Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{8}{5}-\frac{1}{3}=\) ______

Short answer

\( \frac{19}{15} \)

Step-by-step solution

Find a Common Denominator

The fractions involved are \( \frac{8}{5} \) and \( \frac{1}{3} \). To subtract fractions with different denominators, we must first find a common denominator. The denominators are 5 and 3. The least common multiple of 5 and 3 is 15. So, both fractions need to be expressed with a denominator of 15.

Convert Each Fraction to Equivalent Fractions

Convert \( \frac{8}{5} \) to a fraction with a denominator of 15. We do this by multiplying both the numerator and the denominator by 3: \[ \frac{8}{5} \times \frac{3}{3} = \frac{24}{15} \]Now convert \( \frac{1}{3} \) to a fraction with a denominator of 15 by multiplying both the numerator and the denominator by 5:\[ \frac{1}{3} \times \frac{5}{5} = \frac{5}{15} \]

Subtract the Equivalent Fractions

Now that both fractions have a common denominator, we can subtract them:\[ \frac{24}{15} - \frac{5}{15} = \frac{24 - 5}{15} = \frac{19}{15} \]

Check if the Fraction can be Reduced

The fraction \( \frac{19}{15} \) is already in its simplest form since 19 is a prime number and does not share any common factors with 15.

Key concepts

Finding a Common Denominator

When subtracting fractions like \( \frac{8}{5} - \frac{1}{3} \), you first need to deal with their different denominators. Think of the denominator as the number of equal parts a whole is divided into. If they're different, you can't directly subtract because the size of the parts isn't the same.

To get fractions ready for subtraction, you need a **common denominator**. This is a number that both denominators can evenly divide into. You find this by determining the least common multiple (LCM) of the denominators. For 5 and 3, the LCM is 15. Here's how to think about it:

• The multiples of 5 are 5, 10, 15, 20, and so on.
• The multiples of 3 are 3, 6, 9, 12, 15, and so on.

The first number that appears in both lists is 15. That's your common denominator. When both fractions have the same denominator, you're ready to move on to the next step.

Equivalent Fractions

With a common denominator identified, you need to convert each fraction into an **equivalent fraction** that has this new denominator. Equivalent fractions may look different but represent the same value.

To do this, multiply both the numerator (top number) and the denominator (bottom number) of each fraction by the same number.

• For \( \frac{8}{5} \), multiply both numbers by 3: \( \frac{8}{5} \times \frac{3}{3} = \frac{24}{15} \).
• For \( \frac{1}{3} \), multiply both numbers by 5: \( \frac{1}{3} \times \frac{5}{5} = \frac{5}{15} \).

Now both fractions, \( \frac{24}{15} \) and \( \frac{5}{15} \), have the common denominator of 15. You're ready to subtract because they represent equal shares of the same whole.

Simplest Form

After eliminating fractional differences with a common denominator and calculating \( \frac{24}{15} - \frac{5}{15} = \frac{19}{15} \), the next step is simplifying your result. **Simplest form** means the fraction has no common factors other than 1.

Optimization of fractions is done by checking for the greatest common divisor (GCD) of the numerator and denominator:

• 19 is a prime number, meaning its only divisors are 1 and itself.
• 15 is not divided by 19 evenly; they share no factors other than 1.

Since the GCD is 1, \( \frac{19}{15} \) is in its simplest form. Besides, fractions that can't be reduced further uphold mathematical precision and simplicity in calculation. Thus, always check if your answers can become tidier through reduction.