Problem 14 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$\frac{3}{5}+\frac{3}{10}$$

Step-by-Step Solution

Answer

\( \frac{9}{10} \)

Step 1: Find a Common Denominator

To add fractions, we need a common denominator. The denominators are 5 and 10. The least common multiple of 5 and 10 is 10. So, we will use 10 as the common denominator.

Step 2: Convert Fractions

Convert the first fraction, \( \frac{3}{5} \), to have a denominator of 10. To do this, multiply both the numerator and denominator by 2: \( \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \). The second fraction, \( \frac{3}{10} \), already has 10 as the denominator.

Step 3: Add the Fractions

Now, add the fractions with the same denominator: \( \frac{6}{10} + \frac{3}{10} = \frac{6+3}{10} = \frac{9}{10} \).

Step 4: Simplify the Fraction

The fraction \( \frac{9}{10} \) is already in its simplest form, as 9 and 10 have no common divisors other than 1.

Key Concepts

Understanding Common Denominators

When adding or subtracting fractions, fractions must have the same bottom number, known as the denominator. This shared denominator is called a common denominator. Think about a common denominator like finding a common vocabulary to communicate effectively between fractions. Look at the given exercise:

• The fractions are \( \frac{3}{5} \) and \( \frac{3}{10} \).
• We need to adjust these fractions to have the same denominator.

By achieving this, it allows us to combine or subtract fractions seamlessly. Imagine you’re trying to add quarters and dimes; you need to think in cents to do so easily. Finding a common denominator typically revolves around finding the least common multiple (LCM), which we'll discuss next. With a common denominator, both fractions speak the same 'language'. This permits adding their numerators together smoothly, as done in the exercise above. The process doesn’t change their value, just their form. After aligning the denominators, you add or subtract across the numerators.

Converting to the Simplest Form

Once you’ve added or subtracted fractions, simplifying the resulting fraction to its simplest form is crucial. Simplifying a fraction means reducing it to its lowest terms. Think of it as cleaning up your work to make it look nice and tidy. For example, the exercise results in the fraction \[ \frac{9}{10} \]Simplifying a fraction requires checking if both the numerator and the denominator have a common factor other than 1.

• Here, 9 and 10 have no common divisors other than 1.
• Thus, \( \frac{9}{10} \) is already its simplest form.

In essence, a fraction is simplified when no smaller whole number can divide both the numerator and the denominator without leaving a remainder. Simplifying is like ensuring that you've made your answer as neat and condensed as possible. Always double-check your fractions for potential reduction to avoid any overlooked simplifications.

Determining the Least Common Multiple

In order to establish a common denominator, the least common multiple is key. The least common multiple, or LCM, is the smallest number that is a multiple of two or more numbers. Consider the denominators of the fractions in the problem: 5 and 10. To find the LCM:

• List the multiples of 5: 5, 10, 15, 20, 25, ...
• List the multiples of 10: 10, 20, 30, ...
• The smallest multiple common to both sets is 10.

Using the LCM as the common denominator ensures that you're using the smallest possible number to make fraction addition easy. A large number could work, but smaller numbers are efficient. Identifying the LCM saves time and effort, simplifying these operations and allowing fractions to be added or subtracted directly once they have like denominators.