Problem 33 - Prealgebra

Question

Find the following sums. (Add.) \(1 \frac{1}{4}+2 \frac{3}{4}+5\)

Step-by-Step Solution

Answer

The sum is 9.

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For the expression, we have two mixed numbers: \(1 \frac{1}{4}\) and \(2 \frac{3}{4}\).To convert \(1 \frac{1}{4}\), multiply the whole number 1 by the denominator 4 and add the numerator 1, resulting in \(\frac{5}{4}\).To convert \(2 \frac{3}{4}\), multiply the whole number 2 by the denominator 4 and add the numerator 3, resulting in \(\frac{11}{4}\).

Step 2: Rewrite the Expression

Now, rewrite the expression using the improper fractions obtained from the conversion in Step 1. The expression becomes:\[\frac{5}{4} + \frac{11}{4} + 5\]

Step 3: Add the Fractions

Ensure the fractions have a common denominator to add them. Both \(\frac{5}{4}\) and \(\frac{11}{4}\) have the same denominator (4), so we can directly add numerators together:\[\frac{5}{4} + \frac{11}{4} = \frac{16}{4} = 4\]

Step 4: Add the Whole Number

Add the whole number 5 from the original expression to the sum of the fractions obtained (which is 4 in the previous step):\[4 + 5 = 9\]

Step 5: Final Result

The sum of the expression \(1 \frac{1}{4} + 2 \frac{3}{4} + 5\) is 9.

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 value greater than or equal to 1. Improper fractions are useful because they can make calculation processes more straightforward, especially when adding or subtracting fractions.
For example, when we converted the mixed number \(1 \frac{1}{4}\) into an improper fraction, we ended up with \(\frac{5}{4}\). Here, 5 is greater than 4, so it's an improper fraction.
To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator.
• Add the product to the numerator.
• The sum becomes the new numerator, while the denominator stays the same.

By converting mixed numbers into improper fractions, arithmetic operations like addition become more manageable.

Mixed Numbers

Mixed numbers consist of a whole number and a fractional part separated by a space or a plus sign. They are useful for representing numbers that are between whole numbers, providing a clear visual cue for their fractional value.
For instance, in the problem, we started with the mixed numbers \(1 \frac{1}{4}\) and \(2 \frac{3}{4}\). These represent 1 whole plus a quarter, and 2 wholes plus three quarters, respectively.
Although mixed numbers are easy to understand at a glance, calculations like addition and subtraction are simpler with improper fractions.
Thus, converting to improper fractions becomes crucial, especially in adding, as it removes the complexity of dealing with both whole numbers and fractions at the same time.

Common Denominator

A common denominator is essential when adding or subtracting fractions. Having fractions with the same denominator makes it possible to simply add or subtract the numerators and leave the denominator unchanged.
In the exercise, the fractions \(\frac{5}{4}\) and \(\frac{11}{4}\) already have a shared denominator of 4. This means you can directly add the numerators:

• \(5 + 11 = 16\)

This simplifies the expression to \(\frac{16}{4}\), which can be further simplified, as it results in a whole number.
Always ensure to check if fractions have a common denominator before proceeding with addition or subtraction. If they don't, you will need to find a common denominator first by determining a common multiple of the denominators.

Whole Numbers in Fractions

Sometimes, in mathematical problems, whole numbers are presented alongside fractions. In these scenarios, understanding how to deal with whole numbers when performing operations with fractions is necessary.
In our exercise, the whole number 5 appears after adding the fractions. It's crucial to keep whole numbers separate until the process of adding fractions is complete.
Once the fractions \(\frac{5}{4}\) and \(\frac{11}{4}\) are added to become \(\frac{16}{4}\), the result simplifies to 4, a whole number itself. Then, adding this result to the original whole number:

• \(4 + 5 = 9\)

This eventually gives the final answer. Separating whole numbers until the end simplifies calculations and helps avoid confusion.