Problem 77 - Elementary Algebra

Question

In the following exercises, translate and solve. 81 is 75% of what number?

Step-by-Step Solution

Answer

108

Step 1 - Set up the equation

We are given that 81 is 75% of a certain number. Let the unknown number be represented by Let the unknown number be represented by \( x \). The statement 81 is 75% of what number can be written as an equation: \[ 81 = 0.75x \].

Step 2 - Solve for x

To find the value of \( x \), divide both sides of the equation by 0.75:\[ x = \frac{81}{0.75} \].

Step 3 - Perform the division

Now, perform the division:\[ x = \frac{81}{0.75} = 108 \].

Key Concepts

Translating Verbal Statements to Equations

Understanding how to translate verbal statements into equations is crucial for solving percentage problems. Let's break down the process. Typically, words like 'is,' 'of,' and 'what number' give clues on how to form the equation. For example, '81 is 75% of what number?' can be interpreted as:

• '81' is the result or part of the equation.
• 'is' translates to '=' in mathematical terms.
• '75%' translates to '0.75' (since percentages are out of 100, you convert them to decimals by dividing by 100).
• 'of what number?' indicates that we are trying to find a missing value, often represented by a variable like 'x'.

Combining all these clues, the verbal statement '81 is 75% of what number?' becomes the equation: \[81 = 0.75x\]. This step is important for setting up the equation correctly, so make sure to interpret each part of the statement accurately.

Solving Linear Equations

After translating the verbal problem into the equation \[81 = 0.75x\], the next step is solving for the unknown variable 'x'. This process typically involves isolating the variable on one side of the equation. Here's a step-by-step approach:

• Recognize that you need to solve for the variable 'x.'
• Identify that '0.75x' means '0.75' times 'x.'
• To isolate ‘x,’ you need to perform the opposite operation. Since '0.75' is multiplying 'x,' you'll divide both sides of the equation by '0.75.'
• This will leave ‘x’ alone on one side of the equation: \[x = \frac{81}{0.75}\]

Dividing both sides by `0.75` simplifies the equation, which then allows you to find the value of 'x'. Finally, perform the division step to solve for 'x'.

Basic Percentage Calculations

Basic percentage calculations are essential for addressing problems involving proportions and parts of a whole. Here are the key concepts:

• A percentage represents a fraction out of 100. When you see '75%', think of it as \[\frac{75}{100} = 0.75\].
• To convert a percentage to a decimal, divide by 100. For example, '75%' becomes '0.75.'
• Percentage of a number can be found by multiplying the percentage (as a decimal) by that number. For instance, if you need to find 75% of 'x,' you calculate '0.75 * x.'

Applying these steps, you can handle any percentage problem easily. In our exercise, given '81 is 75% of what number?', recognizing how to convert and operate with percentages is crucial. You set up the equation \[81 = 0.75x\] and solve accordingly.