Problem 66 - Algebra 1

Question

RELAY RACE The total time for a two-member team to complete a 5765 -meter relay race was 19 minutes. The first runner averaged 285 meter \(=\) per minute and the second runner averaged 335 meters per minute. Use the verbal model to find how many minutes the first runner ran. [Time for \(1 \text { st runner }+\text { Time for } 2 \text { nd runner }]=\) ]Total time Distance for \(1 ]\text { st runner }]+[\text { Distance for } 2 \text { nd runner }]=\) Total distance]

Step-by-Step Solution

Answer

The first runner ran for approximately 12.4 minutes.

Step 1: Establish the equations

The distance covered by each runner is the product of their speed and time taken. Let the time taken by first runner be \( x \) and second runner be \( y \), we can thus establish 2 equations based on given information: \n\n Equation 1: \( 285x + 335y = 5765 \) (total distance covered by both runners) \n\n Equation 2: \( x + y = 19 \) (total time taken by both runners)

Step 2: Solve the system

We can now solve the system of equations established in the previous step. From Equation 2, we can express \( y \) in terms of \( x \) as \( y = 19 - x \). Substituting this into Equation 1 gives: \( 285x + 335(19 - x) = 5765 \). Solving this equation for \( x \) will yield the time taken by the first runner.

Step 3: Compute the time for the first runner

Solving the equation derived in step 2: \( 285x + 335(19 - x) = 5765 \) simplifies to \( 285x + 6385 - 335x = 5765 \) and then to \( 50x = 620 \). Finally, solving for \( x \) gives \( x = 12.4 \) (rounded to one decimal place).

Key Concepts

Algebraic Equations

At the heart of many mathematical problems lie algebraic equations, which are mathematical statements indicating that two expressions are equal. Usually involving variables (letters representing numbers), constants (specific numbers), and operations (like addition and subtraction), they're essential building blocks for more complex mathematics.

In the scenario of a relay race, the first step is to represent the given information as algebraic equations. For example, if a runner's speed and the time they take to complete their part of the race are known, their distance can be calculated by multiplying these two quantities. The formula essentially is