Problem 30 - Prealgebra

Question

The problems below are a review of the four basic operations with fractions and decimals. Add $$\frac{3}{4}+\frac{1}{6}+\frac{5}{8}$$

Step-by-Step Solution

Answer

The sum is \(1\frac{13}{24}\).

Step 1: Find a Common Denominator

To add fractions, they need to have the same denominator. The denominators here are 4, 6, and 8. The least common multiple (LCM) of these numbers is 24. Therefore, we need to convert each fraction to have a denominator of 24.

Step 2: Convert Fractions

Convert each fraction to an equivalent fraction with a denominator of 24. For \( \frac{3}{4} \), multiply the numerator and the denominator by 6: \( \frac{3}{4} = \frac{18}{24} \). For \( \frac{1}{6} \), multiply by 4: \( \frac{1}{6} = \frac{4}{24} \). For \( \frac{5}{8} \), multiply by 3: \( \frac{5}{8} = \frac{15}{24} \).

Step 3: Add the Fractions

Now, add the fractions with the common denominator: \[ \frac{18}{24} + \frac{4}{24} + \frac{15}{24} = \frac{18 + 4 + 15}{24} = \frac{37}{24}. \]

Step 4: Simplify the Result

The fraction \( \frac{37}{24} \) is an improper fraction. Simplify it by expressing it as a mixed number. Divide 37 by 24 to get 1 remainder 13, so \[ \frac{37}{24} = 1\frac{13}{24}. \]

Key Concepts

Adding Fractions

Adding fractions can be tricky, but once you understand the main steps, it becomes straightforward. When you add fractions, you are essentially combining parts of a whole. However, these parts, or fractions, need to be measured in the same terms, which means they need a common denominator. For example, think of fractions as pieces of pie. If one fraction is in eighths and another in fourths, it's like comparing apples to oranges. You need to convert them so they are in the same "units" before adding them together. Once the denominators are the same, you simply add the numerators. A simple rule to remember is: only numerators are added, and the denominator stays the same.

Least Common Multiple

Before adding fractions with different denominators, you need to find the lowest term in which all denominators can be expressed. This is called the Least Common Multiple (LCM). The LCM is essentially the smallest number divisible by all the denominators involved. Here’s how you can calculate it:

• List the multiples of each denominator.
• Find the smallest multiple that appears in each list.

In the given problem, the denominators are 4, 6, and 8. The LCM of these numbers is 24 since 24 is the smallest number that all three can divide into without leaving a remainder. Once you have the LCM, you convert all fractions to have this as their new denominator.

Improper Fractions

An improper fraction is where the numerator, the top number, is greater than or equal to the denominator, the bottom number. Once you have added your fractions and reached a result, it's essential to check if you have an improper fraction. If your fraction's numerator is larger than the denominator, then it's improper. In this situation, you can convert it to a mixed number to make it easier to understand. For instance, if you finish with \( \frac{37}{24} \), divide 37 by 24. This gives you the mixed number \( 1 \frac{13}{24} \), where 1 is the whole number, and \( \frac{13}{24} \) is the fractional part.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator are as small as they can be while still being the same fraction. In some cases, after adding fractions, the result is already in its simplest form. However, if both your numerator and denominator have a common factor, you can divide them by this factor to simplify. Use a method called "prime factorization" to find common factors, which involves breaking down numbers into their basic prime numbers.
For example, if you have a fraction like \( \frac{20}{28} \), both can be divided by their Greatest Common Divisor (GCD), which is 4. So, simplifying \( \frac{20}{28} \) gives you \( \frac{5}{7} \). Although our example \( \frac{37}{24} \) cannot be simplified further, it's still good practice to always check.