Question

Simplify \(\frac{2 x-5}{10}-\frac{3 x-2}{15}\)

Short answer

Question: Simplify the expression: \(\frac{2x - 5}{10} - \frac{3x - 2}{15}\) Answer: \(\frac{-11}{30}\)

Step-by-step solution

Find the least common multiple (LCM) of the denominators

To combine the two fractions, we first need to find a common denominator. In this exercise, the denominators are 10 and 15. The least common multiple (LCM) of 10 and 15 is 30. This is the smallest number that both 10 and 15 evenly divide into.

Rewrite both fractions with a common denominator

Now that we have the common denominator (30), we will rewrite both fractions with the denominator of 30. To do this, we multiply the numerator and denominator of each fraction by the necessary factor to make the denominator 30: \(\frac{2x - 5}{10} \cdot \frac{3}{3} = \frac{6x - 15}{30}\) \(\frac{3x - 2}{15} \cdot \frac{2}{2} = \frac{6x - 4}{30}\)

Combine the fractions

With both fractions rewritten with a common denominator, we can now combine them: \(\frac{6x - 15}{30} - \frac{6x - 4}{30} = \frac{(6x - 15) - (6x - 4)}{30}\)

Simplify the numerator

Now, we'll simplify the numerator by combining like terms: \((6x - 15) - (6x - 4) = 6x - 15 - 6x + 4 = 0x - 11 = -11\)

Write the final simplified expression

Lastly, we can write the final simplified expression based on the new numerator and denominator, which is: \(\frac{-11}{30}\)

Key concepts

Common Denominator

When working with fractions that have different denominators, it's crucial to find a common denominator to combine them effectively. A common denominator is a number that is a multiple of both denominators you're working with. This concept allows you to compare or combine fractions with ease. In the given exercise, the denominators are 10 and 15. To find a common denominator, we're looking for a number that both of these denominators can divide into evenly.

• The denominators 10 and 15 can both divide into 30 without leaving a remainder, making it a common denominator.
• This idea is vital because it enables accurate addition or subtraction of fractions.

Once the common denominator is identified, we can rewrite each fraction so that they share the same denominator, thus preparing them to be combined through addition or subtraction. This shared base is key to simplifying the arithmetic with fractions.

Least Common Multiple

The least common multiple (LCM) is a term often used in math when working with fractions. It refers to the smallest multiple that two or more numbers can share. In the task of simplifying or comparing fractions, finding the LCM of the denominators ensures we use the smallest possible number as the common denominator, making calculations simpler.
Let's look at the denominators in the exercise:

• For 10, the multiples are: 10, 20, 30, 40, ...
• For 15, the multiples are: 15, 30, 45, 60, ...
• The smallest shared number is 30, which is the LCM.

Using the least common multiple minimizes the need to work with very large numbers, allowing for more straightforward calculations. It's an efficient way to streamline the process of working with multiple fractions.

Combine Like Terms

Combining like terms is a fundamental skill in algebra that involves simplifying expressions. When faced with an expression containing similar variables, we can "combine" them to make the expression less cluttered. This process is especially useful when simplifying fractions or algebraic expressions, as seen in the final steps of the exercise.
Here’s how it works in practice:

• Examine each part of the expression to identify terms with the same variable or constant.
• For example, in the expression ewline (6x - 15) - (6x - 4), observe the terms 6x and -6x.
• These terms can be combined to yield a simpler expression because they cancel each other out, resulting in 0x.
• Similarly, constant numbers -15 and +4 can be combined (-15 + 4 = -11).

This process reduces the expression to its simplest form, aiding clarity and making further calculations, if needed, much easier. The goal here is always to simplify, keeping the work as tidy and direct as possible.