Question

Find each sum or difference, showing each step of your work. Give your answers in lowest terms. If an answer is greater than 1 , write it as a mixed number. $$10 \frac{2}{5}-4 \frac{1}{3}$$

Short answer

6 \(\frac{1}{15}\)

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

Convert both mixed numbers to improper fractions. For 10 \(\frac{2}{5}\): \[ 10 \frac{2}{5} = 10 + \frac{2}{5} = \frac{10 \times 5 + 2}{5} = \frac{52}{5} \] For 4 \(\frac{1}{3}\): \[ 4 \frac{1}{3} = 4 + \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{13}{3} \]

Find a Common Denominator

Determine the least common denominator (LCD) for 5 and 3, which is 15. Convert the improper fractions to have this common denominator: \[ \frac{52}{5} = \frac{52 \times 3}{5 \times 3} = \frac{156}{15} \] \[ \frac{13}{3} = \frac{13 \times 5}{3 \times 5} = \frac{65}{15} \]

Subtract the Fractions

Now subtract the fractions: \[ \frac{156}{15} - \frac{65}{15} = \frac{156 - 65}{15} = \frac{91}{15} \]

Convert the Result to Mixed Number (if needed)

Since \(\frac{91}{15}\) is an improper fraction, convert it to a mixed number. Divide 91 by 15: \[ 91 \text{ divided by } 15 = 6 \text{ R} 1 \] So, \(\frac{91}{15} = 6 \frac{1}{15}\)

Verify the Answer in Lowest Terms

The fraction \(\frac{1}{15}\) is already in lowest terms. Therefore, the final answer is: \(6 \frac{1}{15}\)

Key concepts

Fraction Subtraction

Fraction subtraction involves taking one fraction away from another. To successfully subtract fractions, it's important to follow these steps:

First, ensure both fractions have the same denominator. This requires finding a common denominator if they are different. Only once the fractions share the same denominator can you move forward in subtracting the numerators directly.

Another fundamental aspect is converting mixed numbers to improper fractions before performing the subtraction. Mixed numbers can complicate subtraction if not converted, as they contain both a whole number and a fraction. By converting them into improper fractions, the process becomes simpler and more straightforward.

Converting Mixed Numbers

Mixed numbers feature a whole number combined with a proper fraction. To ease the subtraction process, we convert the mixed numbers into improper fractions.

To convert a mixed number to an improper fraction, use this method:

• Multiply the whole number by the fraction's denominator.
• Add the numerator to this result.
• The sum becomes the new numerator, with the original denominator staying the same.

For example, to convert 10 \(\frac{2}{5}\):

• Multiply 10 by 5 (the denominator): 10 x 5 = 50
• Add the numerator: 50 + 2 = 52
• Place it over the original denominator: \(\frac{52}{5}\)

This same method applies to any mixed number, making it easier to add, subtract, multiply, or divide them.

Least Common Denominator

To subtract fractions effectively, it's essential to have a common denominator. The smallest common denominator that two (or more) fractions can share is known as the Least Common Denominator (LCD). Using the LCD simplifies the fractions consistently.

Here’s how to find the LCD:

• List the multiples of each denominator.
• Identify the lowest multiple common to each denominator.
• Convert each fraction using this common denominator.

For example, for denominators 5 and 3:

• Multiples of 5: 5, 10, 15, 20, ...
• Multiples of 3: 3, 6, 9, 12, 15, ...
• Common multiple: 15

Convert each fraction to have 15 as the denominator:

• \(\frac{52}{5} = \frac{52 \times 3}{5 \times 3} = \frac{156}{15}\)
• \(\frac{13}{3} = \frac{13 \times 5}{3 \times 5} = \frac{65}{15}\)

With the same denominators, subtract the numerators directly to find the answer. This process ensures accuracy and clarity in fraction subtraction.