Problem 32 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{5}{6}+\frac{1}{3}}{\frac{2}{3}-\frac{1}{12}} $$

Step-by-Step Solution

Answer

2

Step 1: Simplify the numerator

Combine the fractions in the numerator. To do this, find a common denominator for \(\frac{5}{6}\) and \(\frac{1}{3}\). The common denominator is 6. \[\frac{5}{6} + \frac{1}{3} = \frac{5}{6} + \frac{2}{6} = \frac{7}{6}\]

Step 2: Simplify the denominator

Combine the fractions in the denominator. To do this, find a common denominator for \(\frac{2}{3}\) and \(\frac{1}{12}\). The common denominator is 12. \[\frac{2}{3} - \frac{1}{12} = \frac{8}{12} - \frac{1}{12} = \frac{7}{12}\]

Step 3: Divide the simplified numerator by the simplified denominator

Now divide the simplified numerator by the simplified denominator. \[\frac{\frac{7}{6}}{\frac{7}{12}} = \frac{7}{6} \times \frac{12}{7} = \frac{7 \cdot 12}{6 \cdot 7} = 2\]

Key Concepts

Adding Fractions

Adding fractions can seem tricky at first, but it becomes much easier once you understand the steps. When adding fractions, the first thing you need to do is find a common denominator. This is a number that both denominators can divide into evenly.

For example, let's add \(\frac{5}{6}\) and \(\frac{1}{3}\). The denominators are 6 and 3. The smallest common denominator between these two numbers is 6.

Next, you adjust the fractions so they both have this common denominator. For \(\frac{1}{3}\), you multiply the numerator and denominator by 2 to get \(\frac{2}{6}\). Now, you have \(\frac{5}{6}\) and \(\frac{2}{6}\).

Once both fractions have the same denominator, you can simply add their numerators: \(\frac{5}{6} + \frac{2}{6} = \frac{7}{6}\). This is your sum!

Subtracting Fractions

Subtracting fractions follows the same basic principles as adding them. The crucial step is to find a common denominator.

For example, let's subtract \(\frac{1}{12}\) from \(\frac{2}{3}\). The denominators are 12 and 3. The smallest common denominator in this case is 12.

Next, adjust the fractions so they both have this common denominator. For \(\frac{2}{3}\), you multiply the numerator and denominator by 4 to get \(\frac{8}{12}\). Now you have \(\frac{8}{12}\) and \(\frac{1}{12}\).

Once the fractions are adjusted, you can subtract the numerators: \(\frac{8}{12} - \frac{1}{12} = \frac{7}{12}\). This is your difference!

Dividing Fractions

Dividing fractions might seem difficult at first, but there's a simple trick to make it easier: multiply by the reciprocal. The reciprocal of a fraction flips the numerator and the denominator.

For example, to divide \(\frac{7}{6}\) by \(\frac{7}{12}\), you multiply \(\frac{7}{6}\) by the reciprocal of \(\frac{7}{12}\), which is \(\frac{12}{7}\).

So the operation looks like this: \(\frac{7}{6} \times \frac{12}{7}\). When you multiply these fractions, the 7s cancel each other out, leaving you with: \(\frac{12}{6} = 2\).

Therefore, \(\frac{\frac{7}{6}}{\frac{7}{12}} = 2\).

Common Denominator

Finding a common denominator is a key step in both adding and subtracting fractions. It is the smallest number that both denominators can evenly divide.

For example, in the fraction problem \(\frac{5}{6} + \frac{1}{3}\), the denominators are 6 and 3. The smallest number both can divide into evenly is 6.

To adjust the fractions, you need to ensure that each fraction has this common denominator. For \(\frac{1}{3}\), multiply both the numerator and the denominator by 2 to get \(\frac{2}{6}\).

Now, you have two fractions: \(\frac{5}{6}\) and \(\frac{2}{6}\). With the same denominator, you can easily add or subtract the numerators.

Using a common denominator makes operations with fractions straightforward!