Problem 99
Question
Write a numerical statement to represent the problem. Then simplify the numerical expression to answer the 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
Verified Answer
Michael lost \( \$ 16.50 \) for the day.
1Step 1: Determine Total Amount Bet
Michael bets \( \\( 5 \) on each of the 9 races. To find the total amount bet, multiply the amount bet per race by the number of races: \( 5 \times 9 = 45 \). Thus, Michael bet a total of \( \\) 45 \) for the day.
2Step 2: Determine Total Winnings
Michael's only winnings were \( \\( 28.50 \). Therefore, his total winnings for the day were \( \\) 28.50 \).
3Step 3: Calculate Net Result
To determine if Michael won or lost money, subtract the total amount bet from his total winnings: \( 28.50 - 45 = -16.50 \). This shows a negative result, indicating a loss.
Key Concepts
Understanding a Numerical StatementSimplifying ExpressionsUnderstanding Arithmetic Operations
Understanding a Numerical Statement
To express Michael's betting and winnings as a numerical statement, let's first recognize what the numerical statement is. A numerical statement is a mathematical sentence involving numbers and sometimes operations, showcasing a problem clearly. Here, we need to assess Michael's total expenses versus his earnings.
The problem tells us Michael spent \(5 on 9 races, and his earnings were \)28.50 from one race. Hence, the numerical statement needs to capture these transactions in number form. We start by writing the full amount Michael bet: this is simply the multiplication of 5 by 9, represented as \( 5 \times 9 \). Such numerical representation condenses and simplifies problem data into a straight calculation. Now, add his only winnings and subtract the total spent to find the net result, ensuring clarity in the statement.
The problem tells us Michael spent \(5 on 9 races, and his earnings were \)28.50 from one race. Hence, the numerical statement needs to capture these transactions in number form. We start by writing the full amount Michael bet: this is simply the multiplication of 5 by 9, represented as \( 5 \times 9 \). Such numerical representation condenses and simplifies problem data into a straight calculation. Now, add his only winnings and subtract the total spent to find the net result, ensuring clarity in the statement.
Simplifying Expressions
Simplifying expressions involves reducing a mathematical expression to its simplest form where it's easier to understand or compute. Let’s walk through how we simplify the expression to solve our problem.
Michael’s total expenses are calculated as \( 5 \times 9 = 45 \). That’s the total amount he spent on betting across all races. If he earned \( 28.50 \), then the task is to simplify this to find net gain or loss. Therefore, the expression we tackle becomes \( 28.50 - 45 \).
This step-by-step simplification helps us get \( -16.50 \), which is the simplest form of representing his financial outcome for that day. Each step, notably, reduces calculations to manageable figures—crystallizing a complete understanding of transactions in a game setting. The outcome, a negative number, is extremely informative as it clearly indicates a loss.
Michael’s total expenses are calculated as \( 5 \times 9 = 45 \). That’s the total amount he spent on betting across all races. If he earned \( 28.50 \), then the task is to simplify this to find net gain or loss. Therefore, the expression we tackle becomes \( 28.50 - 45 \).
This step-by-step simplification helps us get \( -16.50 \), which is the simplest form of representing his financial outcome for that day. Each step, notably, reduces calculations to manageable figures—crystallizing a complete understanding of transactions in a game setting. The outcome, a negative number, is extremely informative as it clearly indicates a loss.
Understanding Arithmetic Operations
Arithmetic operations are the actions we apply to numbers, such as addition, subtraction, multiplication, and division. These form the backbone of arithmetic and algebra, guiding us through problem-solving tasks like determining Michael’s profit or loss at the racetrack.
In our scenario, we used:
By mastering these operations, students gain the ability to interpret and analyze various financial or quantitative scenarios efficiently. Arithmetic operations turn seemingly complex calculations into clear, understandable solutions, showing exactly how operations interact to form meaningful results.
In our scenario, we used:
- Multiplication: To establish the total betting cost as \( 5 \times 9 \).
- Subtraction: To determine the net outcome, subtracting \( 45 \) from Michael’s total winnings of \( 28.50 \).
By mastering these operations, students gain the ability to interpret and analyze various financial or quantitative scenarios efficiently. Arithmetic operations turn seemingly complex calculations into clear, understandable solutions, showing exactly how operations interact to form meaningful results.
Other exercises in this chapter
Problem 98
Write a numerical statement to represent the problem. Then simplify the numerical expression to answer the question. After dieting for 30 days, Ignacio has lost
View solution Problem 99
Answer the question with an algebraic expression. The distance between two cities is \(m\) miles. How far is this, expressed in feet?
View solution Problem 100
Write a numerical statement to represent the problem. Then simplify the numerical expression to answer the question. Max bought a piece of trim molding that mea
View solution Problem 101
Answer the question with an algebraic expression. Explain the difference between simplifying a numerical expression and evaluating an algebraic expression.
View solution