Problem 42 - Algebra 2

Question

Simplify each expression. $$ \frac{3}{4 q}-\frac{2}{5 q}-\frac{1}{2 q} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-3}{20q}\).

Step 1: Identify Common Denominator

To simplify the expression \( \frac{3}{4q} - \frac{2}{5q} - \frac{1}{2q} \), we first need to identify a common denominator for the fractions. The denominators are \(4q\), \(5q\), and \(2q\). The least common multiple of \(4\), \(5\), and \(2\) is \(20\). Since all the terms also involve \(q\), the common denominator is \(20q\).

Step 2: Rewrite Each Fraction

We rewrite each fraction with the common denominator \(20q\). For \(\frac{3}{4q}\), multiply both the numerator and the denominator by \(5\) to get \(\frac{15}{20q}\). For \(\frac{2}{5q}\), multiply both by \(4\) to obtain \(\frac{8}{20q}\). For \(\frac{1}{2q}\), multiply both by \(10\) to get \(\frac{10}{20q}\).

Step 3: Simplify the Expression

Now that all the fractions have the same denominator, we can combine them: \(\frac{15}{20q} - \frac{8}{20q} - \frac{10}{20q}\). Subtract the numerators: \(15 - 8 - 10 = -3\). Thus, the fraction becomes \(\frac{-3}{20q}\).

Step 4: Final Simplified Expression

Since \(-3\) is already a simplified numerator, we conclude that the simplified expression is \(\frac{-3}{20q}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like doing a little spring cleaning in math. You want to combine and reduce parts to make it neat and simple. When you start with something like \(\frac{3}{4q} - \frac{2}{5q} - \frac{1}{2q}\), it might look a bit messy with different denominators.
To simplify, you'll need to:

• Find a common denominator to harmonize your fractions, which means making sure the bottom numbers are the same.
• Once you've done that, you can easily combine the fractions by adding or subtracting the numerators.
• Finally, you'll check if the result can be simplified further by looking for common factors.

Simplified expressions are always easier to work with and understand, just like a tidied-up room is more inviting. It involves turning what's complex into something clear and straightforward.

Least Common Denominator

Finding the least common denominator (LCD) is crucial when dealing with multiple fractions, particularly in simplifying algebraic expressions with different denominators. The LCD is the smallest number that each of the denominators can divide into without leaving a remainder.
When looking at \(4q\), \(5q\), and \(2q\), you find the smallest multiple common to \(4\), \(5\), and \(2\), which is \(20\). But don't forget, each term also has a \(q\), so the LCD here becomes \(20q\).
Consider it like finding the smallest stage where all performers (fractions) can easily play together; they need a common place (denominator) to sound harmonious. Without it, combining fractions would be confusing, like trying to listen to different bands playing different tunes at the same time. With the LCD, each term is given the same footing to work smoothly together.

Combining Fractions

Once you have a common denominator, combining fractions becomes much simpler. Picture this: once all fractions have the same base as a denominator, they are like pieces of the same cake. From there, it's straightforward to add or subtract them.
Using our example, when you rewrite \(\frac{3}{4q}\), \(\frac{2}{5q}\), and \(\frac{1}{2q}\) with the common denominator \(20q\), they become \(\frac{15}{20q}\), \(\frac{8}{20q}\), and \(\frac{10}{20q}\) respectively.
Now it's just basic arithmetic: subtract the numerators: \(15 - 8 - 10 = -3\). You end up with \(\frac{-3}{20q}\).
With a shared denominator, you can easily see which numerators need to be added or subtracted. This is how fractions come together neatly: they share their new common denominator, so everything lines up perfectly, making them much easier to handle.