Problem 19 - Algebra for College Students

Question

Translate each verbal sentence into an equation, using \(x\) as the variable. Then solve the equation. $$ \text { When } \frac{2}{3} \text { of a number is subtracted from } 12, \text { the result is } 10 . \text { Find the number. } $$

Step-by-Step Solution

Answer

The number is 3.

Step 1: Define the variable

Let the variable be defined as the number described in the problem. Let's denote this number by \(x\).

Step 2: Set up the equation

Translate the verbal sentence into a mathematical equation. The sentence says 'When \(\frac{2}{3}\) of a number is subtracted from 12, the result is 10.' This can be written as: 12 - \(\frac{2}{3}x\) = 10.

Step 3: Isolate the variable term

To solve the equation, first isolate the term with the variable on one side. Subtract 12 from both sides: 12 - \(\frac{2}{3}x\) - 12 = 10 - 12 -\(\frac{2}{3}x\) = -2.

Step 4: Solve for the variable

Next, multiply both sides by -1 to simplify the equation: \(\frac{2}{3}x\) = 2. Then, to solve for \(x\), multiply both sides by the reciprocal of \(\frac{2}{3}\): x = 2 \times \frac{3}{2} x = 3.

Key Concepts

Translating Verbal Sentences into Equations

Understanding how to translate verbal sentences into mathematical equations is essential for solving word problems. Read the problem carefully to identify the key elements. In this example, the problem states: 'When \( \frac{2}{3} \) of a number is subtracted from 12, the result is 10. Find the number.'

Here is how you can break it down:

• '\( \frac{2}{3} \) of a number' translates to \( \frac{2}{3} x \), where \( x \) is the unknown number.
• 'is subtracted from 12' means you take 12 and subtract \( \frac{2}{3} x \) from it.
• 'the result is 10' simply means the equation equals 10.

Putting it together, we get: 12 - \( \frac{2}{3} x \) = 10. This equation is a direct translation of the verbal sentence into a mathematical equation.

Isolating Variable Terms

Isolating the variable term is a crucial step in solving equations. This means getting the term with the variable \( x \) alone on one side of the equation. Let's look at our equation from before: 12 - \( \frac{2}{3} x \) = 10.

First, we need to get rid of the constant on the same side as the variable term. We do this by performing the inverse operation:

• Subtract 12 from both sides: 12 - \( \frac{2}{3} x \) - 12 = 10 - 12
• This simplifies to: -\( \frac{2}{3} x \) = -2.

Now we have isolated the variable term on one side.

Arithmetic Operations with Fractions

Handling fractions in equations just involves a few extra steps. You need to manage both the arithmetic and the fractions properly. With our isolated term -\( \frac{2}{3} x \) = -2, we can start solving for \( x \).

First, multiply both sides by -1 to get a positive variable term:

• \( \frac{2}{3} x \) = 2.

To isolate \( x \), multiply both sides by the reciprocal of \( \frac{2}{3} \):

• The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
• So, multiply both sides by \( \frac{3}{2} \): \( x = 2 \times \frac{3}{2} \).

This simplifies to:

• \( x = 3 \).

Thus, the original number is 3.