How to Solve Percentage Word Problems Step by Step
Percentage is basically the ratio of the value of a particular quantity to its total value multiplied by 100. It is generally a way of expressing something as per 100. Further, in this article we will look at various scenarios for which we will find the percentage and learn the steps to solve word problems based on percentage.
Percentage
What is the Percentage?
A percentage is something expressed part per hundred. It is generally a ratio which is given in terms of a fraction of 100. The word percent is taken from the Latin term per centrum which means “by the hundred” . It is denoted by the symbol “\[\% \]”.
What is Percentage
How to Calculate the Percentage?
Percentage is generally calculated by expressing the value whose percentage is to be calculated and the total value as a ratio which is multiplied by 100. The formula for calculating percentage is as follows :
Percentage \[ = \frac{{value}}{{Total\,Value}} \times 100\]
If in case, percentage of a number is to be calculated i.e. if % is given and the value of the quantity whose percentage is given with respect to the total is to be found we use -
For instance,\[{\rm{x\% }}\] of 450 \[ = \]Y and we wish to calculate the value of Y, we can do so by
\[{\rm{Y}} = \frac{{\rm{X}}}{{100}} \times 450\]
E.g. 20% of 800
So, \[\frac{{20}}{{100}} \times 800 = 20 \times 8 = 160\]
Question and Answer based on Percentage
This section contains basic problems based on the concept of percentage which can be solved very easily.
Raju scored 81 out of 90 in mathematics. Convert his marks into percentages.
Ans. Percentage \[ = \] Value \[/\]Total Value \[ \times \]100 \[ = \frac{{81}}{{90}} \times 100 = 9 \times 10 = 90\% \]
In a class of 300 students, there are 75 girls. Calculate the percentage of boys in the class.
Ans. Number of boys \[ = 300 - 75 = 225\]
Percentage of boys \[ = \frac{{225}}{{300}} \times 100 = 75\% \]
Prince spent 75 \[\% \] of the money he had to buy groceries. If he had 4000$ with him initially, what amount did he spend ?
Ans. Amount Spend \[ = \frac{{75}}{{100}} \times 4000 = 3000\]$
Compute 65% of 680
Ans. 65% of 680 \[ = \frac{{65}}{{100}} \times 680 = 442\]
Word Problems based on Percentage
The word problem based on percentage will have some scenarios in which we have to understand the requirements in the given problems and accordingly apply the formula for percentage and find the value of the quantity asked to be found.
1.If a brass article contains 72% of copper. What amount of brass will be required to get 360g of copper?
Ans: Let the quantity of brass required be x g.
Therefore, 72% of x = 360g
\[72\% \, \times \,x = 360g\]
\[0.72 \times x = 360g\]
Therefore, \[x = \frac{{360}}{{0.72}} = 500g\]
So, 500 g of brass is required to get 360g of copper.
2.Karishma appeared for a quiz in which she got 25 answers correct and 15 answers incorrect. What is the percentage of questions that she appeared correctly ?
Ans : Here, in this case 25 answers were correct and 15 were incorrect.
Therefore, total questions \[ = 25 + 15 = 40\]
So, Percentage of correct questions \[ = \frac{{25}}{{40}} \times 100 = 62.5\% \]
Therefore, 62.5\[\% \] of the questions were answered correctly by Karishma.
Solved Examples :
1. What is 30\[\% \] of 450?
Solution : 30 % of 450 \[ = \frac{{30}}{{100}} \times 450 = 0.3 \times 450 = 135\]
Therefore, 30% of 450 is 135.
2. The cost price of a bag is 1500 and the selling price of the same bag is 2100. At what profit is the article sold? What is the profit percentage?
Solution : As , Profit = Selling price - Cost price
Therefore, Profit earned on selling the bag \[ = 2100 - 1500 = 600\]
So, Profit Percentage \[ = \frac{{600}}{{1500}} \times 100 = 40\% \]
The seller earned 40\[\% \] profit on selling the bag.
3. Arun sells an object to Benny at a profit of 15%, Benny sells that object to Chandan for ₹1012 and makes a profit of 10%. At what cost did Arun purchase the object?
Solution: Let the actual cost price at which Arun bought the object be x
When Arun sells the object to Benny
Profit % = 15%
∴ selling price of object \[= \frac{{100 + 15}}{{100}} \times x = 1.15x\]
Now, this cost price of the object for Benny
When Benny sells the object to Chandan
Selling Price = ₹1012
Profit % = 10%
∴ Selling price = \[ = \frac{{100 + 10}}{{100}} \times 1.15x\]
\[ \Rightarrow 1012 = \frac{{100 + 10}}{{100}} \times 1.15x\]
\[ \Rightarrow x = \frac{{1012 \times 1000}}{{11 \times 115}}\]
= ₹800
Therefore, the price at which Arun bought the object is ₹800.
4. In an inter school aptitude test, 200 students appeared. Out of these students; 20 % got A grade, 50 % got B grade and the remaining got C grade. Assuming that no student got a D grade; find the number of students who got a C grade .
Solution : The number of students with A grade \[ = \] 20\[\% \] of 200
\[ = \frac{{20}}{{100}} \times 200 = \frac{{4000}}{{100}}\]
\[ = 40\]
And, the number of students with B grade \[ = \] 50 \[\% \] of 200
\[ = \frac{{50}}{{100}} \times 200\]
\[ = \frac{{10000}}{{100}} = 100\]
Therefore, the number of students who got C grade \[ = 200 - [40 + 100]\]
\[ = 200 - 140 = 60\]
So, 60 students scored C grade.
Conclusion:
Thus, we can say that percentage is a fraction of something expressed as part of 100. So, after understanding the method of calculating percentage and applying it according to the scenario we can easily solve the word problems based on percentage.
FAQs on Percentage Word Problems Made Simple
1. What is the first step to solve any percentage word problem?
The first and most crucial step is to carefully read the problem and identify three key values: the base (the total or original amount), the part (the portion or subset of the base), and the percentage rate (the value with the % symbol). Understanding what you need to find—the part, the base, or the rate—will determine the correct formula to use.
2. How are percentages converted into a usable format for solving word problems?
To perform calculations like multiplication or division in a word problem, a percentage must first be converted into either a decimal or a fraction. To convert to a decimal, divide the percentage by 100 (e.g., 40% becomes 0.40). To convert to a fraction, place the percentage over 100 and simplify (e.g., 40% becomes 40/100, which simplifies to 2/5).
3. What are some common examples of percentage word problems found in real life?
Percentage word problems are frequently used to model real-world scenarios. Some common examples you will encounter include:
Calculating discounts on an item's original price.
Figuring out sales tax or VAT on a purchase.
Determining the interest earned or paid on a loan or investment.
Finding a student's score on a test as a percentage.
Calculating a tip at a restaurant.
4. What is the key difference between finding a 'percentage of a number' and 'percentage increase/decrease'?
The difference lies in the final calculation. Finding '25% of 80' is a simple multiplication: 0.25 × 80 = 20. However, a '25% increase on 80' requires two steps: first find the percentage of the number (0.25 × 80 = 20), and then add this value to the original number (80 + 20 = 100). Similarly, a decrease involves subtracting the value.
5. If an item's price is increased by 10% and then decreased by 10%, why isn't the final price the same as the original?
This is a common misconception. The final price is not the same because the base amount for the percentage calculation changes. For example, if an item costs ₹100, a 10% increase makes it ₹110. The subsequent 10% decrease is calculated on the new price of ₹110 (10% of 110 is ₹11), not the original ₹100. So, the final price becomes ₹110 - ₹11 = ₹99, which is lower than the original price.
6. How do you find the original amount if you are given the final amount after a percentage change?
To find the original amount (the base), you can use a simple algebraic formula. Let the original amount be 'x'. For a percentage increase, the formula is x * (1 + percentage as decimal) = Final Amount. For a decrease, it is x * (1 - percentage as decimal) = Final Amount. For instance, if an item is ₹90 after a 10% discount, the equation is x * (1 - 0.10) = 90, which means 0.90x = 90, so x = ₹100.
7. Why is correctly identifying the 'base' the most critical step in solving percentage word problems?
Identifying the 'base' (the 'whole' or 100% value) is critical because all percentages in the problem are relative to it. A mistake in identifying the base will lead to an entirely wrong answer. For example, in profit and loss problems, the base is almost always the Cost Price (CP), not the selling price. Calculating profit percentage based on the selling price is a common error that occurs from misidentifying the base.
