Finding the Original Number: A Step-by-Step Algebra Solution

Let's break down this word problem to find the original number. We are given a sequence of operations performed on an unknown number, and we know the final result. We can represent the unknown number with a variable and set up an equation to solve for it.

Defining the Unknown Number

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

Translating the Problem into an Equation

We follow the steps described in the question and apply them to \(x\):

1. 7 is added to a certain number: \(x + 7\)
2. The sum is multiplied by 5: \(5 \times (x + 7)\)
3. The product is then divided by 3: \(\frac{5 \times (x + 7)}{3}\)
4. 4 is subtracted from the quotient: \(\frac{5 \times (x + 7)}{3} - 4\)
5. The result comes to 16: \(\frac{5 \times (x + 7)}{3} - 4 = 16\)

So, the equation we need to solve is:

\(\frac{5(x+7)}{3} - 4 = 16\)

Solving the Equation to Find the Original Number

Now, let's solve the equation for \(x\):

Start with the equation:

\(\frac{5(x+7)}{3} - 4 = 16\)

Step 1: Add 4 to both sides of the equation to isolate the term with \(x\).

\(\frac{5(x+7)}{3} - 4 + 4 = 16 + 4\)

\(\frac{5(x+7)}{3} = 20\)

Step 2: Multiply both sides by 3 to eliminate the denominator.

\(\frac{5(x+7)}{3} \times 3 = 20 \times 3\)

\(5(x+7) = 60\)

Step 3: Divide both sides by 5 to isolate the term \((x+7)\).

\(\frac{5(x+7)}{5} = \frac{60}{5}\)

\(x+7 = 12\)

Step 4: Subtract 7 from both sides to solve for \(x\).

\(x + 7 - 7 = 12 - 7\)

\(x = 5\)

Verifying the Solution

Let's check if the original number \(x = 5\) gives the result 16 when the sequence of operations is applied:

1. Add 7: \(5 + 7 = 12\)
2. Multiply by 5: \(12 \times 5 = 60\)
3. Divide by 3: \(\frac{60}{3} = 20\)
4. Subtract 4: \(20 - 4 = 16\)

The final result is 16, which matches the problem statement. Therefore, the original number is indeed 5.

The original number is 5.

Revision Table: Key Steps in Solving Word Problems

Additional Information: Linear Equations

The equation we solved, \(\frac{5(x+7)}{3} - 4 = 16\), is an example of a linear equation in one variable. A linear equation is an equation where the highest power of the variable is 1. These equations can often be solved by isolating the variable using inverse operations (addition/subtraction, multiplication/division) on both sides of the equation, keeping the equation balanced.

• Inverse of adding is subtracting.
• Inverse of subtracting is adding.
• Inverse of multiplying is dividing.
• Inverse of dividing is multiplying.

In our solution, we used inverse operations like adding 4 (inverse of subtracting 4), multiplying by 3 (inverse of dividing by 3), dividing by 5 (inverse of multiplying by 5), and subtracting 7 (inverse of adding 7) to find the value of \(x\).