Problem 112 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 40\left(\frac{3}{8} y-\frac{1}{4}\right)+40\left(\frac{4}{5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 15y + 22 \).

Step 1: Distribute the 40 inside the first parentheses

Start by distributing the 40 into the terms within the first set of parentheses: \( 40\left(\frac{3}{8} y\right) - 40\left(\frac{1}{4}\right) \). This involves multiplying 40 by both of these fractions separately.

Step 2: Calculate the Distribution

Perform the multiplication: \( 40 \times \frac{3}{8} y = 15y \) and \( 40 \times \frac{1}{4} = 10 \). Thus, the expression becomes \( 15y - 10 \).

Step 3: Distribute the 40 into the second expression

Multiply 40 by \( \frac{4}{5} \), which gives \( 40 \times \frac{4}{5} = 32 \).

Step 4: Add it all together

Combine the results of both distributions: \( 15y - 10 + 32 \).

Step 5: Simplify the expression

Combine the constant terms: \( -10 + 32 = 22 \). Therefore, the expression simplifies to \( 15y + 22 \).

Key Concepts

Distribution in Algebra

Distribution is an essential technique in algebra that helps us simplify and solve equations. It involves spreading or distributing a number across terms inside parentheses. In our example, we distribute the number 40 across the expressions inside the parentheses,

• First, we take the expression within the parentheses, \( \frac{3}{8}y - \frac{1}{4} \), and distribute 40 to each term separately.
• This means multiplying 40 by \( \frac{3}{8}y \) and again by \( -\frac{1}{4} \).

This step helps in simplifying the expression, making it easier to handle. The concept of distribution ensures that each term within the parentheses is appropriately scaled by the number outside.

Fraction Multiplication

Multiplying fractions might seem tricky at first, but it's a straightforward process once you understand the rules. When multiplying a fraction by a whole number, such as in our expression, certain steps can help simplify the process:

• Multiply the whole number by the numerator of the fraction.
• Then, divide the result by the denominator of the fraction.

For example, multiplying 40 by \( \frac{3}{8} \) involves:

• Calculate \( 40 \times 3 = 120 \).
• Divide 120 by 8 to get 15.

Apply the same principle for the other terms. This efficient process is easier and quicker, aiding the simplification of expressions where fractions are involved.

Combining Like Terms

Once all terms in an algebraic expression are simplified using distribution and multiplication, the next step is to combine like terms. Like terms are terms that contain the same variables raised to the same power. They make it possible to simplify the expression further:

• Identify terms that have the same variables, like \( 15y \).
• Similarly, look for constants, which are numbers without variables, like -10 and 32.

In our expression \( 15y - 10 + 32 \), the terms \( -10 \) and 32 are both constants. Adding \( -10 \) and 32 results in 22. Therefore, the expression becomes \( 15y + 22 \). This final step of combining like terms streamlines the expression into its simplest form, making problems easier to solve or further manipulate.