Problem 67 - Algebra A Combined Function

Question

When adding the expressions in \(\frac{3 x}{2}+\frac{x}{4},\) can you multiply each term by \(4 ?\) Why or why not?

Step-by-Step Solution

Answer

Yes, multiply by 4 to simplify addition. Result: \(7x\).

Step 1: Recognize Terms in Expression

The given expression is \(\frac{3x}{2} + \frac{x}{4}\). It consists of two fractional terms with different denominators: 2 and 4.

Step 2: Understanding Multiplication with a Common Denominator

When solving problems that involve fractions, it's helpful to have a common denominator to simplify or combine fractions. Multiplying each term by a multiple of the common denominator can help achieve this common base for addition.

Step 3: Multiply Each Term by 4

To eliminate the fractions, we can multiply each term by 4 (the least common denominator of 2 and 4). This operation will simplify the expression by removing the fractional parts. Multiply: \(4 \times \left( \frac{3x}{2} \right) = 4 \times \frac{3x}{2} = \frac{12x}{2} = 6x\) and \(4 \times \left( \frac{x}{4} \right) = 4 \times \frac{x}{4} = \frac{4x}{4} = x\). This simplifies our expression to \(6x + x\).

Step 4: Add the Simplified Expressions

Now add the terms: \(6x + x = 7x\). This gives us the result of adding these two expressions after applying multiplication by the common denominator.

Step 5: Conclusion

Yes, you can multiply each term by 4 because it helps in simplifying the addition of fractions into simple arithmetic addition of whole numbers.

Key Concepts

Common Denominator

When dealing with fractions, the common denominator is fundamental to simplifying or adding them. Fractions need a shared base to combine easily. For example, consider the expression \(\frac{3x}{2} + \frac{x}{4}\). These fractions have different denominators, 2 and 4. To make them compatible for addition, we must first find a common denominator.

• Identify the least common multiple (LCM) of the denominators.
• In this case, the LCM of 2 and 4 is 4.

By using 4 as our common denominator, each term in the expression can be adjusted to allow easy addition. By multiplying each fraction by the appropriate factor, we ensure that the denominators match, resulting in an expression we can work with seamlessly.

Fraction Multiplication

Fraction multiplication, especially when finding a common denominator, is about adjusting the value of fractions without changing the overall value of the expression. Let's see how it works with our expression \(\frac{3x}{2} + \frac{x}{4}\):

• Multiply each fraction by the common denominator, which here is 4.
• This step ensures every term has the same denominator, making them easier to add together.

For \(\frac{3x}{2}\), multiply both the numerator and the denominator by 2 (since 4 divided by 2 is 2), and for \(\frac{x}{4}\), you multiply by 1 (since the denominator is already 4). This multiplication does not change the original value of the fractions but puts them into a form where their denominators are the same. This process allows for simple addition or combination of these fractions.

Simplifying Expressions

After bringing the fractions to a common denominator through multiplication, you simplify the expression to complete the process. Here's how you can simplify the expression \(6x + x\):

• Add the numerators directly because the denominators are now common.
• Simplify any like terms in the resulting expression.

So, from \(6x + x\), add the coefficients of \(x\) (which are 6 and 1) to get \(7x\). This step is crucial because it reduces the expression to its simpler form, expressing the result of the original problem as a neat, singular statement without fractional parts. Simplification is about making expressions easier to understand and use in further calculations.