Problem 74 - Elementary Algebra

Question

\(12\left(\frac{3}{4} x+\frac{4}{15}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \[ 9x + \frac{16}{5} \].

Step 1 - Distribute the 12

Multiply each term inside the parentheses by 12. This involves distributing the 12 to both \( \frac{3}{4} x \) and \( \frac{4}{15} \).

Step 2 - Simplify the First Term

Calculate \( 12 \times \frac{3}{4} x \). This is done by multiplying 12 and \( \frac{3}{4} \), which simplifies to \( 9x \left( \frac{12 \times 3}{4} = 9 \right) \).

Step 3 - Simplify the Second Term

Calculate \( 12 \times \frac{4}{15} \). This is done by multiplying 12 and \( \frac{4}{15} \), which simplifies to \( \frac{48}{15} \). Then simplify \( \frac{48}{15} \) to \( \frac{16}{5} \).

Step 4 - Combine the Results

Add the simplified terms together to get the final simplified expression: \[ 9x + \frac{16}{5} \].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It states that multiplying a number by a sum is the same as doing each multiplication separately. For example, when you see an expression like 12 \left( \frac{3}{4} x + \frac{4}{15} \right), you should distribute the 12 to both terms inside the parentheses:

• First, multiply 12 by \( \frac{3}{4} x \)
• Then, multiply 12 by \( \frac{4}{15} \).

This helps break down complex expressions into simpler parts, making them easier to work with.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms. After distributing in our exercise, we get \( 12 \times \frac{3}{4} x \) and \( 12 \times \frac{4}{15} \). Simplifying these:

• For \( 12 \times \frac{3}{4} x \), you multiply the numerators and divide by the denominator: \[ 12 \times \frac{3}{4} = \frac{12 \times 3}{4} = 9x. \]
• For \( 12 \times \frac{4}{15} \), follow a similar process: \[ 12 \times \frac{4}{15} = \frac{12 \times 4}{15} = \frac{48}{15} \. \]

After that, reduce \( \frac{48}{15} \) by dividing both numerator and denominator by their greatest common divisor, which is 3, resulting in \( \frac{16}{5} \).

Combining Like Terms

Combining like terms means adding or subtracting terms that have identical variables and exponents. In our case from the exercise, after simplifying we have two terms \( 9x \) and \( \frac{16}{5} \):

• Both are already in their simplest form and cannot be combined any further since one is a term with 'x' and the other is a constant (a number without any variables).

Therefore, the final simplified expression is
\[ 9x + \frac{16}{5} \. \]

Multiplication of Fractions

Multiplying fractions is straightforward once you remember a simple rule: Multiply the numerators together and the denominators together. For example, in the exercise’s second step, you need to calculate
\( 12 \times \frac{3}{4} x \):

• Multiply 12 (the whole number) by 3 (the numerator) to get 36.
• Then, divide 36 by 4 (the denominator) to get 9. This is captured as
  \[ 12 \times \frac{3}{4} x = \frac{12 \times 3}{4} x = 9x. \]

The same goes for the second fraction to be multiplied, which is \( 12 \times \frac{4}{15} \):

• Multiply 12 by 4 to get 48.
• Then, divide by 15 to get \( \frac{48}{15} \), and simplify to \( \frac{16}{5} \).

Following these rules makes even the hardest fraction multiplications manageable.