Problem 95 - Algebra for College Students

Question

Michael bet \(\$ 5\) on each of the 9 races at the racetrack. His only winnings were \(\$ 28.50\) on one race. How much did he win (or lose) for the day?

Step-by-Step Solution

Answer

Michael lost $16.50.

Step 1: Calculate Total Money Bet

Michael bet \( \\(5 \) on each race, and there were 9 races. Therefore, the total amount he bet is \( 5 \times 9 = \\)45 \).

Step 2: Determine Total Winnings

Michael won \( \$28.50 \) in one race. This is the only amount he won during the day, as stated in the exercise.

Step 3: Calculate Net Outcome

To find out how much Michael won or lost, subtract his total winnings from the total money he bet. So the calculation is \( 45 - 28.5 = \$16.5 \).

Step 4: Determine the Financial Result

Since the result from Step 3 is a positive number, this means that he lost \( \\(16.5 \) during the day because his total expenditures (\\)45) were more than his winnings (\$28.50).

Key Concepts

Arithmetic Operations

Arithmetic operations are basic math functions that include addition, subtraction, multiplication, and division. In the given exercise, we utilize several arithmetic operations to find the solution. We start with multiplication, where Michael bets \(\\(5\) on each of the 9 races. We calculate the total amount bet by multiplying the bet amount by the number of races:

• Bet per race = \(\\)5\)
• Number of races = 9
• Total money bet = \(5 \times 9 = \\(45\)

Next, we use subtraction to determine Michael's net outcome for the day, where we subtract his total winnings of \(\\)28.50\) from the total amount he bet, \(\$45\). This gives us his total loss. Arithmetic operations allow us to process these quantitative pieces of information step-by-step to reach a clear result.

Problem Solving

Problem solving in mathematics involves breaking down a problem into smaller, manageable steps and solving each part systematically. In this exercise, we need to figure out Michael's financial result for the racetrack day. First, we calculate his total expenditure by looking at his bets placed across all races, which involves multiplication.
Then, we establish his total winnings, paying attention to the fact that he only won in one race.
Finally, calculating the net result involves comparing these two numerical values by subtracting winnings from expenditures. This approach of organizing calculations helps tackle problems that may initially seem complex by simplifying them into a series of smaller tasks.

Word Problems

Word problems require translating a written description into mathematical expressions in order to find a solution. In this exercise, the scenario is set at a racetrack where Michael places bets and wins an amount, and we need to determine if he incurs a net loss or gain. The challenge lies in interpreting the sentences correctly to extract relevant numbers and operations:

• "Michael bet \(\\(5\) on each of the 9 races" translates to multiplying the bet amount by the number of races.
• "His only winnings were \(\\)28.50\) on one race" indicates the total amount he gained.
• "How much did he win or lose for the day?" requires finding the difference between his expenditures and winnings.

Successfully solving word problems depends on a good grasp of arithmetic operations and problem solving strategy, which allows us to convert stories or real-life scenarios into solvable mathematical equations.