Problem 60 - Prealgebra

Question

Multiply and divide as indicated. $$\frac{1}{4}+\frac{1}{12} \cdot 6$$

Step-by-Step Solution

Answer

The final result is \( \frac{3}{4} \).

Step 1: Simplify the Multiplication

First, simplify the expression by performing the multiplication. The expression is \( \frac{1}{4} + \frac{1}{12} \cdot 6 \).Focus on the multiplication part first: \( \frac{1}{12} \cdot 6 = \frac{6}{12} \).Simplify \( \frac{6}{12} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 6, giving \( \frac{1}{2} \). So now the expression is \( \frac{1}{4} + \frac{1}{2} \).

Step 2: Find a Common Denominator

Now add \( \frac{1}{4} \) and \( \frac{1}{2} \). To add fractions, we need a common denominator. The denominators are 4 and 2. The least common multiple of 4 and 2 is 4. Convert \( \frac{1}{2} \) to a fraction with a denominator of 4:\( \frac{1}{2} = \frac{2}{4} \) (multiplied both numerator and denominator by 2).Now we can write the expression as \( \frac{1}{4} + \frac{2}{4} \).

Step 3: Add the Fractions

With both fractions now having the same denominator, simply add the numerators:\( \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \). The denominator stays the same at 4, so the expression evaluates to \( \frac{3}{4} \).

Key Concepts

Multiplying Fractions

Multiplying fractions might seem tricky at first, but it is straightforward. When you multiply fractions, focus on both numerators and denominators. Let's break it down step-by-step:

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.

Here’s an example using the fraction from our exercise: \( \frac{1}{12} \times 6 \).We can rewrite 6 as a fraction \( \frac{6}{1} \). Now, multiply:

• Numerators: 1 and 6. So, \( 1 \times 6 = 6 \).
• Denominators: 12 and 1. So, \( 12 \times 1 = 12 \).

Our result is \( \frac{6}{12} \), a fraction that can be simplified further.

Adding Fractions with Unlike Denominators

Adding fractions with different denominators requires one simple step: finding a common denominator. This unifies the denominators so we can easily add the numerators.
In our problem, we needed to add \( \frac{1}{4} + \frac{1}{2} \). Let's see how to do it:

• Identify the denominators. Here, they are 4 and 2.
• Calculate the least common multiple (LCM) of these denominators, which is 4.
• Convert \( \frac{1}{2} \) to a denominator of 4 by multiplying both top and bottom by 2, giving \( \frac{2}{4} \).

Now, add them:

• Since both fractions now have the same denominator, add the numerators: 1 + 2 = 3.

The sum is \( \frac{3}{4} \), with 4 as the unchanged common denominator.

Simplifying Fractions

Simplifying fractions is the process of making the fraction as simple as possible, or in its lowest terms. After performing operations like multiplication or addition, the result can often be simplified.Here's how you do it:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both by this GCD.

For \( \frac{6}{12} \) from our example, find the GCD of 6 and 12, which is 6:

• Divide the numerator by 6: \( 6 \div 6 = 1 \).
• Divide the denominator by 6: \( 12 \div 6 = 2 \).

The simplified result is \( \frac{1}{2} \). Simplifying makes fractions easier to interpret and use them in further calculations.