Calculating Percentages: A Step-by-Step Guide
Understanding the Concept
Calculating percentages is an essential mathematical skill used in many areas of life. Whether you’re budgeting your finances, comparing statistical data, or figuring out tax rates, breaking down fractions into percentages allows problems to be communicated and understood more concisely. At its core, calculating a percentage involves determining a number’s value relative to 100 or a whole. Percentage problems involve three key pieces of information - the part, the whole, and the percentage rate. The part refers to the specific quantity being measured, such as an amount of money. The whole is the total value that the part is being measured against. Lastly, the percentage rate states what fraction or proportion the part represents out of the whole, which is conventionally expressed out of 100. By understanding these fundamental components and how they relate, percentages become intuitive to work with.
Calculating Percentages Step-by-Step
There are two main calculation methods for determining percentages - finding a percentage of a whole number, and finding a whole number that represents a given percentage. Let’s break down each approach step-by-step:
# Finding a Percentage of a Whole Number
- Identify the whole number and percentage rate
- Change the percentage to a decimal or fraction (e.g. 25% = 0.25 or 1/4)
- Multiply the whole number by the decimal/fraction equivalent of the percentage
For example, to calculate 35% of 80:
1) Whole number is 80, percentage rate is 35%
2) 35% as a decimal is 0.35
3) 80 x 0.35 = 28
# Finding a Whole Number for a Given Percentage
- Identify the part and percentage rate
- Change the percentage to a decimal or fraction
- Divide the part by the decimal/fraction equivalent of the percentage
- Multiply the result by 100
For example, to find the whole if 20 is 80%:
1) Part is 20, percentage rate is 80%
2) 80% as a decimal is 0.8 3) 20 / 0.8 = 25 4) 25 x 100 = 2500 Therefore, the whole is 2500.Applying Percentage Calculations
Being comfortable with these step-by-step methods allows percentages to be applied effectively across many real-world scenarios. Some common examples include calculating sales tax on a purchase, determining employee bonuses and pay raises, analyzing population growth rates, and more. With practice, percentage problems can be solved fluently in the mind for quick estimates and checks. Understanding the relationship between parts, wholes, and rates is key.
Calculating Using Unitary Method
Another approach for calculating percentages is using the unitary method. This strategy is especially helpful when the numbers involved are larger or more complex fractions are present. Let’s revisit the example problem from the introduction using the unitary method step-by-step:
Calculating a Percentage Increase
1) We are given: Out of 25 questions, 20 were answered correctly 2) Represent this ratio as a fraction: 20/25 3) Reduce the fraction to its simplest form: 4/5 4) Now if the total number of questions increases to 35, keep the same ratio of correct answers to total 5) Write the ratio as a fraction of the new total: x/35 6) Set the two fractions equal since they represent the same ratio: 4/5 = x/35 7) Cross multiply: 4 * 35 = 5x 8) Solve for x: 140 = 5x; x = 28 Therefore, using the unitary method, if the total questions increase to 35 keeping the same ratio, the number of correct answers would be 28.
Conclusion
Being comfortable with multiple calculation methods gives flexibility in solving percentage problems. The key takeaways are understanding the relationship between parts, wholes and rates, and practicing conversions between percentages, decimals and fractions. With regular use, percentage calculations can become intuitive for a wide range of real-life applications.
