The ratio of the difference in the amount to its starting value multiplied by 100 is the percentage change (or) successive percentage change. The percentage indicates per \(100\), a number presented as a fraction of \(100\). \({\rm{100 \% }}\) of something can be said the whole of it. We use percentages to compare the quantities.

Percentage change means the increase or decrease in the earlier value. If the present value is more than the earlier value, the percentage increase can determine how much it has increased. If the current value is less than the earlier value, then the percentage decrease can determine how much it has decreased. This article will discuss percentage change successive changes in percentage using formulas and examples.

Learn How to Calculate Percentage

Percentage Definition

In Mathematics, a percentage is a number or ratio expressed as a fraction of \(100\).
Per cent is derived from the Latin word per centum, which means per \(100\). Per cent is expressed by the symbol \({\rm{\% }}\).

Percentages can also be represented in decimal or fraction form such as \({\rm{0}}{\rm{.5 \% , 75 \% }}\) etc.

Formula of Percentage

The percentage is defined as part or amount in every \(100\).
For example, Madhu has a necklace and have \(40\) beads of two different colours. Out of \(40\) beads, \(10\) are red. Hence, out of \(100\), the number of red beads \(\frac{{10}}{{40}} \times 100 = 25\) (out of hundred) \({\rm{ = 25 \% }}\)
\({\rm{1 \% }}\) means \(1\) out of \(100\) or one-hundredths. So, it can be written as \(\frac{1}{{100}}\) or \({\rm{0}}{\rm{.01}}\).
Therefore, \({\rm{25 \% }}\) can be written as \(\frac{{25}}{{100}}\) or \({\rm{0}}{\rm{.25}}\).
\({\rm{Percentage = }}\frac{{{\rm{Actual}}\,{\rm{value}}}}{{{\rm{Total}}\,{\rm{value}}}} \times 100\% \)

Definition of Percentage Change

The percentage change of a quantity is the ratio of the difference in the amount to its initial value multiplied by \(100\).
The phrase per cent change says the difference in the percent of the earlier and present quantity. The percent change produces the difference between quantity out of \(100\).

Percentage Change for the Increment in Percentage

An increment in percentage refers to the percentage change in the value that is increased in a period.
Percentage increase can be computed by using the following formula:

\({\rm{Percentage}}\,{\rm{increase}} = \frac{{{\rm{Initial}}\,{\rm{value}} – {\rm{Final}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)

Therefore, the ratio of the difference between the initial value and the final value (when the final value is subtracted from the initial value) to the initial value is known as percentage change for the increment in percentage.

For example, increment in population, increment in the price of a quantity, increment in the number of students in a class, etc.

Percentage Change for the Decrement in Percentage

A decrement in percentage refers to the percentage change in the value that is decreased in a period.
Percentage decrease can be computed by using the following formula:

\({\rm{Percentage}}\,{\rm{decrease}} = \frac{{{\rm{Initial}}\,{\rm{value}} – {\rm{Final}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)

Therefore, the ratio of the difference between the final value and the initial value( when the initial value is subtracted from the final value) to the initial value is known as percentage change for the decrement in percentage.

For example, a decrement in employment, a discount on the marked price of a quantity, a loss percentage of a quantity, etc.
Note: It can be considered as the negative increment of percentage.

Successive Percentage Change

The percentage change when two or more percentage changes are performed on a quantity repeatedly is called successive percentage change. Here, the ultimate change is not the simple addition of the two or more percentage changes.
There are mainly two types of percentage changes such,

1. Successive increment percentage change

2. Successive decrement percentage change

Successive Increment Percentage Change

The percentage change when two or more increased percentage changes are applied on a quantity repeatedly is called successive increment percentage change.

It means if the price of a quantity is increased by \({\rm{x \% }}\) and then \({\rm{z \% }}\), we have to apply the first percentage change that is \({\rm{x \% }}\) on the quantity, after that, we must apply the second percentage change that is also \({\rm{z \% }}\) on the result that we got from the first applied percentage change.

Let us say the price of the quantity is \(y\).
Now, the first percentage change (increment) \( = y + \left( {y \times \frac{x}{{100}}} \right) = y\left( {1 + \frac{x}{{100}}} \right) = A\)
Second percentage change (increment) \( = \left[ {y\left( {1 + \frac{x}{{100}}} \right) + y\left( {1 + \frac{x}{{100}}} \right) \times \frac{z}{{100}}} \right] = y\left( {1 + \frac{x}{{100}}} \right)\left( {1 + \frac{z}{{100}}} \right) = B\)
Hence, the final percentage change after two successive percentage increment in the price or the net percentage change \( = \frac{{B – y}}{y} \times 100\)
If the value of an object \(y\) is successively increased by \({\rm{x \% , z \% }}\) and then \({\rm{w \% }}\), then final value is \(y\left( {1 + \frac{x}{{100}}} \right)\left( {1 + \frac{z}{{100}}} \right)\left( {1 + \frac{w}{{100}}} \right)\).

Successive Decrement Percentage Change

The percentage change when two or more decreased percentage changes are applied on a quantity repeatedly is called successive decrement percentage change.

It means if the price of a quantity is decreased by \({\rm{x \% }}\) and then \({\rm{z \% }}\), we have to apply the first percentage change that is \({\rm{x \% }}\) on the quantity, after that, we must use the second percentage change that is also \({\rm{z \% }}\) on the result that we got from the first applied percentage change.
Let us say the price of the quantity is \(y\).

Now, the first percentage change (decrement) \( = y – \left( {y \times \frac{x}{{100}}} \right) = y\left( {1 – \frac{x}{{100}}} \right) = A\)

Second percentage change (decrement) \( = \left[ {y\left( {1 – \frac{x}{{100}}} \right) – y\left( {1 – \frac{x}{{100}}} \right) \times \frac{z}{{100}}} \right] = y\left( {1 – \frac{x}{{100}}} \right)\left( {1 – \frac{z}{{100}}} \right) = B\)

Hence, the final percentage change after two successive percentage increment in the price \( = \frac{{B – y}}{y} \times 100\).

If the value of an object \(y\) is successively decreased by \({\rm{x \% , z \% }}\) and then \({\rm{w \% }}\), then the final value is \(y\left( {1 – \frac{x}{{100}}} \right)\left( {1 – \frac{z}{{100}}} \right)\left( {1 – \frac{w}{{100}}} \right)\).

Other Successive Percentage Change

Both the increment and decrement of percentages can be applied to the initial value of an object successively, and the percentage changes can be used multiple times.

If the value of an object \(y\) is successively changed by \({\rm{x \% , z \% }}\) and then \({\rm{w \% }}\), then the final value is \(y\left( {1 \pm \frac{x}{{100}}} \right)\left( {1 \pm \frac{z}{{100}}} \right)\left( {1 \pm \frac{w}{{100}}} \right)\).

Here, the negative sign indicates the decrement, and the positive sign indicates the increment.

Solved Examples

Q.1. If \(60\) is increased to \(90\), find the percentage change.
Ans: Given, \(60\) is increased to \(90\),
The final value \({\rm{ = 90}}\)
The initial value \({\rm{ = 60}}\)
We know that, \({\rm{Percentage}}\,{\rm{increase}} = \frac{{{\rm{Initial}}\,{\rm{value}} – {\rm{Final}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)
Thus, the percentage change \( = \frac{{90 – 60}}{{60}} \times 100\% \)
\( = \frac{{30}}{{60}} \times 100\% \)
\( = \frac{1}{2} \times 100\% = 50\% \)
Hence, if \(60\) is increased to \(90\), the percentage change or the percentage increase is \({\rm{50 \% }}\).

Q.2. The price of a doll is decreased from \({\rm{₹ 25}}\) to \({\rm{₹ 15}}\) after a discount. Find the percentage of the discount.
Ans: Given that the price of a doll is decreased from \({\rm{₹ 25}}\) to \({\rm{₹ 15}}\) after a discount.
The final value \({\rm{ =₹ 15}}\)
The initial value \({\rm{ =₹ 25}}\)
We know that, \({\rm{Percentage}}\,{\rm{decrease}} = \frac{{{\rm{Initial}}\,{\rm{value}} – {\rm{Final}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)
Thus, the percentage change \( = \frac{{25 – 15}}{{25}} \times 100\% \)

\( = \frac{{10}}{{25}} \times 100\% \)

\( = \frac{2}{5} \times 100\% = 40\% \).

Hence, if the price of a doll decreases from \({\rm{₹ 25}}\) to \({\rm{₹ 15}}\) after a discount,the percentage change or the percentage decrease is \(40\% {\rm{ }}\).

Q.3. A number \(54\) is misread as \(45\). Find the percentage error.
Ans: Given that the number \(54\) is misread as \(45\).
The original value \({\rm{ = 54}}\)
The misread value \({\rm{ = 45}}\)
Thus, \({\rm{Percentage}}\,{\rm{error}} = \frac{{{\rm{Original}}\,{\rm{value}} – {\rm{Misread}}\,{\rm{value}}}}{{{\rm{Original}}\,{\rm{value}}}} \times 100\% \)
Thus, the percentage change/error \( = \frac{{54 – 45}}{{54}} \times 100\% \)

\( = \frac{9}{{54}} \times 100\% \)

\( = \frac{1}{6} \times 100\% = 16.67\% \) (approx).

Hence, the percentage error is \(16.67\% \) (approx).

Q.4. A number is increased by \(20\% \) and then the increased number is decreased by \(20\% \). Find the net increase or decrease.
Ans: Given that a number is increased by \(20\% \) and then the increased number is decreased by \(20\% \).

We know that if the value of an object \(y\) is successively changed by \({\rm{x \% , z \% }}\), then the final value is \(y\left( {1 \pm \frac{x}{{100}}} \right)\left( {1 \pm \frac{z}{{100}}} \right)\).
Here, the negative sign indicates the decrement, and the positive sign indicates the increment.
Let us say the original value of a number is \(y\).
The final value of the number \( = y\left( {1 + \frac{{20}}{{100}}} \right)\left( {1 – \frac{{20}}{{100}}} \right) = y \times \frac{{120}}{{100}} \times \frac{{80}}{{100}} = \frac{{96y}}{{100}}\)
Now, the percentage change \({\rm{Percentage}}\,{\rm{change}} = \frac{{{\rm{Initial}}\,{\rm{value}} – {\rm{Final}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)

\( = \frac{{\frac{{96y}}{{100}} – y}}{y} \times 100\% \)

\( = \frac{{\frac{{ – 4y}}{{100}}}}{y} \times 100\% \)

\({\rm{ = – 4 \% }}\)

Here, the negative sign indicated the decrement in price. Hence, the net decrease is \({\rm{4 \% }}\).

Q.5. The population of a city increased by \({\rm{10 \% }}\) in \(2019\) and again decreased by \({\rm{5 \% }}\) in \(2020\). Find the net change in the population before incrementing in \(2019\) as compared to after incrementing in \(2020\).
Ans: Given that the population is increased by \({\rm{10 \% }}\) and then it is increased by \({\rm{5 \% }}\).

We know that if the value of an object \(y\) is successively changed by \({\rm{x \% , z \% }}\), then the final value is \(y\left( {1 \pm \frac{x}{{100}}} \right)\left( {1 \pm \frac{z}{{100}}} \right)\).
Here, the negative sign indicates the decrement, and the positive sign indicates the increment.
Let us say the original population is \(y\).
The final population \( = y\left( {1 + \frac{{10}}{{100}}} \right)\left( {1 – \frac{5}{{100}}} \right)\)
\( = y \times \frac{{110}}{{100}} \times \frac{{95}}{{100}} = \frac{{209y}}{{200}}\)
Now, the percentage change
\({\rm{Percentage}}\,{\rm{change}} = \frac{{{\rm{Initial}}\,{\rm{value}} – {\rm{Final}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)

\( = \frac{{\frac{{209y}}{{200}} – y}}{{\rm{y}}} \times 100\% \)

\( = \frac{{\frac{{9y}}{{200}}}}{{\rm{y}}} \times 100\% = 4.5\% \).

Hence, the net increase is \({\rm{4}}{\rm{.5 \% }}\) in the population before increment in \(2019\) compared to after the increment in \(2020\).

Summary

In this article, we have discussed how to determine the percentage change. We covered the successive percentage change, increment, and decrement of percentage using formulas. We also studied some solved examples to better understand the concept.

Frequently Asked Questions (FAQs)

Q.1. What’s the percent change from \(2000\) to \(3000\)?
Ans: Since \(2000\) is increased to \(3000\),
The final value \({\rm{ = 3000}}\)
The initial value \({\rm{ = 2000}}\)
We know that, \({\rm{Percentage}}\,{\rm{increase}} = \frac{{{\rm{Final}}\,{\rm{value}} – {\rm{Initial}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)
Thus, the percentage change \( = \frac{{3000 – 2000}}{{2000}} \times 100\% = \frac{{1000}}{{2000}} \times 100\% \)

\( = \frac{1}{2} \times 100\% = 50\% \)

Hence, the percentage increase from \(2000\) to \(3000\) is \({\rm{50 \% }}\).

Q.2. Explain the percentage change with examples.
Ans: The percentage change of a quantity is the ratio of the difference in the amount to its initial value multiplied by \(100\).
The phrase percentage change says the difference in the percentage of the earlier and present quantity. So the percentage change produces the difference between quantity out of \(100\).
For example, if \(50\) is increased to \(75\), then,

\({\rm{Percentage}}\,{\rm{increase}} = \frac{{{\rm{Final}}\,{\rm{value}} – {\rm{Initial}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)

Thus, the percentage change \( = \frac{{75 – 50}}{{50}} \times 100\% \)

\( = \frac{{25}}{{50}} \times 100\% \)

\( \Rightarrow \frac{1}{2} \times 100\% = 50\% \)

Q.3. What is the percent of change from \(8000\) to \(10000\)?
Ans: Since \(8000\) is increased to \(10000\),
The final value \({\rm{ = 10000}}\)
The initial value \({\rm{ = 8000}}\)
We know that, \({\rm{Percentage}}\,{\rm{increase}} = \frac{{{\rm{Final}}\,{\rm{value}} – {\rm{Initial}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)
Thus, the percentage change \( = \frac{{10000 – 8000}}{{8000}} \times 100\% \)

\( = \frac{{2000}}{{8000}} \times 100\% \)

\( = \frac{1}{4} \times 100\% = 25\% \)

Hence, the percent change or the percentage increase from \(8000\) to \(10000\) is \(25\% {\rm{ }}\).

Q.4. What is the percent of change from \(74\) to \(35\)?
Ans: Since \(74\) is decreased to \(35\),
The final value \({\rm{ = 35}}\)
The initial value \({\rm{ = 74}}\)
We know that, \({\rm{Percentage}}\,{\rm{decrease}} = \frac{{{\rm{Initial}}\,{\rm{value}} – {\rm{Final}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)
Thus, the percentage change \( = \frac{{74 – 35}}{{74}} \times 100\% \)

\( = \frac{{15}}{{35}} \times 100\% \)

\( = \frac{3}{7} \times 100\% = 52.70\% \) (approx).

Q.5. What is the percent change of \(50\) to \(35\)?
Ans: Since \(50\) is decreased to \(35\),
The final value \({\rm{ = 35}}\)
The initial value \({\rm{ = 50}}\)
We know that, \({\rm{Percentage}}\,{\rm{decrease}} = \frac{{{\rm{Initial}}\,{\rm{value}} – {\rm{Final}}\,{\rm{value}}}}{{{\rm{Initial}}\,{\rm{value}}}} \times 100\% \)
Thus, the percentage change \( = \frac{{50 – 35}}{{50}} \times 100\% \)

\( = \frac{{15}}{{50}} \times 100\% \)

\( = \frac{3}{{10}} \times 100\% = 30\% \)

Hence, the percent change or the percentage decrease from \(50\) to \(35\) is \(30\% {\rm{ }}\).

Hence, the percent change or the percentage decrease from \(74\) to \(35\) is \({\rm{52}}{\rm{.70 \% }}\) (approx).

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