Problem 63 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The sum is \( \frac{5}{6} \).

Step 1: Write down the values of b and c

We are given the values \( b = \frac{1}{6} \) and \( c = \frac{2}{3} \). Our task is to find the sum of these two fractions, \( b + c \).

Step 2: Find a common denominator

To add the fractions \( \frac{1}{6} \) and \( \frac{2}{3} \), we first need a common denominator. The denominators here are 6 and 3. The least common multiple (LCM) of 6 and 3 is 6.

Step 3: Convert fractions to a common denominator

The fraction \( \frac{1}{6} \) already has the denominator 6. To convert \( \frac{2}{3} \) to a denominator of 6, multiply both the numerator and the denominator by 2:\[\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\]

Step 4: Add the fractions

Now, add the fractions with the common denominator:\[\frac{1}{6} + \frac{4}{6} = \frac{1+4}{6} = \frac{5}{6}\]

Step 5: State the final result

The sum of \( b + c \) is \( \frac{5}{6} \).

Key Concepts

Common Denominator

When dealing with fractions, a common denominator is essential for performing addition or subtraction. It simplifies the process, allowing us to combine the fractions smoothly. If two fractions share the same denominator, their denominators become identical, making it possible to simply add or subtract the numerators.

A common denominator is crucial because:

• It aligns the fractions on the same scale.
• Without a common denominator, you can't directly add or subtract fractions.

Finding a common denominator involves locating a multiple that both denominators can easily transform into. This simplifies everything down to dealing only with the numerators, now that the denominators match.

Least Common Multiple

The Least Common Multiple (LCM) is incredibly handy when searching for a common denominator between fractions. The LCM of a set of numbers is the smallest multiple that all the numbers share. For instance, with the denominators 6 and 3, the LCM is 6.

To find the LCM, you can:

• List the multiples of each denominator.
• Select the smallest multiple that is common to both.

In our example, 6 and 3 both fit perfectly into 6, as 6 is the first shared factor when listing their multiples.

Using the LCM as your common denominator makes calculations swift and efficient. It avoids unnecessary complexity as you don't have to deal with larger numbers, which might happen if using a higher multiple.

Numerator and Denominator

Fractions are made of two key parts: the numerator and the denominator.

The numerator sits at the top of the fraction, indicating how many parts of the whole we have. Think of it as the 'counter' for parts.

The denominator is on the bottom, showing into how many equal parts the whole is divided.

• The denominator is crucial for defining the size of each part. A larger denominator means smaller parts and vice versa.
• In addition, for two fractions to be added directly, their denominators must match.

For instance, converting the fraction \( \frac{2}{3} \) into \( \frac{4}{6} \) requires multiplying the numerator and the denominator by the same number, ensuring the fraction's value remains unchanged while adjusting the denominator to the common one.