Problem 59 - Prealgebra

Question

Apply the distributive property, then simplify. $$3\left(\frac{5}{6} a+\frac{4}{9}\right)$$

Step-by-Step Solution

Answer

\(\frac{5}{2}a + \frac{4}{3}\).

Step 1: Apply the distributive property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). In this case, \(a = 3\), \(b = \frac{5}{6}a\), and \(c = \frac{4}{9}\). We distribute \(3\) across the terms inside the parentheses: 3\left(\frac{5}{6}a\right) + 3\left(\frac{4}{9}\right).

Step 2: Multiply each term

First, calculate \(3\left(\frac{5}{6}a\right)\):\(3 \times \frac{5}{6}\) gives \(\frac{15}{6}a\). Then, calculate \(3 \times \frac{4}{9}\):\(3 \times \frac{4}{9}\) gives \(\frac{12}{9}\).

Step 3: Simplify each term

Simplify the coefficient \(\frac{15}{6}\) by dividing the numerator and the denominator by their greatest common divisor, which is 3:\(\frac{15}{6}a = \frac{5}{2}a\).Similarly, simplify \(\frac{12}{9}\) by dividing by 3:\(\frac{12}{9} = \frac{4}{3}\).

Step 4: Combine the terms

After simplifying, our expression becomes:\(\frac{5}{2}a + \frac{4}{3}\). This is the simplified form of the original expression.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in math that makes fractions easier to understand and work with. But what does simplifying really mean? It means reducing a fraction to its simplest form where the numerator and the denominator have no common factors other than 1.
Let's break that down a little more clearly. A fraction like \(\frac{12}{9}\) can be simplified by finding the greatest common divisor (GCD) of the numerator (12) and the denominator (9). In this case, the GCD is 3.

• Divide the numerator by 3: \(12 \div 3 = 4\)
• Divide the denominator by 3: \(9 \div 3 = 3\)

Therefore, \(\frac{12}{9}\) simplifies to \(\frac{4}{3}\). It's that simple! By simplifying fractions, you ensure the fraction is in its simplest form, which makes it easier to add, subtract, multiply, or divide them in equations.

Multiplication of Fractions

Multiplying fractions is straightforward once you know the trick! Unlike addition or subtraction, you don't need a common denominator when multiplying.
Here's the straightforward process:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.

For example, if you multiply \(3\) by \(\frac{5}{6}\), you multiply 3 by 5 for the numerator to get 15, and keep the existing denominator of 6. So you have \(\frac{15}{6}\). This is the result of multiplying, but always remember to simplify if possible.
Multiplication is not limited to fractions alone. If you come across people talking about distributing a number across terms, it essentially involves multiplication, but it just looks a little different when you apply it.

Algebraic Expressions

Algebraic expressions represent numbers using symbols and variables, like \(a\), to communicate unknown or variable amounts. They're like recipes where you find out not only what ingredients, but how much you need.
When dealing with expressions such as \(3(\frac{5}{6}a + \frac{4}{9})\), we use the distributive property to take a number outside of a parenthesis and multiply it with each term inside the parenthesis.

• You multiply \(3\) with \(\frac{5}{6}a\), leading to \(\frac{15}{6}a\).
• Then, multiply \(3\) with \(\frac{4}{9}\), giving \(\frac{12}{9}\).

The task doesn't end there. Simplifying each term is key!
Both understanding and applying operations like these help foster a deeper comprehension of algebra, making equations easier to solve and understand. This is the core power of algebraic expressions in mathematics.