Problem 84 - Prealgebra

Question

Simplify. $$\frac{\frac{5}{7}+\frac{5}{8}}{\frac{5}{7}-\frac{5}{8}}$$

Step-by-Step Solution

Answer

The expression simplifies to 15.

Step 1: Understand the Problem

We need to simplify the given expression, which is a complex fraction: \(\frac{\frac{5}{7}+\frac{5}{8}}{\frac{5}{7}-\frac{5}{8}}\). This involves simplifying both the numerator and the denominator by performing their respective operations.

Step 2: Simplify the Numerator

The numerator is \(\frac{5}{7} + \frac{5}{8}\). To add these fractions, find a common denominator. The least common multiple of 7 and 8 is 56. Convert each fraction: \(\frac{5}{7} = \frac{40}{56}\) and \(\frac{5}{8} = \frac{35}{56}\). Now, add them: \(\frac{40}{56} + \frac{35}{56} = \frac{75}{56}\).

Step 3: Simplify the Denominator

The denominator is \(\frac{5}{7} - \frac{5}{8}\). As with addition, find a common denominator. Using 56 again: \(\frac{5}{7} = \frac{40}{56}\) and \(\frac{5}{8} = \frac{35}{56}\). Subtract the fractions: \(\frac{40}{56} - \frac{35}{56} = \frac{5}{56}\).

Step 4: Divide the Numerator by the Denominator

Now divide \(\frac{75}{56}\) by \(\frac{5}{56}\). When dividing fractions, multiply by the reciprocal: \(\frac{75}{56} \times \frac{56}{5}\). The 56s cancel out, leaving \(\frac{75}{5}\).

Step 5: Final Simplification

Calculate \(\frac{75}{5}\), which simplifies to 15. Thus, the expression simplifies to 15.

Key Concepts

Simplifying Fractions

Fractions are expressions that represent parts of a whole. Simplifying them involves reducing the fraction to its simplest form.
This occurs when there's no common factor between the numerator and the denominator other than 1.

To simplify a fraction:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both by this GCF.
• The fraction simplest form is when it can't be reduced further, meaning the GCF is 1.

Simplifying fractions makes them easier to understand and use in further calculations.

Least Common Multiple

In math, the least common multiple (LCM) is the smallest number that is a multiple of two or more numbers.
We use the LCM to add or subtract fractions with different denominators.

To find the LCM:

• List the multiples of each number.
• Find the smallest multiple they have in common.

For example, the LCM of 7 and 8 is 56, as 56 is a multiple of both.
Using the LCM can simplify the process of working with fractions.

Addition and Subtraction of Fractions

Adding and subtracting fractions require a common denominator.
This is where the LCM is very helpful because it gives us a common denominator to use.

Steps to add or subtract:

• Find the LCM of the denominators to determine the common denominator.
• Convert each fraction to an equivalent fraction with this common denominator.
• Perform the operation (addition or subtraction) on the numerators, keeping the denominator the same.
• If needed, simplify the resulting fraction.

Ensuring a common denominator allows the fractions to be combined accurately.

Reciprocal of a Fraction

The reciprocal of a fraction flips its numerator and the denominator.
It's essentially "turning it upside down."

Reciprocals are particularly useful in division of fractions because dividing by a fraction is the same as multiplying by its reciprocal.

• Given a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
• To divide fractions, multiply by the reciprocal of the divisor.

Using reciprocals makes dividing fractions straightforward and leads to correct calculations.