Question

If 8 is subtracted from 4 times a number the answer is 40. What is the number?

Short answer

The number is 12.

Step-by-step solution

Define the variable

Let the unknown number be represented by the variable \(x\).

Set up the equation

According to the problem, 4 times the number minus 8 equals 40. This translates to the equation: \(4x - 8 = 40\).

Isolate the variable term

Add 8 to both sides of the equation to isolate the term with the variable. \(4x - 8 + 8 = 40 + 8\) simplifies to \(4x = 48\).

Solve for the variable

Divide both sides by 4 to solve for \(x\): \( x = \frac{48}{4} = 12 \).

Check the solution

Substitute \(x = 12\) back into the original equation to verify: \(4 \times 12 - 8 = 48 - 8 = 40\). This confirms that the solution is correct.

Key concepts

algebra

Algebra is a branch of mathematics focused on finding unknown values referred to as variables. In algebra, we form equations using known numbers, variables, and arithmetic operations. The equation can then be solved to find the value of the unknown variable. In this exercise, we're given a situation involving an unknown number where we form an equation to solve for it. This approach is commonly used to solve practical problems involving unknown quantities.

variable isolation

Variable isolation is a key part of solving algebraic equations. It involves manipulating the equation in such a way that the variable or unknown is on one side of the equation, and all constants are on the other side.
For example, in step 3 of the solution, we have the equation \(4x - 8 = 40\). To isolate the variable term, we add 8 to both sides: \(4x - 8 + 8 = 40 + 8\), simplifying it to \(4x = 48\).
After isolating the variable term, the unknown value can be easily solved (as shown in step 4 of the solution). This is a critical step in solving linear equations.

equation verification

Equation verification ensures that the solution obtained is correct. This involves substituting the found value back into the original equation and checking if both sides are equal.
In our exercise, we found that \(x = 12\). To verify this solution, we substitute it back into the original equation: \(4 \times 12 - 8 = 40\). When simplified, we get \(48 - 8 = 40\), which is true.
Thus, verifying the solution confirms its correctness and assures us that we have solved the problem accurately.

basic arithmetic operations

Basic arithmetic operations are foundational in solving algebraic equations. These include addition, subtraction, multiplication, and division.
In this specific problem, we used:

• Addition when we added 8 to both sides of the equation.
• Multiplication in conceptualizing the problem as involving '4 times a number.'
• Division when we divided both sides by 4 to solve for the variable.

Understanding and applying these basic operations correctly is essential for solving algebraic equations and many other math problems.