Problem 99 - Elementary Algebra

Question

For exercises 97-114, evaluate. $$ \frac{3}{4}+\frac{1}{6} \div \frac{1}{2}-\frac{1}{3} $$

Step-by-Step Solution

Answer

\(\frac{3}{4}\)

Step 1 - Evaluate Division

Identify the division operation in the expression. Compute \(\frac{1}{6} \div \frac{1}{2}\). When dividing fractions, multiply by the reciprocal: \(\frac{1}{6} \div \frac{1}{2} = \frac{1}{6} \times \frac{2}{1} = \frac{1\times2}{6\times1} = \frac{2}{6} = \frac{1}{3}\).

Step 2 - Substitution

Substitute the result of the division back into the original expression: \(\frac{3}{4} + \frac{1}{3} - \frac{1}{3}\).

Step 3 - Simplify the Expression

Notice that the \(\frac{1}{3}\) and \(-\frac{1}{3}\) terms cancel each other out. This leaves us with \(\frac{3}{4}\).

Key Concepts

Dividing Fractions

When it comes to dividing fractions, it might seem tricky at first, but there's a simple rule to follow. You need to multiply by the reciprocal of the fraction you are dividing by.

For example, in the exercise, we had to compute \(\frac{1}{6} \div \frac{1}{2}\). The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\). So, we change the division into multiplication by multiplying by this reciprocal:
\[ \frac{1}{6} \times \frac{2}{1} = \frac{1 \times 2}{6 \times 1} = \frac{2}{6} \]
This simplifies to \(\frac{1}{3}\). Remember always to substitute back this result into your original expression, to avoid errors.

Simplifying Expressions

Simplifying an expression means reducing it to its simplest form or combining like terms to make it easier to work with.

In our exercise, after evaluating the division, we substitute \(\frac{1}{3}\) back into the expression, changing it to: \[ \frac{3}{4} + \frac{1}{3} - \frac{1}{3} \]
Notice here, \(\frac{1}{3}\) and \(- \frac{1}{3}\) are like terms and they cancel each other out because adding and then subtracting the same value results in zero. So, the expression simplifies to just:
\[ \frac{3}{4} \]
It's important to keep an eye out for terms that cancel each other out. This makes the expression simpler and easier to solve.

Adding and Subtracting Fractions

Adding and subtracting fractions requires a common denominator, which means the bottom numbers of the fractions must be the same.

In our previous steps, we managed to eliminate the \(\frac{1}{3}\) terms, simplifying our expression significantly. For fractions that do not cancel out, you would make the denominators the same first.

Let's take a quick example: suppose we want to add \(\frac{3}{4} \) and \(\frac{2}{5} \). The least common denominator (LCD) for 4 and 5 is 20. To make denominators the same, we would rewrite these fractions with the LCD:
\[ \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}\]
\[ \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \]
Now that both fractions have the same denominator, we can easily add or subtract them:
\(\frac{15}{20} + \frac{8}{20} = \frac{23}{20}\). Always simplify your final answer if necessary.