Express One Quantity as Percentage of Other: The word per cent is an abbreviation of the Latin phrase per centum, which means per hundred or hundredths. Thus, the term per cent means per hundred or for every hundred. When we say that a man gives \(20\) per cent of his income as a tax, he pays \(₹20\) out of every hundred rupees of his income as tax. The symbol \(\% \) is used for the term per cent.

This article will discuss details about percentage, per cent as a ratio and vice versa, conversion of per cent into fractions and vice versa, conversion of per cent into decimal and vice versa, and express one quantity as a percentage of other and solve some example problems.

What is Per Cent?

A per cent is a fraction expressed with \(100\) as the denominator. An abbreviation of the Latin word percentum which means per hundred or hundredths, is known as a per cent. When we say that a man spends \(30\) per cent of his income on education bills, he gives \(₹30\) out of every hundred rupees of his income. 

A trader makes a profit of \(18\) per cent means that he gains \(₹18\) on every hundred rupees of his investment. A boy who scored \(80\) per cent marks in his final examination means that he obtained \(80\) marks out of every hundred marks. The term per cent is sometimes abbreviated as p.c. The symbol \(\% \) is often used for the term per cent. Thus, \(15\) per cent will be written as \(15%.\)

Per Cent as a Fraction and Ratio

A fraction with its denominator \(100\) is equal to that per cent as the numerator.

Example: \(\frac{{65}}{{100}} = 65\,{\rm{hundredths = 65\% }}\)
A per cent can also be expressed as a ratio with its second term \(100\) and first term equal to the given per cent. We know that, \(12\% = \frac{{12}}{{100}} = 12:100\)

Conversion of a Per Cent into a Fraction

To convert the per cent into a fraction, we will follow the below steps:

1. Obtain the given per cent. Let it be \(y\% \)
2. Drop the per cent sign and divide the numerator by \(100. \Rightarrow y\% = \frac{y}{{100}}\)
   Example: \(36\% = \frac{{36}}{{100}} = \frac{9}{{25}}\)

Conversion of Fraction into a Per Cent

To convert the fraction into a per cent, we will follow the below steps:

1. Obtain the fraction. Let it be \(\frac{x}{y}\)
2. Multiply the fraction by \(100\) and put the per cent sign to obtain the required percentage. Therefore, \(\frac{x}{y} = \left( {\frac{x}{y} \times 100} \right)\% \)
   Example: \(\frac{9}{{20}} = \left( {\frac{9}{{20}} \times 100} \right)\% = 45\% \)

Conversion of Ratio into Percent

To convert the ratio into a per cent, we will follow the below steps.

1. Obtain the ratio, say \(x:y\)
2. Convert the given ratio into a fraction \(\frac{x}{y}\)
3. Multiply the fraction by 100 and put the per cent sign to obtain the required percentage. Therefore, \(\frac{x}{y} = \left( {\frac{x}{y} \times 100} \right)\% \)

Example: \(30:80 = \frac{{30}}{{80}} = \left( {\frac{{30}}{{80}} \times 100} \right)\% = 37.5\% \)

Conversion of Per Cent into Ratio

To convert the per cent into a ratio, we will follow the below steps.

1. Obtain the per cent.
2. Convert the given per cent into a fraction by dividing it by \(100\) and removing per cent sign.
3. Express the fraction obtained in the above step in the simplest form.
4. Express the fraction obtained in the third step as a ratio.

Example: \(0.4\% = \frac{{0.4}}{{100}} = \frac{4}{{1000}} = \frac{1}{{250}} = 1:250\)

Conversion of Per Cent into Decimal

To convert the per cent into a decimal, we will follow the below steps:

1. Obtain the given per cent.
2. Express the given per cent as a fraction with a denominator equal to \(100.\)
3. Write the fraction obtained in the above step in decimal form.

Example: \(18\% = \frac{{18}}{{100}} = 0.18\)

Conversion of Decimal into Per Cent

To convert the decimal into a per cent, we will follow the below steps:

1. Obtain the given decimal.
2. Convert the given decimal into a fraction by removing the decimal part. To remove the decimal point, divide the given decimal by \(10\) or \(100\) or \(1000\) according to the number of digits on the right side of the decimal point.
3. Multiply by \(100\) and put \(\% \) sign.

Example: \(0.004 = \frac{4}{{1000}} = \left( {\frac{4}{{1000}} \times 100} \right)\% = 0.4\% \)

Percentage of a Given Number

In the above section, we have discussed that the per cent is a fraction with a denominator of \(100.\) So, we can find the exact value of a per cent is only when we know the value of the number is part of. Below are the steps to find the per cent of a given number.

1. Obtain the number, say \(y\).
2. Obtain the required per cent, say \(P\% \)
3. Multiply \(y\) by \(P\) and divide by \(100\) to obtain the \(P\% \) of \(y \Rightarrow P\% \) of \(y = \frac{P}{{100}} \times y\)
   Example: \({\rm{16\% }}\) of \(25\,{\rm{litres}} = \left( {\frac{{16}}{{100}} \times 25} \right){\rm{litres}} = 4\,{\rm{litres}}\)

Expressing one Quantity as Percentage of Other

Sometimes we are given two quantities, and we want to find what per cent of one quantity is of the other. In other words, we want to find how many hundredths of one quantity should be taken to equal the second quantity. In such types of problems, we proceed as discussed below:

Let \(x\) and \(y\) be two numbers, and we want to know what per cent of \(x\) is \(y\)?
Let \(a\% \) of \(x\) be equal to \(y.\) Then, \(\frac{a}{{100}} \times x = y \Rightarrow a = y \times \frac{{100}}{x} \Rightarrow a = \frac{y}{x} \times 100\)
Therefore, \(y\) is \(\left( {\frac{y}{x} \times 100} \right)\% \) of \(x.\)

Solved Examples: Express One Quantity as Percentage of Other

Q.1. What per cent of \({\rm{25}}\,{\rm{kg}}\) is \({\rm{3}}{\rm{.5}}\,{\rm{kg}}\)?
Ans: Required per cent\( = \left( {\frac{{3.5\;{\rm{kg}}}}{{25\;{\rm{kg}}}} \times 100} \right)\% = \frac{{3.5 \times 100}}{{25}}\% = \frac{{35 \times 100}}{{250}}\% = \frac{{35 \times 5}}{2}\% = 7 \times 2\% = 14\% \)
Hence, \({\rm{3}}{\rm{.5}}\,{\rm{kg}}\) is \(14\% \) of \({\rm{25}}\,{\rm{kg}}.\)

Q.2. What per cent of \(750\) meters is \(125\) meters?
Ans: Required per cent\( = \frac{{125\,{\rm{ meters }}}}{{750\,{\rm{ meters }}}} \times 100\% = \frac{{125 \times 100}}{{750}}\% = \frac{{1 \times 100}}{6}\% = \frac{{50}}{3}\% = 16.66\% \)
Therefore, \(125\) meters is \(16.66\% \) of \(750\) meters.

Q.3. Express \(75\) paise as a per cent of \(₹5.\)
Ans: We know that \(₹5=500\) paise
Let \(x\% \) of \(₹5\) be \(75\) paise.
Then, \(x\% \) of \(500\) paise\(=75\) paise
\( \Rightarrow \frac{x}{{100}} \times 500 = 75\)
\( \Rightarrow x = \frac{{75 \times 100}}{{500}} = 15\)
Therefore, \(15\% \) of \(₹5\) is \(75\) paise.

Q.4. Express \(18\) hours as a per cent of \(3\) days.
Ans: We know that \(1\) day\(=24\) hours
So, \(3\) days\(=(24×3)\) hours\(=72\) hours
Let \(x\% \) of \(3\) days be \(18\) hours.
Then, \(x\% \) of \(72\) hours\(=18\) hours
\( \Rightarrow \frac{x}{{100}} \times 72 = 18\)
\( \Rightarrow x = \frac{{18 \times 100}}{{72}}\)
\( \Rightarrow x = 25\)
Therefore, \(25\% \) of \(3\) days is equal to \(18\) hours

Q.5. Find: (a) \(10\% \) more than \(₹90\) (b) \(20\% \) less than \(₹70\)
Ans: (a) \(10\% \) more than \( ₹ 90 = ₹ \left( {\frac{{10}}{{100}} \times 90} \right) = ₹ 9\)
Therefore, \(10\% \) more than \(₹90=₹90+₹9=₹99\)
(b) \(20\% \) less than \( ₹ 70 = ₹ \left( {\frac{{20}}{{100}} \times 70} \right) = ₹ 14\)
Therefore, \(20\% \) less than \(₹70=₹(70-14)=₹56\)

Summary

In this article, we have studied the meaning of per cent and its definition. Also, we have studied the conversion of a per cent as a ratio and vice versa, conversion of per cent into fractions and vice versa, conversion of per cent into decimal and vice versa, and express one quantity as a percentage of other and solved some example problems.

Frequently Asked Questions (FAQs)

Frequently asked questions related to percentage is listed as follows

Q.1. How do you express one quantity as a percentage of another?
Ans: Let us consider the below example to express one quantity as a percentage of another.
Let \(x\) and \(y\) be two numbers, and we want to know what per cent of \(x\) is \(y\)?
Let \(a\% \) of \(x\) be equal to \(y.\) Then, \(\frac{a}{{100}} \times x = y \Rightarrow a = y \times \frac{{100}}{x} \Rightarrow a = \frac{y}{x} \times 100\)
Therefore, \(y\) is \(\left( {\frac{y}{x} \times 100} \right)\% \) of \(x.\)

Q.2. Can we express the second quantity as a percentage of the first quantity?
Ans: To express one quantity as a percentage of another, verify that both quantities are expressed within the same units. Write the given quantity as a fraction of the entire and multiply it by \(100\% .\) Then simplify.

Q.3. How to convert ratio into per cent and per cent into ratio?
Ans: To convert the ratio into a per cent, we will follow the below steps.
Obtain the ratio, say \(x:y\)
Convert the given ratio into a fraction \(\frac{x}{y}\)
Multiply the fraction by \(100\) and put the per cent sign to obtain the required percentage. Therefore, \(\frac{x}{y}\left( {\frac{y}{x} \times 100} \right)\% \)
To convert the per cent into a ratio, we will follow the below steps.
Obtain the per cent.
Convert the given per cent into a fraction by dividing it by \(100\) and removing per cent sign.
Express the fraction obtained in the above step in the simplest form.
Express the fraction obtained in the third step as a ratio.

Q.4. How to convert the fraction into per cent and per cent into a fraction?
Ans: To convert the fraction into a per cent, we will follow the below steps:
Obtain the fraction. Let it be \(\frac{x}{y}\)
Multiply the fraction by \(100\) and put the per cent sign to obtain the required percentage. Therefore, \(\frac{x}{y} = \left( {\frac{x}{y} \times 100} \right)\% \)
To convert the per cent into a fraction, we will follow the below steps:
Obtain the given per cent. Let it be \(y\% \)
Drop the per cent sign and divide the numerator by \(100. \Rightarrow y\% = \frac{y}{{100}}\)
Simplify the fraction if required

Q.5. How to convert a decimal to per cent and the per cent to decimal?
Ans: To convert the per cent into a decimal, we will follow the below steps:
1. Obtain the given per cent.
2. Express the given per cent as a fraction with a denominator equal to \(100.\)
3. Write the fraction obtained in the above step in decimal form.
To convert the decimal into a per cent, we will follow the below steps:
1. Obtain the given decimal.
2. Convert the given decimal into a fraction by removing the decimal part. To remove the decimal point, divide the given decimal by \(10\) or \(100\) or \(1000\) according to the number of digits on the right side of the decimal point.
3. Multiply by \(100\) and put \(\% \) sign.

We hope this detailed article on how to express one quantity as a percentage of other helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!