Problem 46 - Prealgebra

Question

\(5 \frac{5}{6}+3 \frac{5}{6}-2 \frac{1}{6}\)

Step-by-Step Solution

Answer

The result is \(7 \frac{1}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Start by turning each mixed number into an improper fraction. \(5 \frac{5}{6} = \frac{35}{6}\), \(3 \frac{5}{6} = \frac{23}{6}\), and \(2 \frac{1}{6} = \frac{13}{6}\).

Step 2: Add the Improper Fractions

Add the first two improper fractions: \(\frac{35}{6} + \frac{23}{6}\). Since they have the same denominator, simply add the numerators: \(35 + 23 = 58\). So, the sum is \(\frac{58}{6}\).

Step 3: Subtract the Third Fraction

Subtract \(\frac{13}{6}\) from \(\frac{58}{6}\). Since they have the same denominator, subtract the numerators: \(58 - 13 = 45\). So, the result is \(\frac{45}{6}\).

Step 4: Simplify the Resulting Fraction

Simplify the fraction \(\frac{45}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3: \(\frac{45}{6} = \frac{15}{2}\).

Step 5: Convert Back to Mixed Number

Convert the improper fraction \(\frac{15}{2}\) back to a mixed number. Divide 15 by 2, which is 7 with a remainder of 1, so \(\frac{15}{2} = 7 \frac{1}{2}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often used in day-to-day life to represent quantities that are more than a whole but less than two wholes.
For example, when we say we walked 2 miles and another half, we essentially mean 2 mixed with 1/2, which is a mixed number. To work with mixed numbers in mathematical operations like addition or subtraction, it's simpler to convert them to improper fractions first.

• The whole number part tells you how many whole parts there are.
• The fraction tells you about any extra parts.
• The improper fraction representation makes mathematical operations easier.

Understanding mixed numbers is fundamental, particularly when performing tasks such as cooking or measuring, where portions may not be whole.

Improper Fractions

An improper fraction is where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, \(\frac{15}{2}\) is an improper fraction because 15 is greater than 2.
This type of fraction signifies an amount that is more than or equal to a whole. Improper fractions can easily be converted back and forth into mixed numbers.To convert a mixed number into an improper fraction, follow these steps:

• Multiply the whole number by the denominator.
• Add the numerator to the result.
• Place this sum over the original denominator.

For example, converting \(5 \frac{5}{6}\) to improper fractions results in \(\frac{35}{6}\). This allows for straightforward operations in addition or subtraction.

Simplifying Fractions

Simplifying a fraction is the process of making the fraction as simple as possible. We do this by finding the greatest common divisor (GCD) of the numerator and the denominator.
Then, we divide both by this number to reduce the fraction to its simplest form.For example, to simplify \(\frac{45}{6}\):

• Find the GCD of 45 and 6, which is 3.
• Divide the numerator and the denominator by their GCD.
• This results in \(\frac{15}{2}\).

Simplifying fractions helps in recognizing proportions more easily and makes numbers less cumbersome to work with in calculations.

Addition and Subtraction of Fractions

Adding and subtracting fractions involves a few important steps, especially when dealing with mixed numbers and improper fractions.
First, ensure that the fractions have the same denominator. This makes addition or subtraction straightforward. If they don't share the same denominator, you will need to find a common one.Here’s how you add or subtract fractions:

• Add or subtract the numerators together while keeping the denominator the same.
• In events where fractions have different denominators, convert them to common denominators first.
• Finally, simplify the resulting fraction if necessary.

In the given problem, because all fractions share a denominator of 6, adding \(\frac{23}{6}\) to \(\frac{35}{6}\) and subtracting \(\frac{13}{6}\) from the result, can be executed by merely focusing on the numerators, then simplifying the final outcome.