Problem 23 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$-10 \frac{2}{3}+9 \frac{7}{12}$$

Step-by-Step Solution

Answer

The answer is \(-\frac{13}{12}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. The number \(-10 \frac{2}{3} = -10 - \frac{2}{3} = -\frac{30}{3} - \frac{2}{3} = -\frac{32}{3}\). Similarly, \(9 \frac{7}{12} = \frac{108}{12} + \frac{7}{12} = \frac{115}{12}\).

Step 2: Find a Common Denominator

To add the fractions \(-\frac{32}{3}\) and \(\frac{115}{12}\), we need a common denominator. The least common multiple of 3 and 12 is 12. We convert \(-\frac{32}{3}\) to \(-\frac{128}{12}\). Now both fractions have the same denominator.

Step 3: Add the Fractions

Now that we have \(-\frac{128}{12}\) and \(\frac{115}{12}\), we can add these fractions:\[-\frac{128}{12} + \frac{115}{12} = \frac{-128 + 115}{12} = \frac{-13}{12}\].

Step 4: Write the Result in Simplest Form

The fraction \(\frac{-13}{12}\) is already in its simplest form because 13 and 12 have no common factors other than 1.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that consist of two parts: a whole number and a fraction. They are particularly useful when representing values greater than or less than a whole number.
For instance, in the exercise, you encountered mixed numbers such as \(-10 \frac{2}{3}\) and \(-9 \frac{7}{12}\). When simplifying or performing operations with these numbers, we often convert them to improper fractions because it streamlines the arithmetic processes.

• The whole number part of the mixed number shows how many full parts are there.
• The fractional part represents any leftover portions.

This transformation to improper fractions helps as it simplifies mathematical operations like addition or subtraction.

Improper Fractions

An improper fraction is a type of fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). In our exercise, we converted mixed numbers into improper fractions.
Understanding improper fractions is crucial because they make it simpler to deal with arithmetic operations like addition and subtraction, especially with different denominators.

• Take the whole number part and multiply it by the denominator of the fraction part.
• Add this result to the numerator of the fractional part.
• The sum becomes the new numerator, and the denominator stays the same.

For example, converting \(9 \frac{7}{12}\) into an improper fraction gives us \(\frac{115}{12}\). This method ensures all fractions are in the same format, aiding our calculations.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions.
A common denominator is basically a shared multiple of the denominators of the fractions in question, which allows for straightforward addition or subtraction.

• Identify the least common multiple (LCM) of all the denominators in the fractions you are working with.
• Convert each fraction so that they all have the same denominator as the LCM.

In our exercise, the least common multiple of 3 and 12 is 12, so we converted both fractions to have a denominator of 12. This ensures comparability and simplifies addition or subtraction.

Simplest Form

The simplest form of a fraction occurs when both the numerator and the denominator have no common factors other than 1.
This step comes in handy after performing arithmetic operations, ensuring that the fraction is as reduced as possible, offering a clear representation of the value.

• To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD to reduce the fraction.

In our exercise, the resulting fraction \(\frac{-13}{12}\) was already in its simplest form as 13 and 12 share no common factors. Always check if the final result can be simplified further, as it makes subsequent calculations easier.