First Principles Thinking for Entrepreneurs

Percentages Made Simple

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Mental Percentages: A Study Guide

Quiz

Instructions: Answer the following questions in 2-3 sentences each.

Explain the literal meaning of the term “percent” and how it relates to calculating percentages.
Describe the “divide by 100” trick for calculating percentages mentally. Why does this trick work?
How can dividing by 10 twice simplify percentage calculations? Provide an example.
When calculating percentages mentally, how can you leverage your knowledge of 50% and 10%?
Explain how the word “of” in percentage problems relates to mathematical operations.
Why is it incorrect to simply multiply a percentage by a number without considering the percent sign?
How can understanding fractions be helpful in solving percentage problems?
Describe the steps involved in calculating a percentage using the fraction form of the percent.
Explain the alternative method for calculating percentages mentally using decimals. How is it different from using fractions?
Can percentages be greater than 100%? Provide an example and explain what a percentage over 100% represents.

Quiz Answer Key

The term “percent” means “out of one hundred.” It represents a fraction where the denominator is 100. To calculate percentages, we express the given percentage as a fraction out of 100 and then multiply it by the number.
The “divide by 100” trick involves dividing both the percentage and the number by 100 before multiplying. This works because dividing both terms of a multiplication problem by the same number doesn’t change the result.
Dividing by 10 twice is equivalent to dividing by 100. This can simplify calculations by making the numbers smaller and easier to work with. For example, to find 25% of 80, we can divide 80 by 10 twice to get 0.8 and then multiply by 25.
Knowing that 50% is half and 10% is one-tenth of a number can be very useful. For example, to find 60% of 42, we can calculate 50% of 42 (half of 42 is 21) and then add 10% of 42 (4.2).
The word “of” in percentage problems indicates multiplication. For instance, “25% of 20” means 25/100 multiplied by 20.
A percentage represents a fraction out of 100. Multiplying directly without considering the percent sign would lead to an incorrect result as it doesn’t account for the division by 100.
Fractions and percentages are closely related. Understanding how to work with fractions can make it easier to convert percentages to fractions and perform calculations.
To calculate a percentage using fractions: (1) convert the percentage to a fraction by putting it over 100, (2) multiply this fraction by the given number, and (3) simplify the resulting fraction if necessary.
The alternative method uses the decimal form of the percentage. This involves moving the decimal point two places to the left and then multiplying the resulting decimal by the given number. This method often proves simpler when using a calculator.
Yes, percentages can be greater than 100%. For example, 120% of a number represents a value greater than the original number. It indicates an increase of 20% compared to the original.

Essay Questions

Explain the different methods of calculating percentages, including the “divide by 100” trick, the fractional method, and the decimal method. Discuss the advantages and disadvantages of each method.
Discuss the real-world applications of percentages, providing specific examples of how percentages are used in everyday life.
Explain how understanding the relationship between percentages, fractions, and decimals is essential for solving percentage problems effectively.
Analyze the common errors students make when solving percentage problems and suggest strategies for avoiding these mistakes.
Explore the concept of percentages exceeding 100%. Explain how these percentages are calculated and provide examples of scenarios where percentages greater than 100% might be used.

Glossary

Percent: A fraction or ratio with a denominator of 100, represented by the symbol “%”.

Percentage: A portion or share of a whole, expressed as a percent.

Fraction: A numerical quantity that represents a part of a whole, expressed as a numerator over a denominator.

Decimal: A number system using a base of 10, where digits represent different powers of 10 and are separated by a decimal point.

Of: In percentage problems, the word “of” indicates multiplication.

Simplify: To reduce a fraction or expression to its lowest terms or simplest form.

Improper fraction: A fraction where the numerator is greater than or equal to the denominator.

Mental Math: Performing calculations in one’s head without the aid of external tools like calculators or paper and pencil.