Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{4}{7}-\frac{1}{3}=\) ______

Short answer

The difference is \(\frac{5}{21}\).

Step-by-step solution

Find a Common Denominator

To subtract fractions, we need to have the same denominator for both fractions. The denominators given are 7 and 3. To find a common denominator, calculate the least common multiple (LCM) of 7 and 3, which is 21. So, we will convert both fractions to have a denominator of 21.

Convert Fractions to Equivalent Fractions

Now that we have the common denominator (21), convert each fraction to an equivalent fraction with this common denominator.\(\frac{4}{7} = \frac{4\times 3}{7\times 3} = \frac{12}{21}\)\(\frac{1}{3} = \frac{1\times 7}{3\times 7} = \frac{7}{21}\)

Subtract the Equivalent Fractions

With the fractions now having the same denominator, subtract the numerators while keeping the denominator the same:\(\frac{12}{21} - \frac{7}{21} = \frac{12 - 7}{21} = \frac{5}{21}\)

Simplify the Fraction

Check if the resulting fraction \(\frac{5}{21}\) can be simplified. Since 5 and 21 have no common factors other than 1, \(\frac{5}{21}\) is already in its simplest form.

Key concepts

Common Denominator

When subtracting fractions, having the same denominator makes the operation straightforward. This is known as finding a common denominator. Think of the denominator like a base that groups parts of the whole. Different fractions with different bases require a uniform base for easy computation. In our example with denominators 7 and 3, we must adjust them so they share this base.
One way to achieve this is by finding a common multiple of both denominators. This brings us to the next topic: least common multiple.

Least Common Multiple

The least common multiple (LCM) is the smallest number that both denominators can divide evenly without leaving a remainder. To find the LCM of 7 and 3 in our problem, list the multiples of each:

• Multiples of 7: 7, 14, 21, 28, ...
• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...

The smallest number that appears in both lists is 21. Using 21 as a common denominator is perfect because it simplifies our subtraction process without altering the value of the fractions.

Equivalent Fractions

Once we have identified the common denominator, we need to convert each fraction to equivalent forms with that common denominator. Equivalent fractions are different-looking fractions that represent the same value. In our example:

• \( \frac{4}{7} \) becomes \( \frac{12}{21} \) by multiplying both the numerator and the denominator by 3.
• \( \frac{1}{3} \) becomes \( \frac{7}{21} \) by multiplying both by 7.

This transformation ensures that both fractions are ready for subtraction, and it involves simple multiplication, ensuring no change to the original value represented by the fraction. All equivalent fractions look different but are numerically similar.

Simplifying Fractions

After the subtraction, the final step is simplification. Simplifying fractions means reducing them to their smallest form. A fraction is simplified when the numerator and the denominator cannot be divided by the same number other than 1. In our result \( \frac{5}{21} \), the numbers 5 and 21 cannot be further divided by any number except 1, which means this fraction is already as simple as it can be.
Simplifying fractions is essential because it gives the clearest picture of the quantity, helping to easily compare with other simplified fractions.