Problem 54 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or as a mixed number in simplest form. $$ 1 \frac{1}{4}+2 \frac{5}{8} $$

Step-by-Step Solution

Answer

The simplest form of the sum is \(3 \frac{7}{8}\)

Step 1: Identify the components

In this problem, we have two mixed numbers: 1 and 1/4 is the first mixed number while 2 and 5/8 is the second mixed number. A mixed number is a whole number plus a fraction.

Step 2: Add the whole numbers

Let's add the whole numbers first. This will be 1 + 2 = 3.

Step 3: Add the fractions

Next, add the fractions. This requires the fractions to have a common denominator. The least common multiple (LCM) of the denominators 4 and 8 is 8. The fraction 1/4 needs to be converted to have 8 as the denominator. Multiply both the numerator and denominator of 1/4 by 2 to get 2/8. Now, add 2/8 and 5/8 together to get 7/8.

Step 4: Combine the sums

Now, add the total from the fractions to the whole numbers. With 3 as the sum of the whole numbers and 7/8 as the sum of the fractions, adding these together gives 3 7/8 as the final answer. This is already in simplest form, so no further simplification is necessary.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction placed together. It looks like this: say you have something like 1 \(\frac{1}{4}\). Here, the '1' is your whole number, and \(\frac{1}{4}\) is your fraction.
When dealing with mixed numbers, the first step is always to handle the whole numbers separately from the fractions. This means you add the whole parts together and then deal with the fractional parts separately.
This separation helps keep the numbers straightforward and clear as you work through the problem, ultimately recombining them at the end to find your final answer. So anytime you're faced with mixed numbers, think of them as two distinct parts that only come together at the very end.

Least Common Multiple

Finding the least common multiple (LCM) is crucial when you're adding fractions with different denominators. The LCM helps you find the smallest number that both denominators can divide into evenly. This number becomes your new common denominator.
Let's take an example with the denominators 4 and 8. You list the multiples of each number until you find the smallest shared one:

• Multiples of 4: 4, 8, 12, 16, ...
• Multiples of 8: 8, 16, 24, ...

In this case, 8 is the first shared multiple both 4 and 8 divide into evenly, so it's your LCM. Now, knowing the LCM allows you to convert each fraction to have this common denominator, making it so much easier to add those fractions together.
Once you've moved your fractions to the common denominator, you can proceed to add their numerators, since the number on the bottom (denominator) is the same.

Simplifying Fractions

Once you've added fractions together, the next step is often to simplify the result. Simplifying means reducing your fraction to its smallest form while making sure that it still represents the same value.
To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator (the top number) and the denominator (the bottom number). For instance, if you end up with a fraction like \(\frac{10}{20}\), you'd look for the largest number that divides both 10 and 20 without leaving a remainder, which is 10 in this case.
Divide both the numerator and the denominator by their GCD:

• \(\frac{10}{20} = \frac{10 \div 10}{20 \div 10} = \frac{1}{2}\)

This is the simplified form of the fraction, meaning it's as "small" as it can get. Simplified fractions are always preferable because they're easier to read and work with, especially in larger calculations.