Problem 33 - Prealgebra

Question

Find \(1 \frac{1}{4}+4 \frac{5}{6} .\) Write in simplest form.

Step-by-Step Solution

Answer

The sum is \(6 \frac{1}{12}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers into improper fractions. For \(1 \frac{1}{4}\), multiply the whole number 1 by the denominator 4 and add the numerator 1 to get \( \frac{5}{4} \). For \(4 \frac{5}{6}\), multiply the whole number 4 by the denominator 6 and add the numerator 5 to get \( \frac{29}{6} \).

Step 2: Find a Common Denominator

To add \( \frac{5}{4} \) and \( \frac{29}{6} \), find a common denominator. The smallest common multiple of 4 and 6 is 12. Convert \( \frac{5}{4} \) to \( \frac{15}{12} \) and \( \frac{29}{6} \) to \( \frac{58}{12} \).

Step 3: Add the Improper Fractions

Add the fractions with the common denominator. This gives \( \frac{15}{12} + \frac{58}{12} = \frac{73}{12} \).

Step 4: Simplify the Improper Fraction

Convert the improper fraction \( \frac{73}{12} \) into a mixed number. Divide 73 by 12 to get 6 with a remainder of 1. Therefore, \( \frac{73}{12} \) can be simplified to \(6 \frac{1}{12}\).

Key Concepts

Improper Fractions

An improper fraction is a fraction where the numerator is larger than or equal to the denominator. This means the fraction represents a number greater than or equal to one. Improper fractions can easily be spotted because the top number is big compared to the bottom number.
To understand them, think of eating pizza slices. If each pizza is cut into quarters, 5 slices would make more than one whole pizza or one 'improper fraction'. For instance, if you have 5 out of 4 slices, you have an improper fraction, \(\frac{5}{4}\).
Improper fractions are crucial when performing arithmetic operations. They help simplify the process of addition and subtraction by allowing math to focus on numerators primarily. When working with problems like \(1 \frac{1}{4} + 4 \frac{5}{6}\), converting to improper fractions simplifies operations involving these mixed numbers.

Mixed Numbers

Mixed numbers are composed of a whole number and a fraction. An example is \(1 \frac{1}{4}\), which means one whole unit plus one part out of four parts.
The mix of whole numbers and fractions make them nice to look at but can be tricky when doing calculations.
To add or subtract mixed numbers, it's often easier to first convert them to improper fractions. This makes the math more straightforward. After the calculation, converting them back into mixed numbers gives a clear result, which is easier to understand. Mixed numbers tell us precisely how much 'more than whole' a quantity is, which is why you'll often see recipes use them.

Common Denominator

When adding or subtracting fractions, finding a common denominator is essential. The denominator is the bottom part of the fraction, and it shows the number of equal parts the whole is divided into.
To easily add fractions, the denominators must be the same so that the pieces being counted are the same size. This shared value is known as the 'common denominator'. For instance, to add \(\frac{5}{4}\) and \(\frac{29}{6}\), we find their common denominator to be 12.
Here's why it's important:

• It helps align the fractions to have the same 'size' parts.
• Having the same denominator allows direct addition or subtraction of the fractions.

Once both fractions have the common denominator, sum them up by adding the numerators.

Simplifying Fractions

Simplifying a fraction means reducing it to its smallest form, making it as clear and concise as possible. For improper fractions, this often involves converting them into mixed numbers.
Take \(\frac{73}{12}\) as an example. When you divide 73 by 12, you get 6 with a remainder. This remainder becomes the numerator of the fractional part, and the divisor (12) is the denominator.
Why simplify? Well:

• Simplified fractions are easier to read and understand.
• They provide a complete visual representation of the number.
• Simplification is often required for final answers in math problems.

For mixed numbers like \(6 \frac{1}{12}\), it’s clear that there are 6 wholes and 1 out of 12 parts in addition.