Problem 70 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{13}{15}-\frac{8}{45} $$

Step-by-Step Solution

Answer

\(\frac{31}{45}\)

Step 1: Find a common denominator

Identify the least common denominator (LCD) for the fractions. The denominators are 15 and 45. The LCD for 15 and 45 is 45.

Step 2: Convert fractions to have the same denominator

Convert \(\frac{13}{15}\) to a fraction with denominator 45. To do this, multiply both the numerator and the denominator by 3: \(\frac{13 \times 3}{15 \times 3} = \frac{39}{45}\). The fraction \(\frac{8}{45}\) already has the denominator 45, so it remains unchanged.

Step 3: Perform the subtraction

Now subtract the fractions: \(\frac{39}{45} - \frac{8}{45}\). Since the denominators are the same, subtract the numerators: \(\frac{39 - 8}{45} = \frac{31}{45}\).

Step 4: Simplify the fraction, if possible

Check if \(\frac{31}{45}\) can be simplified. The greatest common divisor (GCD) of 31 and 45 is 1, so \(\frac{31}{45}\) is already in its simplest form.

Step 5: Verify using a calculator

Use a calculator to verify the result: perform \( \frac{13}{15} - \frac{8}{45} \), which equals approximately 0.6889 - 0.1778 = 0.5111. Converting \(\frac{31}{45}\) to a decimal also gives approximately 0.6889 - 0.1778 = 0.5111.

Key Concepts

Least Common Denominator

To subtract fractions, the first step is to find the least common denominator (LCD) of the fractions involved. The LCD is the smallest number that both denominators divide into evenly. Think of it as finding a common base to make the fractions comparable.

For example, consider the problem \(\frac{13}{15}-\frac{8}{45}\). Here, the denominators are 15 and 45. Since 45 is a multiple of 15, it serves as the LCD.

• First, list the multiples of each denominator.
• Then, identify the smallest multiple they share.

In this case:

• Multiples of 15: 15, 30, 45, 60...
• Multiples of 45: 45, 90, 135...

As you can see, the smallest common multiple is 45.

Converting Fractions

Next, Convert fractions to have the same denominator. This is a crucial step for making sure that the fractions are comparable. Using the LCD makes the subtraction process straightforward.

To convert \(\frac{13}{15}\) to a fraction with denominator 45, multiply both the numerator and the denominator by 3:
\(\frac{13 \times 3}{15 \times 3} = \frac{39}{45}\).

• Numerator (top number): 13 * 3 = 39
• Denominator (bottom number): 15 * 3 = 45

The fraction \(\frac{8}{45}\) already has the denominator 45, so it remains unchanged.

Fraction Subtraction

Now that the fractions have the same denominator, you can perform the subtraction. When dealing with fractions that have a common denominator, simply subtract the numerators and place the result over the common denominator.

For \(\frac{39}{45} - \frac{8}{45}\), subtract the numerators (39 - 8):
\(\frac{39 - 8}{45} = \frac{31}{45}\).

• Subtract the numerators: 39 - 8 = 31
• Keep the denominator the same: 45

Simplifying Fractions

After performing the subtraction, check if the resulting fraction can be simplified. Simplifying fractions involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

For \(\frac{31}{45}\), check to see if there are any common factors between 31 and 45.

• The greatest common divisor of 31 and 45 is 1.

Since 31 and 45 have no common divisor other than 1, the fraction \(\frac{31}{45}\) is already in its simplest form.

Common Divisor

In the context of simplifying fractions, a common divisor is any number that evenly divides both the numerator and the denominator. Finding the greatest common divisor (GCD) helps determine if a fraction can be simplified.

Here's how to find the GCD:

1. List the factors of each number.
2. Identify the largest factor they both share.

For 31 and 45:

• Factors of 31: 1, 31
• Factors of 45: 1, 3, 5, 9, 15, 45

Since the only common factor is 1, the fraction \(\frac{31}{45}\) is already simplified.