Problem 84 - Prealgebra

Question

The following problems review addition and subtraction with fractions and mixed numbers. $$\frac{7}{39}-\frac{2}{65}$$

Step-by-Step Solution

Answer

The solution is \( \frac{29}{195} \).

Step 1: Find the Least Common Denominator (LCD)

The fractions \( \frac{7}{39} \) and \( \frac{2}{65} \) have different denominators. To subtract these fractions, we first find the least common denominator (LCD). The denominators are 39 and 65. We factor each number: 39 = 3 × 13, and 65 = 5 × 13. The LCD is the least common multiple of both: 3 × 5 × 13 = 195.

Step 2: Convert Fractions to Equivalent Fractions with the LCD

Convert each fraction to an equivalent fraction with the denominator 195. For \( \frac{7}{39} \), multiply both numerator and denominator by 5 to get \( \frac{35}{195} \). For \( \frac{2}{65} \), multiply both numerator and denominator by 3 to get \( \frac{6}{195} \).

Step 3: Subtract the Equivalent Fractions

Now that both fractions have the same denominator, subtract them: \( \frac{35}{195} - \frac{6}{195} = \frac{29}{195} \).

Step 4: Simplify the Result (if possible)

Check if \( \frac{29}{195} \) can be simplified further. Since 29 is a prime number, and there are no common factors between 29 and 195, the fraction is already in its simplest form.

Key Concepts

Least Common Denominator

Before we can add or subtract fractions, they need to have the same denominator. This common denominator is known as the Least Common Denominator (LCD). The LCD is the smallest multiple that two denominators share.
To find the LCD, first factor each denominator into its prime factors.

• For 39, the prime factors are 3 and 13, since 39 can be broken down into 3 × 13.
• For 65, the prime factors are 5 and 13, since 65 equals 5 × 13.

Once you have the prime factors of each denominator, you combine them, using each factor the greatest number of times it appears in any of the factorizations. Here, the prime factors we use are 3, 5, and 13. Therefore, the LCD for 39 and 65 is 3 × 5 × 13, which results in 195.
This step allows us to rewrite each fraction so they share this denominator, preparing them for the addition or subtraction process.

Equivalent Fractions

To add or subtract fractions, you need to create equivalent fractions with the common denominator you found, which is 195 in this case. An equivalent fraction is another name for the same amount, just with a different denominator and numerator.

• For the fraction \( \frac{7}{39} \), multiply both the numerator and the denominator by 5 to keep the value of the fraction the same. This transforms it into \( \frac{35}{195} \).
• For \( \frac{2}{65} \), multiply both the numerator and the denominator by 3 to convert it into \( \frac{6}{195} \).

By converting each of these fractions to \( \frac{35}{195} \) and \( \frac{6}{195} \), they become equivalent fractions sharing the same denominator, making it possible to perform the arithmetic operation.

Simplifying Fractions

Once a subtraction is performed with fractions that have the same denominator, it’s crucial to simplify the result if possible. Simplifying means to make the fraction as simple as possible, usually by reducing it to its lowest terms.
The process involves finding any common factors between the numerator and the denominator and dividing them out.

• For example, after subtracting \( \frac{35}{195} \) from \( \frac{6}{195} \), the result is \( \frac{29}{195} \).
• Examine \( \frac{29}{195} \) to see if it can be reduced further. Since 29 is a prime number and has no common factors with 195, the fraction is already in its simplest form.

Simplification helps to keep fractions neat and tells us when we've reached the simplest form, reducing errors in further calculations.