Problem 60 - Prealgebra

Question

Apply the distributive property, then simplify. $$2\left(\frac{3}{4} a-\frac{5}{6}\right)$$

Step-by-Step Solution

Answer

\( \frac{3}{2}a - \frac{5}{3} \)

Step 1: Distribute the Coefficient

Apply the distributive property by multiplying the coefficient \(2\) by each term inside the parentheses: \( \frac{3}{4}a \) and \( -\frac{5}{6} \).The expression becomes:\( 2 \times \frac{3}{4}a + 2 \times \left(-\frac{5}{6}\right) \).

Step 2: Perform the Multiplication

Calculate each multiplication step separately:1. \( 2 \times \frac{3}{4}a = \frac{6}{4}a = \frac{3}{2}a \) after simplifying the fraction.2. \( 2 \times \left(-\frac{5}{6}\right) = -\frac{10}{6} = -\frac{5}{3} \) after simplifying the fraction.

Step 3: Write the Simplified Expression

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

Key Concepts

Simplifying Fractions

Fraction simplification makes using fractions much easier. It occurs by finding the numerator and the denominator's greatest common divisor (GCD) and dividing them both by it. This process reduces the fraction to its simplest form.
For example, in the fraction \( \frac{6}{4} \), both numbers can be divided by 2. Hence, it simplifies to \( \frac{3}{2} \). Similarly, \( \frac{10}{6} \) simplifies to \( \frac{5}{3} \), as 2 is the GCD.

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

Once the fraction is simplified, it becomes easier to add, subtract, or multiply with other fractions.

Multiplication of Fractions

Multiplying fractions is quite straightforward. You need to multiply the numerators together and the denominators together. Then, simplify the resulting fraction if possible.
Take the operation \( 2 \times \frac{3}{4} \). Here it is as simple as multiplying the whole number by the numerator. So, \( 2 \times 3 = 6 \). The denominator remains the same, giving us \( \frac{6}{4} \). Just like this example, always check if you can simplify further. Here, \( \frac{6}{4} \) becomes \( \frac{3}{2} \), post simplification.

• Multiply the numerators together: numerator \(1 \times \) numerator \(2\).
• Multiply the denominators together: denominator \(1 \times \) denominator \(2\).
• Simplify the resulting fraction if needed.

For fractions with whole numbers, convert the whole number by giving it a denominator of 1, and then proceed with the above steps.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They do not have an equal sign like equations do. Understanding these expressions is key when applying the distributive property.
For instance, in the expression \( 2 \left( \frac{3}{4} a - \frac{5}{6} \right) \), the operation inside the parentheses involves subtracting fractions, and outside, we distribute the coefficient 2. By using the distributive property, we expand and simplify the expression.

• Terms: parts of the expression separated by \( + \) or \( - \) signs.
• Coefficients: numbers multiplied by variables (e.g., 3 in \( 3a \)).
• Distributive Property: multiplies a term outside the parentheses by each term inside (e.g., \( 2 \times \frac{3}{4}a \)).

Breaking down algebraic expressions into smaller parts helps to simplify complex problems. Remember to always combine like terms to get your final expression simplified.