Percents of Numbers and Percent Equations Word Problems

Percents of Numbers and Percent Equations Word Problems: A Step-by-Step Guide

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Navigating percentages and numbers in word problems can be challenging, but once understood, it can become an interesting part of mathematics. Let's unravel the process step-by-step.

Step 1: Understand the Basics of Percents

Percents are a fraction or decimal of 100. It's essentially a ratio with 100 as the denominator. When a quantity is expressed as a percentage, it means "per hundred". For instance, 30% is 30 per 100, or $0.3$ as a decimal, or $\frac{3}{10}$ as a fraction.

Step 2: Learn to Translate Word Problems to Equations

Word problems involving percentages require you to translate the problem into an equation. The three components you will often see are the part, the whole, and the percent. The basic formula for percent problems is:

$$\frac{\text{Percent}}{100} = \frac{\text{Part}}{\text{Whole}}$$

Step 3: Identify the Knowns and Unknowns

The first step in solving word problems is to identify what you know and what you need to find out. Word problems often provide you with two of the three components (percent, part, or whole), and your task is to find the missing component.

Step 4: Solve for the Unknown

Once you have set up your equation, use algebraic manipulation to solve for the unknown. This can involve cross-multiplication and then solving the resulting equation.

Step 5: Check Your Answer

Always check your answers in word problems. Substitute your found value back into the original equation to ensure everything balances out.

A store is offering a 20% discount on a jacket that originally cost $100. How much will you save?

Identify the knowns and unknowns. The whole is $100, and the percent is 20. You need to find the part, which is the amount of discount.

Translate into equation:

$$\frac{20}{100} = \frac{\text{Part}}{100}$$

Solve for the part:

$$\text{Part} = \frac{20}{100} \times 100 = \$20$$

So, you will save $20.

By following these steps, you can successfully solve word problems involving percentages. It requires practice, but over time, solving these problems will become second nature.

Exercises

1) What is $30\%$ of $80$?

2) If $40$ is $20\%$ of what number, what is the number?

3) If $15\%$ of a number is $9$, what is the number?

4) A coat was reduced by $25\%$ in a sale. The coat now costs $$60$. What was its original price?

5) A student got $85\%$ of the questions correct in an exam. If she answered $255$ questions correctly, how many questions were there in the exam?

6) The population of a town increased by $8\%$ this year to a total of $27,000$. What was the population of the town last year?

7) A shirt is on sale for $40\%$ off and now costs $$36$. What was the original price of the shirt?

8) If a population decreases from $5000$ to $4500$, what is the percent decrease?

9) A car's value depreciates by $15\%$ each year. If the car is worth $$20,000$ now, how much will it be worth next year?

10) A bag of candy costs $$2.40$. If the price of the candy increased by $20\%$, how much does the candy cost now?