Solving Percentage Problems: Find the Number

This problem asks us to find an unknown number when we are given a specific percentage of that number. The question states that 80% of a certain number is equal to 100. We need to figure out what that original number is.

Let's represent the unknown number with the variable \(x\). The problem can be translated into a mathematical equation.

The phrase "80% of a number" means \(80\% \times x\). The word "is" typically means equals (\(= \)). So, the equation is:

\(80\% \times x = 100\)

To solve for \(x\), we first need to convert the percentage into a decimal or a fraction. Converting a percentage to a decimal is done by dividing the percentage by 100.

\(80\% = \frac{80}{100} = 0.80\)

Now substitute the decimal into the equation:

\(0.80 \times x = 100\)

To find \(x\), we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 0.80.

\(x = \frac{100}{0.80}\)

Now, let's perform the division:

\(x = 125\)

Alternatively, we could convert the percentage to a simplified fraction:

\(80\% = \frac{80}{100} = \frac{4}{5}\)

The equation becomes:

\(\frac{4}{5} \times x = 100\)

To solve for \(x\), multiply both sides by the reciprocal of \(\frac{4}{5}\), which is \(\frac{5}{4}\):

\(x = 100 \times \frac{5}{4}\)

\(x = \frac{100 \times 5}{4}\)

\(x = \frac{500}{4}\)

\(x = 125\)

Both methods give us the same result. The number is 125.

Verification

Let's check if 80% of 125 is indeed 100:

\(80\% \times 125 = 0.80 \times 125\)

\(0.80 \times 125 = 100\)

The calculation is correct. 80% of 125 is 100.

Summary of Calculation

Let the number be \(x\).

• Set up the equation: \(80\% \text{ of } x = 100\).
• Convert percentage to decimal: \(0.80 \times x = 100\).
• Solve for \(x\): \(x = \frac{100}{0.80}\).
• Calculate the result: \(x = 125\).

The number is 125.

Revision Table: Percentage Calculations

Additional Information: Understanding Percentages

Percentages are a fundamental concept in mathematics used widely in everyday life, such as in finance, statistics, and retail (discounts!). Understanding how to work with percentages, especially converting between percentages, decimals, and fractions, is crucial for solving problems like finding the original number when a percentage of it is known.

• Converting a percentage to a decimal: Divide by 100. Example: \(50\% = \frac{50}{100} = 0.50\).
• Converting a decimal to a percentage: Multiply by 100. Example: \(0.75 = 0.75 \times 100\% = 75\%\).
• Converting a percentage to a fraction: Write the percentage over 100 and simplify. Example: \(25\% = \frac{25}{100} = \frac{1}{4}\).
• Converting a fraction to a percentage: Convert the fraction to a decimal, then multiply by 100. Example: \(\frac{3}{4} = 0.75 = 0.75 \times 100\% = 75\%\).

In this problem, we used the conversion from percentage to decimal (\(80\% = 0.80\)) to solve the equation \(0.80x = 100\). This is a standard approach for solving such percentage problems.