Problem 10 - Prealgebra

Question

Find each difference. Write in simplest form. $$-9 \frac{3}{4}-\left(-5 \frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The difference is \(-4 \frac{1}{4} \).

Step 1: Understand the problem

The exercise requires us to subtract two mixed numbers. The problem is presented as \(-9 \frac{3}{4} - \left(-5 \frac{1}{2}\right)\). The goal is to find this difference and simplify it if necessary.

Step 2: Simplify the subtraction using addition

The expression \(-9 \frac{3}{4} - \left(-5 \frac{1}{2}\right)\) can be simplified by remembering that subtracting a negative is equivalent to adding a positive. So, we rewrite this as \(-9 \frac{3}{4} + 5 \frac{1}{2}\).

Step 3: Convert mixed numbers to improper fractions

First convert \(-9 \frac{3}{4}\) to an improper fraction. Multiply 9 by 4 and add 3: \(-9 \frac{3}{4} = -\left( (9 \times 4) + 3 \right)/4 = -39/4\).Next, convert \(5 \frac{1}{2}\) to an improper fraction.Multiply 5 by 2 and add 1:\(5 \frac{1}{2} = \left( (5 \times 2) + 1 \right)/2 = 11/2\).For consistency, convert \(11/2\) to a denominator of 4:\(11/2 = (11 \times 2)/(2 \times 2) = 22/4\).

Step 4: Add the improper fractions

Now add the improper fractions:\(-39/4 + 22/4\).Combine the numerators:\((-39) + 22 = -17\).So, the result is \(-17/4\).

Step 5: Simplify the fraction

Since \(-17/4\) is already in its simplest form (as 17 and 4 have no common factors), there's no need for further simplification. However, if needed, we can convert this back into a mixed number:\(-17/4 = -4\frac{1}{4}\).

Key Concepts

mixed numbers

Mixed numbers are a combination of a whole number and a proper fraction. In our exercise, we encounter negative mixed numbers, such as

• -9 \(\frac{3}{4}\)
• 5 \(\frac{1}{2}\)

Mixed numbers are often used because they provide an easier way of interpreting quantities that are more than one but not whole. The whole number and fraction parts are added together to express these values. To calculate using them, it's generally easier to first convert them into improper fractions. But remember to keep track of the sign, especially when dealing with negative numbers.

improper fractions

Improper fractions occur when the numerator, which is the top part of a fraction, is larger than the denominator, the bottom part. This is a necessary conversion for simplifying calculations with fractions.

• For example, to convert -9 \(\frac{3}{4}\) to an improper fraction, you multiply the whole number by the denominator and then add the numerator: \(9 \times 4 + 3 = 39\), resulting in \(-\frac{39}{4}\).
• Similarly, 5 \(\frac{1}{2}\) becomes \(\frac{11}{2}\) after converting.

These conversions make calculations like addition or subtraction straightforward—a uniform denominator helps in the process.

simplifying fractions

Simplifying fractions is cutting down a fraction to its smallest possible form. This involves reducing the fraction to the point where the numerator and denominator have no common factors besides 1.
In our example, we combined the fractions \(-\frac{39}{4} \) and \(\frac{22}{4}\), which resulted in the fraction \(-\frac{17}{4}\). Since 17 is a prime number with no common factors with 4, the fraction is already simplified.
If the fraction was not in its simplest form, we would divide both the numerator and the denominator by their greatest common factor. This process ensures clarity and conciseness in mathematical expressions.

addition of fractions

Adding fractions requires a common denominator so you can directly add the numerators. In this exercise, converting mixed numbers to improper fractions allowed us to maintain a common denominator for straightforward addition.
First, ensure both fractions have the same lower number. For instance, converting \(\frac{11}{2}\) to \(\frac{22}{4}\) aligns it with our \(-\frac{39}{4}\) denominator. With the same base, you simply add the tops:

• \(-39 + 22\)

This results in \(-\frac{17}{4}\). Aligning denominators is key since it ensures that the fractions are pieces of the whole divided in the same way, allowing you to sum these pieces properly.