Problem 62 - Prealgebra

Question

Apply the distributive property, then simplify. $$-4\left(\frac{5}{12} y+\frac{3}{8}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{5}{3}y - \frac{3}{2}\).

Step 1: Understand the Distributive Property

The distributive property states that for any numbers \( a, b, \) and \( c \), \( a(b+c) = ab + ac \). In this problem, \( a = -4 \), \( b = \frac{5}{12}y \), and \( c = \frac{3}{8} \). We will distribute \( -4 \) to both terms inside the parentheses.

Step 2: Distribute -4 to Each Term

Apply the distributive property: multiply \(-4\) by each term inside the parentheses: 1. \(-4 \times \frac{5}{12}y = -\frac{20}{12}y\) 2. \(-4 \times \frac{3}{8} = -\frac{12}{8}\).

Step 3: Simplify the Resulting Expression

Simplify each fraction from the distribution: 1. For \(-\frac{20}{12}y\), simplify by dividing the numerator and the denominator by 4: \(-\frac{20}{12}y = -\frac{5}{3}y\)2. For \(-\frac{12}{8}\), simplify by dividing the numerator and the denominator by 4: \(-\frac{12}{8} = -\frac{3}{2}\).

Step 4: Combine the Results

Combine the two simplified terms from Step 3 to form the simplified expression: \(-\frac{5}{3}y - \frac{3}{2}\).

Key Concepts

Fraction Simplification

Simplifying fractions is about making them as simple as possible. This process helps to make working with algebraic expressions easier. You do this by finding the greatest common divisor (GCD) of both the numerator and the denominator. Then you divide them both by this number.

For example, take the fraction \(-\frac{20}{12}\). The GCD of 20 and 12 is 4. So, you divide the numerator and the denominator by 4 to get \(-\frac{5}{3}\).

Similarly with \(-\frac{12}{8}\). The GCD of 12 and 8 is 4 too. So, by dividing both the numerator and the denominator by 4, you simplify it to \(-\frac{3}{2}\). Simplifying fractions like this makes complex algebraic expressions more manageable.

Algebraic Expressions

Algebraic expressions include numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. They are expressions of numbers and symbols used to represent a certain quantity.

In our exercise, the expression \(-4\left(\frac{5}{12} y+\frac{3}{8}\right)\) involves fractions, variables, and a negative coefficient. Applying the distributive property allows you to multiply each term inside the brackets by \(-4\). More specifically:

• First multiply \(-4\) by \(\frac{5}{12}y\) to get \(-\frac{20}{12}y\).
• Then multiply \(-4\) by \(\frac{3}{8}\) to get \(-\frac{12}{8}\).

This process expands the expression into separate terms, which can then be simplified further. When dealing with algebraic expressions, always make sure to accurately apply operations and simplify fractions when possible, for a cleaner result.

Negative Numbers

Understanding negative numbers is crucial, especially in algebra, as they often appear as coefficients or constants. A negative number is any number less than zero. When you multiply or distribute negative numbers, they follow specific rules:

• Multiplying a positive number by a negative number results in a negative number.
• Multiplying two negative numbers results in a positive number.

For the problem \(-4(\frac{5}{12} y + \frac{3}{8})\), \(-4\) is multiplied by each term inside the parentheses, converting them into negative values:

• \(-4 \times \frac{5}{12}y = -\frac{20}{12}y\)
• \(-4 \times \frac{3}{8} = -\frac{12}{8}\)

It's important to keep the sign in mind to ensure your final expression accurately reflects all negative interactions.