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Chapter 5: Problem 2

Find each sum. Write in simplest form. $$\frac{1}{10}+\frac{1}{3}$$

Short Answer

Expert verified
The sum is \( \frac{13}{30} \) in simplest form.

Step by step solution

01

Identify the Problem

We need to find the sum of two fractions: \( \frac{1}{10} \) and \( \frac{1}{3} \), and then express the result in its simplest form.
02

Find a Common Denominator

To add fractions, they need to have the same denominator. The denominators here are 10 and 3. The least common multiple (LCM) of 10 and 3 is 30, so we will use 30 as the common denominator.
03

Convert Fractions to a Common Denominator

Convert \( \frac{1}{10} \) to a fraction with 30 as the denominator by multiplying both the numerator and denominator by 3: \[ \frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30} \]. Convert \( \frac{1}{3} \) by multiplying both the numerator and denominator by 10: \[ \frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30} \].
04

Add the Fractions

Now, add the two fractions with a common denominator: \[ \frac{3}{30} + \frac{10}{30} = \frac{3 + 10}{30} = \frac{13}{30} \].
05

Simplify the Result

The fraction \( \frac{13}{30} \) is in its simplest form since 13 and 30 have no common divisors other than 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When you want to add fractions with different denominators, finding a common denominator is essential. A *common denominator* is a shared multiple of the denominators of the fractions you are working with. Imagine it's like finding a common language for two people to communicate.

Here's how you do it:
  • First, identify the denominators of your fractions. In our example, they are 10 and 3.
  • Next, determine the least common multiple (LCM) of these two numbers, which will be used as the common denominator.
  • Then, adjust each fraction so that its denominator matches the LCM.
Using a common denominator allows you to directly add the numerators of your fractions together. This makes the process of finding their sum simple and straightforward!
Simplifying Fractions
Simplifying fractions is the process of reducing a fraction to its simplest form. This means converting it so that the numerator and the denominator have no common divisors, other than 1. Simplifying a fraction not only makes it look neater but often easier to understand and work with further.

Here's the way to simplify:
  • Check if the numerator and the denominator have any common factors.
  • Divide both the numerator and the denominator by their greatest common factor (GCF).
In our exercise, the result \( \frac{13}{30} \) is already in its simplest form. This is because 13 is a prime number, which only has 1 and 13 as factors, while 30 does not share any of these. Thus, no further simplification is needed.
Least Common Multiple (LCM)
Finding the least common multiple (LCM) is a key step in operating with fractions, particularly when adding or subtracting them. The LCM is the smallest number that is a multiple of each of the numbers in question. It鈥檚 like choosing a playground where all children with different hobbies can play together.

To find the LCM:
  • List a few multiples of each number. For 10, the multiples are 10, 20, 30, 40, etc. For 3, they are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, etc.
  • Identify the smallest multiple they have in common. In our example, both 10 and 3 share the number 30.
The LCM ensures both numbers can "meet" at an equivalent value, allowing you to adjust the fractions so they have this shared denominator. Finding an LCM streamlines the addition process and ensures accuracy in calculation.

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