Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(39 \frac{5}{8}-13 \frac{1}{3}=\) ______

Short answer

The difference is \(\frac{631}{24}\).

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For the first number, \(39 \frac{5}{8}\), multiply the whole number (39) by the denominator (8) and add the numerator (5). This gives \(39 \times 8 + 5 = 317\), so the improper fraction is \(\frac{317}{8}\). Similarly, for \(13 \frac{1}{3}\), multiply the whole number (13) by the denominator (3) and add the numerator (1). This gives \(13 \times 3 + 1 = 40\), so the improper fraction is \(\frac{40}{3}\).

Find a Common Denominator

To subtract the fractions, we need a common denominator. The denominators are 8 and 3, and their least common multiple is 24. Convert \(\frac{317}{8}\) to a denominator of 24 by multiplying by \(\frac{3}{3}\), which gives \(\frac{951}{24}\). Convert \(\frac{40}{3}\) to a denominator of 24 by multiplying by \(\frac{8}{8}\), which gives \(\frac{320}{24}\).

Subtract the Fractions

With a common denominator, subtract \(\frac{320}{24}\) from \(\frac{951}{24}\): \(\frac{951}{24} - \frac{320}{24} = \frac{631}{24}\).

Simplify the Fraction

Check if the fraction \(\frac{631}{24}\) can be simplified. Since 631 and 24 have no common factors other than 1, \(\frac{631}{24}\) is in its simplest form.

Key concepts

Mixed Numbers

Mixed numbers are numbers that consist of both a whole number and a fraction part. For instance, in the mixed number \(39 \frac{5}{8}\), 39 is the whole number and \(\frac{5}{8}\) is the fractional part.
This format is common when dealing with measurements or quantities in everyday situations. To better manipulate mixed numbers, especially in the case of operations like addition or subtraction, it's often easier to first convert them to improper fractions.
A mixed number can seamlessly transition to an improper fraction by multiplying the whole number by the denominator of the fraction and adding the numerator to the result. This conversion is crucial in the process of fraction subtraction, as it simplifies computations.

Improper Fractions

An improper fraction is one where the numerator is larger than the denominator, such as \(\frac{317}{8}\). Although not always intuitive for representation, these fractions are quite useful in calculations.
When you convert a mixed number like \(39 \frac{5}{8}\) into an improper fraction, you follow these steps:

• Multiply the whole number by the denominator (39 \(\times\) 8) = 312.
• Add the numerator to this product (312 + 5) to get 317.
• Place the result over the original denominator, resulting in \(\frac{317}{8}\).

Improper fractions allow for straightforward operations like addition and subtraction, which lead to a more effortless calculation process.

Common Denominator

To subtract fractions, especially if they are improper, finding a common denominator is a necessary step. This involves identifying a common multiple for the denominators involved.
In this exercise, for the fractions \(\frac{317}{8}\) and \(\frac{40}{3}\), the common denominator—the least common multiple of 8 and 3—is 24.
Converting to a common denominator involves:

• Identifying the least common multiple of the denominators.
• Adjusting each fraction to have this common denominator. In our case, convert \(\frac{317}{8}\) to \(\frac{951}{24}\) and \(\frac{40}{3}\) to \(\frac{320}{24}\).

This allows for the easy operation of subtraction, as both fractions are now parts of identical wholes.

Simplifying Fractions

Once the subtraction is complete, the resulting fraction \(\frac{631}{24}\) should be checked for simplification. Simplification involves reducing the fraction to its simplest form, meaning it cannot be further divided by any common numbers except for 1.
This processes involves:

• Finding the greatest common divisor (GCD) of the numerator and the denominator.
• If no common factors are found, the fraction is already simplified.

In this exercise, 631 and 24 have no common divisors other than 1, indicating \(\frac{631}{24}\) is already in its simplest form. Simplifying helps in keeping the results comprehensible and handy for further usage or interpretation.