Problem 65 - Algebra 2

Question

Find each sum if \(a=\frac{1}{2}, b=\frac{1}{6}, c=\frac{2}{3},\) and \(d=\frac{3}{4}\) $$ b+d $$

Step-by-Step Solution

Answer

The sum is \(\frac{11}{12}\).

Step 1: Identify the Values

We are given the variables: - \(b = \frac{1}{6}\) - \(d = \frac{3}{4}\)Our task is to find the sum \(b + d\).

Step 2: Find a Common Denominator

To add the fractions \(b\) and \(d\), we need a common denominator. The denominators, 6 and 4, have a least common multiple (LCM) of 12. We will use 12 as the common denominator.

Step 3: Convert Fractions to Common Denominator

Convert \(b = \frac{1}{6}\) to have a denominator of 12: \[ \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \]Convert \(d = \frac{3}{4}\) to have a denominator of 12:\[\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

Step 4: Add the Fractions

Now that both fractions have the same denominator, we can add them directly:\[\frac{2}{12} + \frac{9}{12} = \frac{11}{12} \]The sum of \(b + d\) is \(\frac{11}{12}\).

Key Concepts

Least Common Multiple (LCM)

In the world of fractions, finding a common base is crucial for many calculations. The least common multiple, or LCM, is a key concept here. The LCM of two numbers is the smallest number that is a multiple of both. It helps in synchronizing different pieces and is especially handy when dealing with fractions. For instance, in the exercise provided, the denominators are 6 and 4. By finding the LCM, we can convert these fractions into a compatible form for addition. The LCM of 6 and 4 is 12. Therefore, 12 becomes the common ground or platform on which the fractions can interact with ease.
It acts as a mutual meeting point for denominators, facilitating smoother calculations.

Common Denominator

A common denominator allows you to combine fractions seamlessly. The concept arises from the need to have equal-sized parts when adding, subtracting, or comparing fractions. To find a common denominator, you typically seek a number that all the denominators divide into evenly. The LCM, as discussed earlier, often serves this purpose. In our exercise, we changed the fractions \(\frac{1}{6}\) and \(\frac{3}{4}\) so that they both shared the denominator of 12.
Turning \(\frac{1}{6}\) into \(\frac{2}{12}\) and \(\frac{3}{4}\) into \(\frac{9}{12}\) allows us to easily perform arithmetic operations.

• It ensures each fraction represents comparable quantities.
• It makes the process of addition straightforward.

Finally, with a common denominator in place, you can proceed with your calculations confidently.

Adding Fractions

Adding fractions might seem tricky, but with the right tools, it becomes simple and straightforward. The main aim is to arrive at equivalent fractions with a shared denominator.
Once you've converted fractions to have a common denominator, the process becomes as easy as adding whole numbers. You simply add the numerators (the top part of the fractions) and maintain the common denominator.
For instance, in our solution:

• We combined \(\frac{2}{12}\) and \(\frac{9}{12}\) by adding their numerators: 2 + 9 = 11.
• The denominator remains the same, which is 12.

Thus, the resulting sum is \(\frac{11}{12}\). This demonstrates how efficient fraction addition becomes once fractions share a common ground, or denominator.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the numerator is any integer and the denominator is a non-zero integer. In practice, they encompass fractions like \(\frac{1}{6}\) and \(\frac{3}{4}\) from our exercise.
These numbers can represent parts of whole items, making them essential in various real-life contexts, from cooking measurements to dividing resources.

• They allow for expressing repeated decimals, such as \(\frac{1}{3}=0.333\cdots\).
• They are involved in operations that include addition, subtraction, multiplication, and division (except by zero).

Understanding rational numbers is fundamental for grasping more complex mathematical concepts. Moreover, while dealing with rational numbers, the skill of managing their addition or comparison is rooted in finding common denominators and LCMs. Mastery of these connections lays a solid foundation for future mathematical success.