Problem 49 - Prealgebra

Question

Simplify each of the following complex fractions. [Examples 5–7] $$\frac{3 \frac{1}{4}+5 \frac{1}{6}}{2 \frac{1}{3}+3 \frac{1}{4}}$$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{101}{67}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to change the mixed numbers into improper fractions so that the operations are manageable. \(3 \frac{1}{4}\) can be expressed as \(\frac{13}{4}\) because \(3 \times 4 + 1 = 13\). Likewise, \(5 \frac{1}{6}\) becomes \(\frac{31}{6}\), \(2 \frac{1}{3}\) becomes \(\frac{7}{3}\), and \(3 \frac{1}{4}\) becomes \(\frac{13}{4}\).

Step 2: Add Fractions in the Numerator

Now, add the fractions in the numerator: \(\frac{13}{4} + \frac{31}{6}\). To add these fractions, find a common denominator, which is 12. Convert and add: \(\frac{39}{12} + \frac{62}{12} = \frac{101}{12}\).

Step 3: Add Fractions in the Denominator

Similarly, add the fractions in the denominator: \(\frac{7}{3} + \frac{13}{4}\). Again, the common denominator is 12. Convert and add: \(\frac{28}{12} + \frac{39}{12} = \frac{67}{12}\).

Step 4: Divide the Resultant Fractions

Divide the simplified numerator by the simplified denominator: \(\frac{101}{12} \div \frac{67}{12}\). This is equivalent to multiplying by the reciprocal of the divisor: \(\frac{101}{12} \times \frac{12}{67} = \frac{101}{67}\).

Step 5: Simplify the Result

Finally, check if \(\frac{101}{67}\) can be simplified. Since there are no common factors between 101 and 67, it remains \(\frac{101}{67}\).

Key Concepts

Mixed Numbers

A mixed number combines a whole number and a fraction. For example, in the mixed number \(3 \frac{1}{4}\), 3 is the whole number, and \(\frac{1}{4}\) is the fraction part. Mixed numbers are often used in everyday life to represent quantities like "3 and a quarter". They're easier for some to understand at a glance because they show the whole and fractional parts clearly.

However, when performing arithmetic operations like addition or multiplication, converting them to improper fractions makes the calculation more straightforward. You convert a mixed number to an improper fraction by multiplying the whole number by the denominator of the fraction part, then adding the numerator. For \(3 \frac{1}{4}\), multiply 3 by 4, then add 1, giving you 13. The improper fraction is \(\frac{13}{4}\).

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

Improper Fractions

Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, \(\frac{13}{4}\) is an improper fraction because 13 is greater than 4.

Improper fractions are useful because they can simplify arithmetic operations like addition, subtraction, multiplication, and division with fractions. Converting a mixed number to an improper fraction allows for easier manipulation, especially when finding common denominators or performing division of complex fractions.

In this exercise, mixed numbers like \(3 \frac{1}{4}\) and \(5 \frac{1}{6}\) were converted to improper fractions (\(\frac{13}{4}\) and \(\frac{31}{6}\)) to aid in the addition and division processes.

Adding Fractions

Adding fractions involves finding a common denominator so that you can combine the numerators (the top numbers). When the denominators are different, as seen with \(\frac{13}{4}\) and \(\frac{31}{6}\), you need to find the least common multiple (LCM) of the denominators to create equivalent fractions with the same denominator.

For these fractions, the LCM of 4 and 6 is 12, so we convert them to \(\frac{39}{12}\) and \(\frac{62}{12}\), respectively. Once they share a common denominator, simply add the numerators: \(39 + 62 = 101\). The result is \(\frac{101}{12}\).

Always remember:

• Find the lowest common denominator (LCD).
• Convert each fraction to an equivalent fraction with the LCD.
• Add the numerators, keeping the denominator the same.

Reciprocal

The reciprocal of a fraction is what you get when you switch the numerator and the denominator. In simpler terms, if you have a fraction like \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). The reciprocal concept is crucial when dividing fractions because dividing by a fraction is the same as multiplying by its reciprocal.

In the given exercise, once the fractions in the numerator and denominator are simplified, you need to divide \( \frac{101}{12} \) by \( \frac{67}{12} \). Instead of dividing, multiply \( \frac{101}{12} \) by the reciprocal of \(\frac{67}{12}\), which is \(\frac{12}{67}\). This simplifies division into multiplication: \(\frac{101}{12} \times \frac{12}{67} = \frac{101}{67}\).

Using the reciprocal:

• Simplifies complex division of fractions.
• Transforms division into a multiplication problem.

Remember, multiplying fractions is more straightforward than dividing them, making this technique highly efficient.