Question

Evaluate each expression. $$ \frac{6}{35}-\frac{3}{14} $$

Short answer

\[ \frac{-3}{70} \]

Step-by-step solution

Find the Least Common Denominator (LCD)

Identify the least common denominator for the fractions \(\frac{6}{35}\) and \(\frac{3}{14}\). The denominators are 35 and 14. Find the LCD by determining the least common multiple (LCM) of 35 and 14. The LCM of 35 and 14 is 70.

Convert the Fractions

Convert each fraction to have the same denominator (70). Multiply both the numerator and denominator of \(\frac{6}{35}\) by 2, and \(\frac{3}{14}\) by 5:\[ \frac{6}{35} \times \frac{2}{2} = \frac{12}{70} \]\[ \frac{3}{14} \times \frac{5}{5} = \frac{15}{70} \]

Subtract the Fractions

With both fractions having the same denominator, subtract the numerators: \[ \frac{12}{70} - \frac{15}{70} = \frac{12-15}{70} = \frac{-3}{70} \]

Simplify the Fraction

Make sure the fraction is in its simplest form. In this case, \(\frac{-3}{70}\) is already simplified.

Key concepts

least common denominator

When subtracting fractions, finding the least common denominator (LCD) is crucial. The LCD is the smallest number that is a common multiple of the denominators. For example, in the fractions \(\frac{6}{35}\) and \(\frac{3}{14}\), the denominators are 35 and 14. We need to find the least common multiple (LCM) of these denominators. Once we have the LCM, we use it as our common denominator for the subtraction. In this case, the LCM of 35 and 14 is 70, which becomes our LCD.

least common multiple

The least common multiple (LCM) is the smallest positive integer that is divisible by both numbers. Finding the LCM involves a few steps:
1. List the multiples of each number.
2. Identify the smallest multiple that appears in both lists.
For 35 and 14, listing multiples:

• Multiples of 35: 35, 70, 105, ...
• Multiples of 14: 14, 28, 42, 56, 70, ...

The smallest common multiple is 70. Hence, 70 is the LCM.

fraction simplification

After performing subtraction or addition of fractions, simplifying the result is essential.
This means reducing the fraction to its simplest form. A fraction is simplified when the numerator and denominator have no common factors other than 1. In the expression \(\frac{-3}{70}\), the greatest common divisor of 3 and 70 is 1, so the fraction \(\frac{-3}{70}\) is already in its simplest form.

subtraction of fractions

To subtract fractions, you first need to ensure both fractions have the same denominator. Using our example:
\(\frac{6}{35} - \frac{3}{14}\).
We convert the fractions to have a common denominator of 70:

• Multiply both the numerator and denominator of \(\frac{6}{35}\) by 2 to get \(\frac{12}{70}\)
• Multiply both the numerator and denominator of \(\frac{3}{14}\) by 5 to get \(\frac{15}{70}\)

Now, subtract the numerators: \(\frac{12}{70} - \frac{15}{70} = \frac{12-15}{70} = \frac{-3}{70}\). The result is your final fraction, ready to be simplified if needed.