Problem 88 - Intermediate Algebra

Question

Simplify. \(\frac{\frac{1}{y}+\frac{4}{5}}{-\frac{3}{20}}\)

Step-by-Step Solution

Answer

\( \frac{-20 - 16y}{3y} \)

Step 1: Combine the fractions in the numerator

The numerator of the given expression is \( \frac{1}{y} + \frac{4}{5} \). To combine them, we need a common denominator. The common denominator for \( y \) and \( 5 \) is \( 5y \). Thus, rewrite the fractions as \( \frac{5}{5y} + \frac{4y}{5y} = \frac{5 + 4y}{5y} \).

Step 2: Simplify the overall fraction

Now substitute back into the original expression: \( \frac{\frac{5 + 4y}{5y}}{-\frac{3}{20}} \). This can be simplified by multiplying by the reciprocal of the denominator: \( \frac{5 + 4y}{5y} \times \frac{20}{-3} \).

Step 3: Multiply the fractions

When multiplying the fractions, multiply the numerators together and the denominators together:\[\frac{(5 + 4y) \times 20}{5y \times (-3)} = \frac{20(5 + 4y)}{-15y}.\]

Step 4: Distribute and simplify

Distribute the \( 20 \) across \( (5 + 4y) \) to get \( 100 + 80y \). The expression becomes:\[\frac{100 + 80y}{-15y}.\]Factor out a common factor of \( 5 \) from the numerator: \( 5(20 + 16y) \), so the expression is:\[\frac{5(20 + 16y)}{-15y}.\]

Step 5: Cancel out the common factor

Notice both numerator and denominator have a common factor of \( 5 \). Cancel it out:\[\frac{20 + 16y}{-3y}.\]

Step 6: Final Simplification

Distribute the negative from the denominator to get:\[-\frac{20 + 16y}{3y} = \frac{-20 - 16y}{3y}.\]

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra that helps reduce complex expressions to their simplest form. By simplifying, you make fractions easier to understand and work with. This involves reducing both the numerator and the denominator by their greatest common divisor. In our exercise, we began by simplifying the numerator, which was a sum of two fractions, before dealing with the division.

• Firstly, identify any common factors between the numerator and the denominator.
• If they share a factor, divide both by this number to make the fraction smaller.
• Continue simplifying until no further common factors are found.

It's important to keep an eye on the signs. Negative signs can change the fraction significantly when placed in different parts of the fraction. Here, we moved on to simplify by writing the fraction with a simpler expression.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions, as it allows you to combine them properly. The common denominator is a multiple of both denominators in each fraction. In this case, we dealt with the fractions \( \frac{1}{y} \) and \( \frac{4}{5} \). Here's how finding a common denominator works:

• Identify the least common multiple (LCM) of the denominators involved.
• Rewrite each fraction with this new denominator without changing the value of the fractions.
• Transform \( \frac{1}{y} \) to \( \frac{5}{5y} \) and \( \frac{4}{5} \) to \( \frac{4y}{5y} \).

Once the fractions have the same denominator, you can easily add their numerators. This leads to a simpler fraction before you move on to other operations, like division or multiplication.

Multiplying Fractions

Multiplying fractions is a straightforward process that involves multiplying the numerators together and the denominators together. In our original exercise, once the fractions have been combined and simplified, the next step was to multiply:

• First, multiply the numerators together: \((5 + 4y) \times 20 \).
• Next, multiply the denominators: \(5y \times (-3)\).

The result was \(\frac{20(5 + 4y)}{-15y}\). Multiplying fractions often means handling negative numbers, so be cautious to maintain the correct sign through multiplication. This operation sometimes involves reducing the resulting fraction further by cancelling out common factors, simplifying it more.

Factoring Polynomials

Factoring polynomials is like unwrapping a gift to see what smaller products are hiding inside. It's a process that helps in further simplifying or solving equations where polynomials are part of the expression. In our solution, this step was necessary to identify and remove common factors.

• Look for terms you can pull out of the polynomial.
• In our case, the numerator \(100 + 80y\) was factored as \(5(20 + 16y)\).
• Factoring can allow for factors to be cancelled out with those in the denominator.

Once the polynomial is factored and common factors are cancelled, it usually leads to a simpler expression that is often easier to interpret or work with in future calculations. This is a key step in reducing fraction expressions in algebra.