To win a contest, the number of beans in a jar has to be guessed within 20 of the actual number. If the number of beans in the jar is 645, which equation can be used to find the minimum and maximum number of beans that will win the contest, and what is the maximum guess that could win?
A. $|x-645|=20$; maximum guess: 635 beans
B. $|x-645|=20$; maximum guess: 665 beans
C. $|x-20|=645$; maximum guess: 635 beans
D. $|x-20|=645$; maximum guess: 665 beans
Answer (1)
Define the variable: Let x represent the guess.
Formulate the equation: The absolute difference between the guess and the actual number is ∣ x − 645∣ = 20 .
Solve for the maximum guess: x = 645 + 20 = 665 .
The maximum guess that could win is 665 beans.
Explanation
Understanding the Problem Let x be the guess for the number of beans in the jar. The problem states that the guess must be within 20 of the actual number of beans, which is 645. This means the difference between the guess and the actual number must be less than or equal to 20. We can write this as an absolute value inequality: ∣ x − 645∣ l e 20 . However, the options provided use an equality, so we consider the boundary condition ∣ x − 645∣ = 20 .
Finding the Maximum Guess To find the maximum winning guess, we solve the equation x − 645 = 20 . Adding 645 to both sides, we get x = 645 + 20 = 665 .
Finding the Minimum Guess To find the minimum winning guess, we solve the equation x − 645 = − 20 . Adding 645 to both sides, we get x = 645 − 20 = 625 .
Conclusion The equation that represents the situation is ∣ x − 645∣ = 20 , and the maximum winning guess is 665.
Examples
Imagine you're organizing a school fair and need to estimate the number of candies in a jar for a contest. Knowing that the actual number is 645, and the winning guess must be within 20, this problem helps determine the range of acceptable guesses. This concept is useful in various estimation scenarios, such as inventory management, resource allocation, or even predicting outcomes within a certain margin of error. By understanding the range, you can set appropriate boundaries for acceptable answers, ensuring fairness and accuracy in your estimations.
