Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(48 \frac{6}{11}-24=\) ______

Short answer

The answer is \(\frac{270}{11}\).

Step-by-step solution

Convert Mixed Number to Improper Fraction

First, convert the mixed number \(48 \frac{6}{11}\) to an improper fraction. Multiply the whole number \(48\) by the denominator \(11\) and add the numerator \(6\):\[48 \times 11 + 6 = 528 + 6 = 534.\] Now, place this result over the original denominator: \(\frac{534}{11}\).

Subtract Whole Number from Improper Fraction

We need to perform \(\frac{534}{11} - 24\). To subtract these, we express 24 as a fraction with the same denominator: \(24 = \frac{264}{11}\). Now, subtract the fractions:\[\frac{534}{11} - \frac{264}{11} = \frac{534 - 264}{11} = \frac{270}{11}.\]

Reduce the Fraction to Lowest Terms

To reduce \(\frac{270}{11}\) to its lowest terms, note that 11 is a prime number and does not divide evenly into 270. Thus, \(\frac{270}{11}\) is already in its simplest form.

Key concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often used in everyday situations because they are easier to understand at a glance. For example, if you eat "1 and a half" pizzas, it makes more sense to say "1 pizza and 1/2" rather than an improper fraction like "3/2".

To perform operations like addition, subtraction, or multiplication on mixed numbers, it's usually best to convert them into improper fractions first. This is because improper fractions are easier to manage arithmetically, as they lack the addition of the whole number part and fraction part. In our example, the mixed number \(48 \frac{6}{11}\) can be converted to an improper fraction \(\frac{534}{11}\). This conversion allows us to subtract and add fractions smoothly.

Improper Fractions

Improper fractions have a numerator larger than or equal to the denominator, meaning the fraction's value is equal to or greater than 1. They are especially useful for calculations involving mixed numbers because they leverage a single top-heavy fraction.

To convert a mixed number into an improper fraction, multiply the whole number by the denominator and add the numerator. Place the result over the original denominator. For example, with the mixed number \(48 \frac{6}{11}\), you multiply 48 by 11 and add 6 to get 534, resulting in the improper fraction \(\frac{534}{11}\).

Improper fractions ensure a straightforward process for operations like subtraction, making it simpler by dealing with one type of number rather than two.

Reducing Fractions

Reducing fractions means simplifying them to their most basic form, where the numerator and denominator have no common factors other than 1. This is sometimes called simplifying or putting a fraction into its lowest terms.

To reduce a fraction, find the greatest common factor (GCF) of the numerator and the denominator, then divide both by this number. However, in our example, since the denominator is 11, a prime number, we simply need to check if it divides evenly into the numerator, which it doesn't in this case. So, \(\frac{270}{11}\) is already in its simplest form.

Remember, reducing fractions can make your answers clearer and are often expected in final results.

Lowest Terms

Bringing a fraction to its lowest terms means simplifying it as much as possible. This makes it more readily comparable and easier to understand, similar to simplifying mathematical equations.

A fraction is in its lowest terms when you can't divide the numerator and denominator by any number other than 1. In problems that ask for the answer in lowest terms, it's crucial to check this final step. Let's say we ended up with \(\frac{270}{11}\); since 11 doesn't divide 270 evenly, we know that fraction is in lowest terms.

• Check divisibility by smallest primes first, like 2, 3, 5, etc.
• Ensure no further common factors exist.

Keeping fractions in their simplest form helps make mathematical communication consistent and understandable.