Problem 22 - Prealgebra

Question

Solve each of the following problems. 90 is \(80 \%\) of what number?

Step-by-Step Solution

Answer

90 is 80% of 112.5.

Step 1: Understanding the Question

We are given that 90 is 80% of a certain number, and we need to find out what that number is. This is typically expressed in terms of percentages, where 80% of a number is equal to 90.

Step 2: Setting the Equation

To solve this, we write the equation that represents the problem. Since 80% of a number is written as \(0.8 \times x\) and we know it equals 90, the equation is: \(0.8 \times x = 90\).

Step 3: Solving for the Number

We need to solve the equation \(0.8x = 90\) for \(x\). To do this, divide both sides of the equation by 0.8: \[x = \frac{90}{0.8}\]

Step 4: Calculating the Value

Perform the division to find the value of \(x\): \[x = 112.5\] Thus, the number we are looking for is 112.5.

Key Concepts

Equation Solving

Solving equations is like cracking a puzzle. Everything depends on finding the unknown number, which we often call "x." In the problem "90 is 80% of what number," we're trying to uncover that hidden number which forms 80% of another number. To do this, we set up an equation, a mathematical sentence that says both sides are equal.
When we're given "80% of a number is 90," we write it mathematically as \(0.8 \times x = 90\). This equation tells us that 80% of some number \(x\) equals 90. To "solve" this equation, we rearrange it to make \(x\) stand alone on one side and everything else on the other. This is the essence of equation solving. It's all about performing operations to isolate the variable and find its value.

• Consider the simplest way to isolate \(x\) on one side.
• Manipulate the equation by performing the same mathematical operations on both sides.
• Verify your solution by substituting the value back into the original equation.

Proportion Problems

Proportion problems involve understanding how two ratios or fractions relate to one another. When we talk about percentages, we are essentially dealing with a type of proportion. In the equation \(0.8 \times x = 90\), the 0.8 represents 80%, and shows a proportion of the whole number we're trying to find. To solve proportion problems, we use cross-multiplication or straightforward algebra techniques. Here, the main task is identifying how part of the whole relates to the entire value. By setting up the proportion correctly, solving the problem becomes simpler and more systematic.

• Identify what part (percentage) relates to the whole.
• Set up an equation to represent this relationship.
• Solve for the unknown to find the whole by rearranging the proportion.

Understanding and solving proportion problems is key to mastering percentages, allowing you to work out what a certain percentage of a number represents.

Division in Algebra

Division in algebra isn't just about splitting numbers; it's a fundamental operation used to solve equations. When we have the equation \(0.8 \times x = 90\), and we need to find \(x\), division helps us isolate \(x\). By dividing both sides by 0.8, we make \(x\) the subject of the formula.

This step is crucial because it simplifies our equation and reveals the unknown number. Performing the division \(x = \frac{90}{0.8}\) gives us \(x = 112.5\). This shows that 112.5 is the number of which 90 is 80%. The principles of division in algebra emphasize careful handling of operations to maintain equality on both sides of the equation. Always remember, whatever you do to one side, you must do to the other.

• Use division to isolate the variable.
• Ensure operations are consistently applied to both sides of the equation.
• Check your results by plugging values back into the original equation.

Division in algebra is a powerful tool for unraveling the mysteries of equations and solving for variables.