Problem 55 - Algebra A Combined Function

Question

If the distance formula \(d=r \cdot t\) is solved for \(t,\) ther \(t=\frac{d}{r} .\) Use this formula to find \(t\) if distance \(d\) is \(\frac{20 x}{3}\) miles and rate \(r\) is \(\frac{5 x}{9}\) miles per hour. Write \(t\) if simplified form.

Step-by-Step Solution

Answer

\(t = 12\) hours.

Step 1: Substitute the Values

We are provided with the distance formula \(d = r \cdot t\) and its rearranged form \(t = \frac{d}{r}\). Plug the given values \(d = \frac{20x}{3}\) and \(r = \frac{5x}{9}\) into this formula for \(t\).

Step 2: Setup the Fraction Division

We need to calculate \(t = \frac{\frac{20x}{3}}{\frac{5x}{9}}\). This can be simplified by multiplying the first fraction, \(\frac{20x}{3}\), by the reciprocal of the second fraction, \(\frac{9}{5x}\).

Step 3: Perform the Multiplication

Multiply the numerators and denominators: \(t = \frac{20x \times 9}{3 \times 5x}\). This results in \(t = \frac{180x}{15x}\).

Step 4: Simplify the Expression

Simplify \(t = \frac{180x}{15x}\). Cancel out the \(x\) from the numerator and the denominator, and simplify the fraction \(\frac{180}{15}\), which equals \(12\).

Step 5: Verify the Solution

After simplification, we get \(t = 12\). Double-check each step carefully to confirm the answer is consistent and all calculations are correct.

Key Concepts

Fraction Division

When working with fractions, dividing one fraction by another involves flipping the second fraction and then multiplying. This process is called multiplying by the reciprocal. When you have a division problem like \( \frac{\frac{20x}{3}}{\frac{5x}{9}} \), the key is to turn the complex division into a multiplication:

• First, find the reciprocal of the denominator fraction. The reciprocal of \( \frac{5x}{9} \) is \( \frac{9}{5x} \).
• Next, multiply the first fraction \( \frac{20x}{3} \) by this reciprocal.

This simplifies to \( \frac{20x}{3} \times \frac{9}{5x} \).
By multiplying the numerators together and then the denominators together, you'll transform the division into a straightforward multiplication process, setting up the problem for easy simplification.

Simplification

Simplification of expressions reduces them into their simplest form. Once you have performed the multiplication of fractions, like in our scenario \( \frac{20x \times 9}{3 \times 5x} \), you can simplify further. Follow these steps:

• Multiply the numerators and the denominators separately: \( 20x \times 9 = 180x \) and \( 3 \times 5x = 15x \).
• Cancel common terms in the numerator and denominator. Here 'x' appears in both.
• Dividing the terms \( \frac{180x}{15x} \) cancels 'x', leaving \( \frac{180}{15} \).

After cancellation,
simplify \( \frac{180}{15} \) by dividing both numbers by 15, resulting in 12. This is the simplest form of our expression.

Algebraic Manipulation

Algebraic manipulation involves using algebraic rules to rearrange and simplify expressions. In the provided problem, we start with a formula for distance, \( d = r \cdot t \), already solved for \( t \). Rearrange to \( t = \frac{d}{r} \). First, plug in the complex expressions for \( d \) and \( r \) that involve fractions, and then use the laws of algebra

• Substitute known values to set up the fraction \( \frac{\frac{20x}{3}}{\frac{5x}{9}} \).
• Algebraic laws allow us to flip the denominator and multiply, transforming the problem into the easier multiplication \( \frac{20x}{3} \times \frac{9}{5x} \).
• Finally, carry out the multiplication, and simplify as needed.

This demonstrates algebraic manipulation by transitioning from division to multiplication, and then into a cleaner, reduced form. A clear understanding of these steps ensures precise and efficient solutions.