Problem 76 - Elementary Algebra

Question

In the following exercises, translate and solve. 36 is 25% of what number?

Step-by-Step Solution

Answer

The number is 144.

Step 1: Understand the problem

We need to find a number such that 25% of this number is equal to 36.

Step 2: Set up the equation

Let the unknown number be represented by \( x \). The equation based on the problem statement is: \( 0.25x = 36 \).

Step 3: Solve for \( x \)

To isolate \( x \), divide both sides of the equation by 0.25: \( x = \frac{36}{0.25} \).

Step 4: Calculate the result

Perform the division: \( x = 144 \).

Key Concepts

Setting Up Equations

Setting up equations is a crucial step in algebra, especially for problems dealing with percentages. In the given problem, you're asked to find out what number 36 is 25% of. To start, we need to represent the unknown number as a variable, let’s say \( x \). Next, we turn the verbal statement into a mathematical equation. Since 25% of \( x \) equals 36, we convert 25% into its decimal form which is 0.25. This transforms the problem statement into the equation: \( 0.25x = 36 \). This makes setting up equations essential because it simplifies complex verbal problems into solvable mathematical expressions.

Solving for Unknowns

Once the equation \( 0.25x = 36 \) is set up, the next step is solving for the unknown variable \( x \). Solving for an unknown means isolating the variable on one side of the equation. To do this, you'll need to perform operations that reverse the operations affecting \( x \). In our case, \( x \) is multiplied by 0.25. To isolate \( x \), you'll need to divide both sides of the equation by 0.25. This gives us: \( x = \frac{36}{0.25} \). Isolating the variable simplifies the equation and makes it easier to find the value of \( x \).

Division in Algebra

In algebra, division is used frequently to isolate variables and solve equations. For the problem \( 0.25x = 36 \), we need to divide both sides by 0.25 to isolate \( x \). This division process is straightforward: \( x = \frac{36}{0.25} \). To perform this division, you can either use a calculator or do it manually. Manual division involves multiplying both the numerator and the denominator by 100 to remove the decimal: \( x = \frac{36 \times 100}{0.25 \times 100} = \frac{3600}{25} \), which simplifies to \( x = 144 \). Division helps to simplify and solve equations, making it an indispensable tool in algebra.