A percentage is a fraction or a ratio in which the value of the whole is always 100. The word percentage is originated from the Latin word “Per centum”, which describes the value equals per hundred. We can say that percentages are nothing but fractions, with their denominator always equal to a hundred. Generally, they are represented by the symbol \(\%.\)A word problem on percentage consists of a few sentences describing a real-life scenario where a mathematical calculation of percentage must use to solve a problem.

We have various applications of the percentage. We have to follow some methods or formulas to solve different problems related to percentages in mathematics. In this article, we will discuss how to solve word problems on percentage.

Definition of Percentage

In mathematics, a percentage is a number that can be written in the form of a fraction, with the denominator equal to a cent value (hundred). The percentage defines the part per cent value (Hundred). The word percentage originated from the Latin word “Per centum”, which describes the value equals per hundred.
Generally, the percentage is represented by the symbol \(\%.\) The percentage is said to be a dimensionless number as it has no units. In general percentage of a number can be expressed in the fractional form or decimal form.
Example: \(\frac{2}{3}\% ,\,0.3\% ,\,75\% \) etc.

Definition of Word Problem on Percentage

As we know, percentage defines the part per cent value (Hundred). We have various problems associated with the percentage in real life. A word problem on percentage consists of a few sentences describing a real-life scenario where a mathematical calculation of percentage must be used to solve a problem.

Word problem on percentage tells the applications of percentage in our daily life. The various types of word problems on percentage are listed below:

1. Word problem on a percentage of a number
2. Word problems on percentage increase
3. Word problems on percentage decrease
4. Word problems on parts of percentage
5. Word problems on the conversion of percentages to fractions, ratio, decimals and vice-versa.
6. Word problems on percentage profit and loss
7. Word problems on the percentage of marks
8. Word problems on percentage errors

Word Problems on a Percentage of a Number

We can find the percentage of a number by dividing the given number by the whole and multiplying it by a hundred.
Percentage \( = \frac{{{\text{actual}\;\text{number}}}}{{{\text{total}\;\text{number}}}} \times 100\)

Example:
Calculate the percentage of marks of Keerthi in Maths. She got \(99\) marks out of \(100.\)
The percentage received by Keerthi in Maths is given by
\(\frac{{99}}{{100}} \times 100 = 99{\%}\)

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Word Problems in Percentage Increase

The problems related to the increase in the country’s population, increase in the number of species, increase in the commodity etc., are the problems related to percentage increase.
Percentage increase means the percentage change in the given value when it is increased for a given period of time, and it can be calculated by using the formula:
\({\mathbf{Percentage}}\;{\mathbf{Increase}} = \frac{{{\mathbf{increased}}\;{\mathbf{value}} – {\mathbf{original}}\;{\mathbf{value}}}}{{{\mathbf{original}}\;{\mathbf{value}}}} \times 100\)

Example:
The population of the town in \(2000\) is \(1,00,000\) and in the year \(2010\) is \(1,50,000\). Find the percentage increase in the population.
The percenatge increase in population \( = \frac{{1,50,000 – 1,00,000}}{{1,00,000}} \times 100 = 50\%\)

Word Problems in Percentage Decrease

The problems related to a decrease in the number of patients in the hospital, a decrease in the level of rainfall, etc., are related to percentage decrease.
Percentage decrease means the percentage change in the given value when it is decreased for a given period of time, and it can be calculated by using the formula:
\({\mathbf{Percentage}}\;{\mathbf{decrease}} = \frac {{{\mathbf{original}}\;{\mathbf{value}}} – {{{\mathbf{decreased}}\;{\mathbf{value}}}}}{{\mathbf{original}}\;{\mathbf{value}}} \times 100\)

Example:
The rainfall is decreased in a city from \(200\,\rm{cm}\) to \(150\,\rm{cm}\). Find the percentage decrease in the rainfall.
Percentage decrease in rainfall \( = \frac{{200 – 150}}{{200}} \times 100 = 25\;\% \)

Word Problem on Percentage

The \(x\%\) of \(y\) or \(y\%\) of \(x\) can be written as \(\frac{{xy}}{{10}}.\)

Example:
Keerthi paid \(25\%\) of her income \(Rs. 20,000\) to insurance. Find the percentage of the amount she paid to insurance.
Percentage of income paid to insurance \( = 25\%\) of \(20,000 = 25 \times \frac{{20,000}}{{100}} = {\text{Rs}}.\,5000\)

Word Problems on Profit and Loss Percentage

Profits (and losses) are generally calculated in the form of profit per cent, and it tells how much profit or loss each business/individual gets.
Profit \(= {\rm{selling}}\,{\rm{price}} – {\rm{cost}}\,{\rm{price}}.\)
Profit percentage is given by \({\% }\,{\text{profit = }}\frac{{{\text{profit}}}}{{{\text{cost}}\,{\text{price}}}} \times 100\)
As in the case of loss, the selling price is less than that of the cost price.
Loss \(= {\rm{cost}}\,{\rm{price}} – {\rm{selling}}\,{\rm{price}}.\)
Loss percentage is given by \({\% }\,{\text{loss = }}\frac{{{\text{loss}}}}{{{\text{cost}}\,{\text{price}}}} \times 100\)

Example:
Keerthi sold her old T.V at the cost of \(Rs. 10,000\) to the person, which she bought for \(Rs. 15,000.\) Find the loss or percentage?
Here, S.P. \( = Rs. 10,000\) and C.P. \( = Rs. 15,000\).
As \(\rm{S.P} > \rm{C.P}\) Keerthi got a profit, and the percentage loss is given by
\(\% \,{\text{loss}} = \frac{{15000 – 10000}}{{15000}} \times 100 = 33.3\% \)

Word Problems on Percentage of Discount

We know that discount is the price reduced on the marked price of an item. It is equal to the difference between the marked price and the selling price.
Discount percentage \( = \frac{{{\text{discount}}}}{{{\text{marked}\;\text{price}}}} \times 100\)

Example:
Keerthi sold an item to a person at a discount of \(Rs. 5\), which is marked at \(Rs. 20\). Find the discount percentage?
The discount percentage \( = \frac{5}{{20}} \times 100 = 25\% \)

Word Problems on Calculation of Marks Percentage

To find the percentage of marks secured by a student in an examination, we have to divide the total marks of the student (in all subjects) by the maximum marks and multiply it by \(100\).
\({\text{Percentage}}\,{\text{marks}} = \frac{{{\text{marks}}\,{\text{obtained}}\,{\text{in}}\,{\text{all}}\,{\text{subjects}}}}{{{\text{maximum}}\,{\text{marks}}\,{\text{in}}\,{\text{all}}\,{\text{subjects}}}} \times 100\)

Example:
Keerthi has got \(95\) out of \(100\) in Maths, \(85\) out of \(100\) in Physics, and \(75\) out of \(100\) in Chemistry. Find the overall percentage.
The total marks secured by the Keerthi \( = \left( {95 + 85 + 75} \right) = 255\)
Maximum marks is \(\left( {100 + 100 + 100} \right) = 300.\)
Therefore, the percentage of marks obtained by the Keerthi is
\(\left( {\frac{{255}}{{300}}} \right) \times 100\% = 85\% .\)

Word Problems on Percentage Errors

The percentage errors are used to know the calibration or manufacturing errors in the measuring instruments. Percentage error is the difference between the approximate value and actual value.
\\({\rm{Percentage}}\,{\rm{error}} = \left\{ {\frac{{{\rm{approximate}}\,{\rm{or}}\,{\rm{observed}}\,{\rm{value}} – {\rm{exact}}\,{\rm{value}}}}{{{\rm{exact}}\,{\rm{or}}\,{\rm{actual}}\,{\rm{value}}}}} \right\} \times 100\)

Example:
Keerthi measures the temperature of the room with an instrument. She observed the reading was \(22.35^\circ \rm{C}\), but the actual reading was \(22.25^\circ \rm{C}\). Find the percentage error.
The percentage error \( = \frac{{22.35^\circ {\text{C}} – 22.25^\circ {\text{C}}}}{{22.25^\circ {\text{C}}}} \times 100 = 0.45\% \) (Approx)

Solved Examples – Word Problem on Percentage

Q.1. A class contains a total of \(50\) students. On a particular day, only \(14\%\) of the students are absent from the class. Find the number of students present in the class on that particular day.
Ans: Given the total number of students in a class \(=50\) and \(14\%\) of students are absent from the class.
So, the number of students absent for the class \(14\%\) of \(50 = 14 \times \frac{{50}}{{100}} = 7\)
Students present in the class \(=\) total students \(-\) number of students absent \( = 50 – 7 =43\)
Hence, there are \(43\) students present in the class.

Q.2. Keerthi has got \(99\) out of \(100\) in Aptitude, \(98\) out of \(100\) in general knowledge and \(100\) out of \(100\) in reasoning in a Public exam. Find the percentage of marks secured by the Keerthi in the Public exam.
Ans: Total marks obtained by Keerthi \(98 + 99 + 100 = 297\)
Maximum marks in the exam \(100 + 100 + 100 = 300\)
The percentage of marks obtained by the Keerthi is given by the ratio of total marks obtained to the maximum marks of the exam, and that is multiplied by \(100.\)
\( = \% \;{\text{marks}} = \frac{{297}}{{300}} \times 100 = 99\% \)
Hence, the percentage of marks obtained by the Keerthi is \(99\% .\)

Q.3. In a plot, only \(4500\,\rm{sq.m}.\) is allowed for construction out of \(6000\,\rm{sq.m}.\) What is per cent of the plot to remain without construction?
Ans: The total area of the plot \( = 6000\,\rm{sq.m}.\)
The area allowed for the construction \( = 4500\,\rm{sq.m}.\)
Thus, the area not allowed for the construction \( = 6000\,\rm{sq.m} – 4500\,\rm{sq.m} = 1500\,\rm{sq.m}.\)
The percentage of the area of plot not allowed for the construction \( = \frac{{1500}}{{6000}} \times 100 = 25\% \)
Therefore, \(25\%\) of the plot is not required for the construction.

Q.4. In a class, \(40\%\) of girls are there, in a total of \(50\) students. Find the number of girls and boys in the class.
Ans: Given the total number of students \(= 50\)
The percentage of girls in a class is \(40\%\)
So, the number of girls in the class \( = 40\%\) of \(50 = 40 \times \frac{{50}}{{100}} = 20.\)
The number of boys in the class is the difference of total students and the number of girls \( = 50 – 20 = 30\)
Hence, the total number of boys and girls in the class are \(30,\;20\) respectively.

Q.5. According to sources, in a given year, it snowed \(13\) days. What is the percentage of days that year during which it snowed? (Assume non-leap year)
Ans: We need to know that there is a total of \(365\) days in a year (assuming that it is a non-leap year). There are given \(13\) snowed days.
The required percentage is given by \(\frac{{13}}{{365}} \times 100 = 3.5\% \) (approximately)

Summary

In this article, we have studied the definitions of percentage and the formulas of percentage. This article gives the word problems on percentages in various cases like percentage error, percentage increase, percentage decrease, profit or loss percentage, percentage of marks, etc.
In this article, we have discussed the word problems on percentage with mathematical solutions that help us understand the concept and solve them easily.

Frequently Asked Questions (FAQs)- Word Problem on Percentage

Q.1. What is a word problem on percentage?
Ans: A word problem on percentage consists of a few sentences describing a real-life scenario where a mathematical calculation of percentage must be used to solve a problem.

Q.2. How do you solve the word problem on percentage?
Ans: Word problem on percentage can be solved by using the percentage formula: \(\frac{{{\text{actual}\;\text{number}}}}{{{\text{total}\;\text{number}}}} \times 100.\)

Q.3. How do you find the word problem on percentage errors?
Ans: Word problem on percentage error can be calculated by using:
\({\rm{Percentage}}\,{\rm{error}} = \left\{ {\frac{{{\rm{approximate}}\,{\rm{or}}\,{\rm{observed}}\,{\rm{value}} – {\rm{exact}}\,{\rm{value}}}}{{{\rm{exact}}\,{\rm{or}}\,{\rm{actual}}\,{\rm{value}}}}} \right\} \times 100\)

Q.4. How do you solve the word problem on percentage change?
Ans: The word problem on percentage change can be done by using the formula:
\({\text{percentage}}\,{\text{change}} = \left( {\frac{{{\text{new}}\,{\text{value}} – {\text{old}}\,{\text{value}}}}{{{\text{old}}\,{\text{value}}}}} \right) \times 100\)

Q.5. What is the percentage?
Ans: The percentage defines the part per cent value (Hundred).

Learn How To Calculate Percentage Here

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