Problem 101 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( 6y \).

Step 1: Distribute the Constant

To simplify the expression, distribute the constant value 8 over the fraction \( \frac{3}{4} y \). This means multiplying 8 with \( \frac{3}{4} \) and \( y \).

Step 2: Simplify the Fraction Multiplication

Multiply the constant 8 by the fraction \( \frac{3}{4} \). Use the property of fractions: \( a \times \frac{b}{c} = \frac{a \times b}{c} \). This gives \( \frac{8 \times 3}{4} y \).

Step 3: Perform the Multiplication

Calculate \( 8 \times 3 = 24 \). So now the expression becomes \( \frac{24}{4} y \).

Step 4: Simplify the Fraction

Divide 24 by 4 to simplify the fraction. This gives \( 6 \), making the expression \( 6y \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that makes simplifying expressions easier. It states that for any numbers or variables, the expression \( a(b + c) \) can be expanded to \( ab + ac \). This property allows us to multiply a single term by each term inside a parenthesis individually.

• In our original problem, \( 8(\frac{3}{4} y) \), we use the distributive property to multiply 8 by both \( \frac{3}{4} \) and \( y \).


• This results in the expression: \( 8 \times \frac{3}{4} \times y \).

By applying the distributive property, we ensure that each term is correctly multiplied, paving the way for further simplification.

Fraction Multiplication

Multiplying fractions is straightforward when you follow the rules! When you encounter the task of multiplying a fraction by a whole number, as in our expression \( 8 \times \frac{3}{4} y \), you treat 8 as \( \frac{8}{1} \):

• Multiply the numerators together: \( 8 \times 3 = 24 \)


• Multiply the denominators together: \( 1 \times 4 = 4 \)

Thus, \( 8 \times \frac{3}{4} \) simplifies to \( \frac{24}{4} \), which is then attached to the variable \( y \) resulting in \( \frac{24}{4} y \).
Remember, the goal is to simplify by performing multiplication at the numerator level, then manage the fractions as needed.

Simplifying Expressions

Simplifying expressions is about making them easier to read and solve. From our fraction \( \frac{24}{4} y \), we focus on reducing it by performing the division. This results in the simplest form.

• Divide 24 by 4, which equals 6.


• The variable \( y \) remains, giving us the final simplified expression \( 6y \).

Remember, every time you simplify, you aim to achieve the most reduced form of the expression, thereby making calculations straightforward in future problems. Keep the rules of arithmetic—especially division—a priority in simplifying fractions to streamline your solution.