Problem 72 - Intermediate Algebra

Question

Transportation. If a bus travels a distance \(d_{1}\) at a speed \(s_{1},\) and then travels a distance \(d_{2}\) at a speed \(s_{2},\) the average (mean) speed \(\bar{s}\) is given by the following formula. Simplify the complex fraction. $$ \bar{s}=\frac{d_{1}+d_{2}}{\frac{d_{1}}{s_{1}}+\frac{d_{2}}{s_{2}}} $$

Step-by-Step Solution

Answer

\(\bar{s} = \frac{(d_{1} + d_{2})s_{1}s_{2}}{d_{1}s_{2} + d_{2}s_{1}}\).

Step 1: Understanding the Formula

The formula for average speed given is \(\bar{s} = \frac{d_{1} + d_{2}}{\frac{d_{1}}{s_{1}} + \frac{d_{2}}{s_{2}}}\). This formula compares the total distance traveled to the total time taken for the journey.

Step 2: Break Down the Denominator

The denominator \(\frac{d_{1}}{s_{1}} + \frac{d_{2}}{s_{2}}\) represents the total time spent traveling the distances \(d_1\) and \(d_2\) at speeds \(s_1\) and \(s_2\) respectively.

Step 3: Simplify the Denominator

To simplify the denominator, find a common denominator. The common denominator for \(\frac{d_{1}}{s_{1}}\) and \(\frac{d_{2}}{s_{2}}\) is \(s_{1}s_{2}\). Thus, rewrite the terms as: \(\frac{d_{1}s_{2}}{s_{1}s_{2}} + \frac{d_{2}s_{1}}{s_{2}s_{1}}\).

Step 4: Combine Denominator Terms

Combine the terms in the denominator: \(\frac{d_{1}s_{2} + d_{2}s_{1}}{s_{1}s_{2}}\). This simplifies the expression for total time.

Step 5: Substitute and Simplify the Entire Fraction

Substitute the combined denominator back into the original fraction: \(\bar{s} = \frac{d_{1} + d_{2}}{\frac{d_{1}s_{2} + d_{2}s_{1}}{s_{1}s_{2}}}\). This is a complex fraction, so multiply top and bottom by \(s_{1}s_{2}\) to simplify.

Step 6: Simplify the Expression

Multiply \((d_{1} + d_{2})\) by \(s_{1}s_{2}\) and divide by \(d_{1}s_{2} + d_{2}s_{1}\) to simplify: \(\bar{s} = \frac{(d_{1} + d_{2})s_{1}s_{2}}{d_{1}s_{2} + d_{2}s_{1}}\).

Key Concepts

Average Speed

When a vehicle, like a bus, travels over different distances at varying speeds, calculating the average speed isn't as simple as taking the arithmetic mean of the speeds. Instead, the formula for average speed in such situations relates to the total distance traveled divided by the total time taken. This formula can be written as: \[ \bar{s} = \frac{d_{1} + d_{2}}{t_{1} + t_{2}} \] where \(d_1\) and \(d_2\) are the distances and \(t_1\) and \(t_2\) are the times taken for those distances.

• Total Distance: Simply add up all the distances; \(d_1 + d_2\).
• Total Time: Find the time for each segment: \(t_1 = \frac{d_1}{s_1}\) and \(t_2 = \frac{d_2}{s_2}\).

The understanding of average speed requires both the concept of distance added and the total time calculated, which may involve more complex calculations, especially when speeds differ.

Fraction Simplification

Fraction simplification is a key skill in solving equations, especially those involving complex fractions. A complex fraction has a fraction in its numerator, denominator, or both. In our equation, we have: \[ \bar{s} = \frac{d_{1} + d_{2}}{\frac{d_{1}}{s_{1}} + \frac{d_{2}}{s_{2}}} \] To simplify this, it's important to first tackle the denominator separately. Let's break it down:

• Common Denominator: For \(\frac{d_{1}}{s_{1}}\) and \(\frac{d_{2}}{s_{2}}\), the common denominator is \(s_{1} s_{2}\).
• Rewriting Terms: Express each term over the common denominator: \(\frac{d_{1}s_{2}}{s_{1}s_{2}} + \frac{d_{2}s_{1}}{s_{2}s_{1}}\).
• Combine: Add these fractions: \(\frac{d_{1}s_{2} + d_{2}s_{1}}{s_{1}s_{2}}\).

Now, to simplify the entire expression, multiply the complex fraction's numerator and denominator by the denominator \(s_{1}s_{2}\) to get rid of the fraction within a fraction, enabling a clearer solution.

Algebraic Expressions

Understanding algebraic expressions is fundamental when working with formulas and simplifications. An algebraic expression is a combination of numbers, variables, and arithmetic operations. In the context of our problem, expressions such as \(d_{1}+d_{2}\) and \(\frac{d_{1}s_{2} + d_{2}s_{1}}{s_{1}s_{2}}\) form the basis of our calculations.

• Variables: Symbols like \(d_1\), \(d_2\), \(s_1\), and \(s_2\) represent specific quantities.
• Operators: Addition, multiplication, and division are key operations used to build expressions.
• Simplification: This involves combining like terms, finding common factors, or using algebraic identities to simplify expressions.

The manipulation of these expressions through addition, multiplication, and factoring is what allows us to simplify complex fractions and solve for unknowns in equations efficiently.