Problem 40 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{3 y}{4}+\frac{7 y}{5}$$

Step-by-Step Solution

Answer

The sum is \(\frac{43y}{20}\).

Step 1: Find a Common Denominator

To add two fractions, we must first have a common denominator. The denominators in this problem are 4 and 5. The least common denominator (LCD) of 4 and 5 is 20.

Step 2: Convert Each Fraction to the Common Denominator

Convert \(\frac{3y}{4}\) to \(\frac{3y \times 5}{4 \times 5} = \frac{15y}{20}\). Convert \(\frac{7y}{5}\) to \(\frac{7y \times 4}{5 \times 4} = \frac{28y}{20}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add them by adding the numerators and keeping the common denominator: \(\frac{15y}{20} + \frac{28y}{20} = \frac{43y}{20}\).

Step 4: Simplify the Resulting Fraction

The fraction \(\frac{43y}{20}\) is already in its simplest form, as 43 is a prime number and does not have any common factors with 20.

Key Concepts

Common Denominator

To add fractions, it's essential to have a common denominator. Think of the denominator as the 'bottom number'. It's the part of the fraction that shows the total number of equal parts. For example, with \( \frac{3y}{4} \) and \( \frac{7y}{5} \), the denominators are 4 and 5. They represent the number of pieces that make up a whole in each fraction. However, trying to add these fractions directly doesn't work since we're dealing with different "wholes".
To solve this, we find the least common denominator (LCD). The LCD is the smallest number that both denominators can divide into without leaving a remainder.

• The denominators here are 4 and 5.
• The smallest number that 4 and 5 can both divide into is 20.

Finding the LCD involves listing the multiples of each number and choosing the smallest one they share. This step is crucial because it aligns the fractions, making them compatible for addition.

Adding Fractions

Once the fractions have a common denominator, adding them becomes straightforward. Imagine having equal-sized pieces, now you can simply combine them. With both fractions now expressed over 20, our task is to add \( \frac{15y}{20} \) and \( \frac{28y}{20} \).
To do this,

• Keep the denominator the same, since all pieces are now of a uniform size.
• Add the numerators, which are the top numbers, together: \( 15y + 28y \).

The result is \( \frac{43y}{20} \). The denominator remains 20, showing that all parts are still uniform. Meanwhile, the newly summed numerator tells us how many of these parts you have altogether.

Simplest Form

After adding the fractions, it's important to check if the result can be simplified. Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator share no common factors other than 1.

• In our example, the fraction is \( \frac{43y}{20} \).
• To determine if simplification is possible, check the greatest common factor (GCF) of the numerator and denominator.

Since 43 is a prime number and has no factors in common with 20 other than 1, \( \frac{43y}{20} \) is already in simplest form. To summarize: a fraction in simplest form ensures clarity and makes it easier to understand and use the resulting fraction in further calculations.