Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{4}{6}-\frac{3}{8}=\) ______

Short answer

\(\frac{7}{24}\)

Step-by-step solution

Find a Common Denominator

To subtract fractions, we first need a common denominator. The denominators we have are 6 and 8. The least common multiple (LCM) of 6 and 8 is 24. So, we will convert both fractions to have a denominator of 24.

Convert to Equivalent Fractions

Convert \(\frac{4}{6}\) and \(\frac{3}{8}\) to fractions with a denominator of 24. For \(\frac{4}{6}\), multiply both numerator and denominator by 4 to get \(\frac{16}{24}\). For \(\frac{3}{8}\), multiply both numerator and denominator by 3 to get \(\frac{9}{24}\).

Subtract the Numerators

Now that both fractions have the same denominator, subtract the numerators: \(16 - 9 = 7\). This gives us \(\frac{7}{24}\).

Simplify if Possible

Check if \(\frac{7}{24}\) can be simplified. The greatest common divisor (GCD) of 7 and 24 is 1, so \(\frac{7}{24}\) is already in its simplest form.

Key concepts

Least Common Multiple

In fraction subtraction, finding the Least Common Multiple (LCM) is important for aligning denominators. This ensures that fractions can be compared and subtracted easily. If the denominators are different, as in the fractions \(\frac{4}{6}\) and \(\frac{3}{8}\), we can't directly subtract them. That's where the LCM comes in.

• The LCM of two numbers is the smallest number that both denominators can divide into without leaving a remainder.
• For the numbers 6 and 8, we see that the least common multiple is 24.

Once you've identified the LCM, you change the fractions so they both have this common denominator. This step is crucial for performing operations on fractions.

Equivalent Fractions

The concept of equivalent fractions is all about expressing fractions with a common denominator. After determining the least common multiple (24 in this exercise), the next step is converting the fractions to equivalent ones with this denominator.

• To convert \(\frac{4}{6}\), multiply the numerator and the denominator by 4. This gives \(\frac{16}{24}\).
• For \(\frac{3}{8}\), multiply both by 3 to obtain \(\frac{9}{24}\).

These conversions ensure that both fractions have the same denominator, allowing us to then perform subtraction smoothly. It is an essential skill for dealing with fractions, especially in operations like addition and subtraction. By understanding and applying the concept of equivalent fractions, students can tackle a wide variety of problems with greater ease.

Numerator Subtraction

After converting fractions to have a common denominator, the next step in the subtraction process is quite straightforward: subtracting the numerators.

• With \(\frac{16}{24}\) and \(\frac{9}{24}\), you just subtract the numerators: 16 - 9 = 7.
• The denominator remains the same (24) since we've already adjusted the fractions to share a common foundation.

This results in a new fraction \(\frac{7}{24}\). Numerator subtraction is simple once you have equivalent fractions, and it's a key step in solving fraction subtraction problems. Always ensure that numerators are being subtracted from fractions with a shared denominator.

Simplifying Fractions

Once the subtraction is complete, the final step is simplifying the fraction. Simplifying a fraction means reducing it to its lowest terms, where the numerator and the denominator have no common factors other than 1.

• In the exercise, after subtracting, we end up with \(\frac{7}{24}\).
• To simplify, check for any common divisors of 7 and 24. Here, the greatest common divisor is 1.

Since there are no common factors other than 1, \(\frac{7}{24}\) is already in its simplest form. Simplification makes it easier to compare and work with fractions in further calculations. Being adept at recognizing when a fraction is fully simplified is an important part of mastering fractions and arithmetic.