Problem 16 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(5 \frac{4}{5}-3 \frac{1}{3}\)

Step-by-Step Solution

Answer

The result is \(2 \frac{7}{15}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For the first mixed number, \(5 \frac{4}{5}\), multiply the whole number (5) by the denominator (5), and add the numerator (4): \(5 \cdot 5 + 4 = 29\). So, \(5 \frac{4}{5}\) becomes \(\frac{29}{5}\). For the second mixed number, \(3 \frac{1}{3}\), multiply the whole number (3) by the denominator (3), and add the numerator (1): \(3 \cdot 3 + 1 = 10\). So, \(3 \frac{1}{3}\) becomes \(\frac{10}{3}\).

Step 2: Find a Common Denominator

Find the least common denominator (LCD) of the fractions \(\frac{29}{5}\) and \(\frac{10}{3}\). The denominators are 5 and 3; hence, the LCD is 15.

Step 3: Convert Fractions to Have the Common Denominator

Convert each fraction to have the common denominator of 15. For \(\frac{29}{5}\), multiply both numerator and denominator by 3 to get \(\frac{87}{15}\). For \(\frac{10}{3}\), multiply both numerator and denominator by 5 to get \(\frac{50}{15}\).

Step 4: Subtract the Improper Fractions

Subtract the fractions: \(\frac{87}{15} - \frac{50}{15} = \frac{37}{15}\).

Step 5: Convert the Result to a Mixed Number

Convert \(\frac{37}{15}\) back into a mixed number. Divide 37 by 15, which gives 2 as the whole number and a remainder of 7. So, \(\frac{37}{15}\) converts to \(2 \frac{7}{15}\).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). They often appear when converting mixed numbers, which combine a whole number with a proper fraction.
To understand improper fractions, think about having a whole pizza cut into slices. If each pizza has 8 slices, and you have 10 slices in total, then you technically have more than one pizza, represented as the improper fraction \(\frac{10}{8}\).
In our exercise, converting mixed numbers to improper fractions is a crucial first step. For \(5 \frac{4}{5}\), multiply the whole number 5 by the denominator 5 to get 25, then add the numerator 4, giving us \(\frac{29}{5}\). Similarly, \(3 \frac{1}{3}\) becomes \(\frac{10}{3}\) after calculation. This sets the stage for operations like adding or subtracting fractions.

Least Common Denominator

The least common denominator (LCD) is the smallest number that both denominators of two or more fractions can divide into without leaving a remainder. It is essential in fraction arithmetic to ensure the fractions have the same base for addition or subtraction.
To find the LCD, list the multiples of each denominator and identify the smallest common multiple. In our example:

• For 5: 5, 10, 15, 20, 25, ...
• For 3: 3, 6, 9, 12, 15, ...

Here, 15 is the smallest number common to both lists. Once the LCD is found, fractions are converted to have this denominator, facilitating straightforward comparison and arithmetic.

Converting Fractions

Converting fractions to have a common denominator is essential when adding or subtracting fractions with different denominators. This process permits operations to occur seamlessly by ensuring each fraction "speaks the same language."
In practice, converting the fractions \(\frac{29}{5}\) and \(\frac{10}{3}\) to a denominator of 15 involves multiplying each fraction appropriately.

• Multiply \(\frac{29}{5}\) by \(\frac{3}{3}\) (essentially multiplying by one) to get \(\frac{87}{15}\).
• Similarly, multiply \(\frac{10}{3}\) by \(\frac{5}{5}\) to yield \(\frac{50}{15}\).

These resulting fractions can now be easily added or subtracted, owing to their shared denominator.

Fraction Subtraction

Fraction subtraction follows similar rules to subtraction of whole numbers, but it requires fractions to have the same denominator. Once fractions have been converted to have the same denominator, subtraction is straightforward.
In our exercise, we have \(\frac{87}{15} - \frac{50}{15}\). Both fractions share a denominator of 15, allowing us to subtract their numerators while keeping the denominator constant:
\(87 - 50 = 37\), thus the result is \(\frac{37}{15}\).
Sometimes, simplifying or converting the result into a mixed number is necessary for clarity or standard form. By dividing 37 by 15, we get 2 with a remainder of 7, so \(\frac{37}{15}\) becomes \(2 \frac{7}{15}\). This final form might resemble the initial problem but concludes our subtraction with a straightforward representation.

• Start by ensuring like denominators.
• Subtract the numerators.
• Simplify or convert the result if necessary.