &times;100= 1.20C 1.50C &times;100 = 1.20 1.50 &times;100=0.8&times;100=80%

The first number is 80% of the second.

Example 10: Comparing Two Numbers Less than a Third Number
Two numbers are respectively 32% and 20% less than a third number. What percent is the first of the second?

Let the third number be C.

The first number is 32% less than C, so:
A = C - 0.32C = 0.68C
The second number is 20% less than C, so:
B = C - 0.20C = 0.80C

(adsbygoogle = window.adsbygoogle || []).push({});

Now, find $A$ as a percentage of $B$:

A B &times;100= 0.68C 0.80C &times;100 = 0.68 0.80 &times;100=0.85&times;100=85%

The first number is 85% of the second.

D Price Increase and Change in Consumption

If the price of a commodity increases or decreases, there is a corresponding change in consumption to maintain the same expenditure.

Price Increase and Reduced Consumption
	If the price of a commodity increases by P%, the reduction in consumption (to maintain the same expenditure) is:
	Reduction in consumption = P P+100 &times;100%
	
	Explanation
	- Suppose the original price is $P_0$ and the new price after an increase of $P\%$ is $P_1 = P_0(1 + \frac{P}{100})$.
	- To maintain the same expenditure, the consumption must decrease proportionally to the price increase. The new consumption will be inversely proportional to the price increase, leading to the formula.
	 
	Price Decrease and Increased Consumption
	If the price of a commodity decreases by P%, the increase in consumption (to maintain the same expenditure) is:
	Increase in consumption = P 100&minus;P &times;100%
	
	Explanation
	- If the price decreases, the consumer can afford to buy more. The increase in consumption is inversely proportional to the price decrease, which leads to this formula.

Example 11: Price Decrease and Increase in Consumption
If the price of a commodity decreases by 25%, by how much percent should its consumption be increased to maintain the same expenditure?

Using the formula for a price decrease:

Increased in consumption = P 100&minus;P &times;100 = 25 100&minus;25 &times;100 = 25 75 &times;100=33.33%

Consumption should be increased by 33.33%.

(adsbygoogle = window.adsbygoogle || []).push({});

Example 12: Price Decrease and Increase in Consumption
If the price of a commodity decreases by 20%, by how much percent should its consumption be increased to maintain the same expenditure?

Using the formula for a price decrease:

Increase in consumption = P 100&minus;P &times;100 = 20 100&minus;20 &times;100 = 20 80 &times;100=25%

Consumption should be increased by 25%.

Conclusion

In this first part of cracking percentages, we've covered the basic formulas, conversions, and some essential problem-solving techniques. Whether you&rsquo;re calculating a percentage increase or decrease, converting fractions to percentages, or solving real-world problems involving percentages, understanding these concepts is critical. Stay tuned for Part Two, where we&rsquo;ll dive deeper into advanced percentage problems and applications.

Exercises

If A is 30% of B and B is 40% of C, what percentage is A of C?
	Two numbers are respectively 25% and 35% less than a third number. What percentage is the first number of the second?
	If a number increases by 15%, find the percentage decrease in consumption required to keep expenditure constant.
	A commodity&rsquo;s price increased by 10%, but the expenditure remained the same. By what percentage did the consumption decrease?
	If the price of a product is reduced by 12%, by how much percent should consumption increase to maintain the same expenditure?
	Two numbers are 40% and 60% more than a third number. What percentage is the first number of the second?
	If A is 15% less than B, and B is 25% more than C, what percentage is A of C?
	The price of a commodity increased by 25%. What percent decrease in consumption will keep the total expenditure unchanged?
	Two numbers are respectively 18% and 22% less than a third number. What percentage is the first number of the second?
	If A is 45% of B and B is 60% of C, what percentage is A of C?
	The price of a product decreases by 15%. By what percentage should the consumption increase to keep the total expenditure constant?
	If a product's price decreases by 30%, calculate the percentage increase in consumption required to maintain the same expenditure.
	Two numbers are 50% and 70% more than a third number. What percentage is the first number of the second?
	If a number is reduced by 25%, how much percent must it be increased to return to its original value?
	If the price of a commodity decreases by 20%, what percent should the consumption increase so that the total expenditure remains unchanged?

(adsbygoogle = window.adsbygoogle || []).push({});