Problem 83 - Prealgebra

Question

Simplify. $$\frac{\frac{3}{5}+\frac{3}{7}}{\frac{3}{5}-\frac{3}{7}}$$

Step-by-Step Solution

Answer

The expression simplifies to 6.

Step 1: Identify the Expression

The given expression is \( \frac{\frac{3}{5}+\frac{3}{7}}{\frac{3}{5}-\frac{3}{7}} \). This is a complex fraction, which consists of fractions in both its numerator and denominator. Our task is to simplify this fraction.

Step 2: Find Common Denominators

To simplify both the numerator and the denominator, we first find a common denominator for each. For \( \frac{3}{5} + \frac{3}{7} \) and \( \frac{3}{5} - \frac{3}{7} \), the common denominator is 35, as it is the least common multiple of 5 and 7.

Step 3: Simplify the Numerator

Rewriting the numerator with a common denominator, we have \( \frac{3}{5} = \frac{21}{35} \) and \( \frac{3}{7} = \frac{15}{35} \). The sum is: \( \frac{21}{35} + \frac{15}{35} = \frac{36}{35} \).

Step 4: Simplify the Denominator

Similarly, rewrite the denominator with a common denominator: \( \frac{3}{5} = \frac{21}{35} \) and \( \frac{3}{7} = \frac{15}{35} \). The difference is: \( \frac{21}{35} - \frac{15}{35} = \frac{6}{35} \).

Step 5: Simplify the Complex Fraction

Now substitute back into the complex fraction: \( \frac{\frac{36}{35}}{\frac{6}{35}} \). This simplifies to \( \frac{36}{35} \times \frac{35}{6} \).

Step 6: Multiply and Simplify

When multiplying, the 35s cancel out, and we are left with \( \frac{36}{6} \), which simplifies to 6. Therefore, the simplified form of the expression is 6.

Key Concepts

Least Common Multiple

To simplify a complex fraction like \( \frac{\frac{3}{5} + \frac{3}{7}}{\frac{3}{5} - \frac{3}{7}} \), it is crucial to understand the concept of the least common multiple (LCM). When you want to add or subtract fractions, you need a common denominator. The common denominator must be a multiple of each fraction's original denominator. Here, the two denominators are 5 and 7. The LCM of these two numbers is the smallest number that both 5 and 7 can divide into without leaving a remainder.

To find the LCM of 5 and 7, you can list their multiples:

• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, ...
• Multiples of 7: 7, 14, 21, 28, 35, ...

The smallest number common in both lists is 35, making 35 the LCM. Hence, both fractions in the numerator and in the denominator need to be rewritten with 35 as their common denominator before performing addition or subtraction.

Simplifying Fractions

Once we have a common denominator, the next step is simplifying the fractions by combining the numerators. Take the expression \( \frac{21}{35} + \frac{15}{35} \) in the numerator, for example. The denominator is already common, so add the numerators directly:

• Add \( 21 + 15 \) to get 36. Thus, the resulting fraction is \( \frac{36}{35} \).

Similarly, in the denominator \( \frac{21}{35} - \frac{15}{35} \), you subtract the numerators:

• Subtract 15 from 21 to get 6. This simplifies to \( \frac{6}{35} \).

After these simplifications, you plug the simplified fractions back into the complex fraction. At this point, simplify further by treating the problem as a multiplication as fractions divide by flipping the fraction in the denominator. This means \( \frac{\frac{36}{35}}{\frac{6}{35}} \) becomes \( \frac{36}{35} \times \frac{35}{6} \). Simplifying gives \( \frac{36}{6} = 6 \).
This method shows that simplifying fractions often involves both finding a common denominator and then performing basic arithmetic on the numerators.

Numerator and Denominator

In any fraction, the numerator is the top part that tells you how many parts you have, while the denominator is the bottom part that specifies into how many parts the whole is divided. For the complex fraction \( \frac{\frac{3}{5} + \frac{3}{7}}{\frac{3}{5} - \frac{3}{7}} \), both the main numerator, \( \frac{3}{5} + \frac{3}{7} \), and the main denominator, \( \frac{3}{5} - \frac{3}{7} \), themselves contain another set of numerators and denominators.

• In \( \frac{3}{5} + \frac{3}{7} \), 3 is the numerator for both fractions, while 5 and 7 are individual denominators.
• In \( \frac{3}{5} - \frac{3}{7} \), again, 3 acts as the numerator for each, with 5 and 7 acting as denominators.

Simplifying the complex fraction requires not only simplifying each numerator and denominator as separate fractions but combining them appropriately to simplify the overall expression. Understanding the roles of numerator and denominator can help you visualize how operations on fractions are applied in algebraic expressions.