Tanya guessed there were 65 beans in a jar. Her guess was off by 20. Bryan guessed there were 35 beans in the jar. He was off by 10. How many beans are in the jar?
And how would I show my work?
65 - 20 = 45
35 + 10 = 45
To find the number of beans in the jar, we can use the method of trial and error.
First, let's calculate the difference between Tanya's guess and the actual number of beans: 65 (Tanya's guess) - 20 (off by) = 45
Now, let's calculate the difference between Bryan's guess and the actual number of beans: 35 (Bryan's guess) - 10 (off by) = 25
Since Tanya's guess was 45 off and Bryan's guess was 25 off, we can consider the actual number of beans to be the average of their guesses minus the average of their differences:
Calculation: (65 + 35) / 2 - [(45 + 25) / 2] = 100 / 2 - 70 / 2 = 50 - 35 = 15
Therefore, there are 15 beans in the jar.
To show your work, you can write:
Tanya's guess - off by = difference
65 - 20 = 45
Bryan's guess - off by = difference
35 - 10 = 25
Average of Tanya's guess and Bryan's guess:
(65 + 35) / 2 = 100 / 2 = 50
Average of Tanya's difference and Bryan's difference:
(45 + 25) / 2 = 70 / 2 = 35
Actual number of beans in the jar:
Average of the guesses - Average of the differences
50 - 35 = 15
To find the number of beans in the jar, we can use the information given about Tanya and Bryan's guesses. We know that Tanya guessed 65 beans and was off by 20, while Bryan guessed 35 beans and was off by 10.
Let's break down the information given:
- Tanya's guess: 65 beans, off by 20.
- Bryan's guess: 35 beans, off by 10.
This means that the actual number of beans in the jar is either 20 less or 20 more than Tanya's guess, and either 10 less or 10 more than Bryan's guess.
To find the possible range of bean counts, we can subtract and add the respective amounts Tanya and Bryan were off by:
For Tanya's guess:
- Lower bound: 65 (Tanya's guess) - 20 (off by) = 45 beans.
- Upper bound: 65 (Tanya's guess) + 20 (off by) = 85 beans.
For Bryan's guess:
- Lower bound: 35 (Bryan's guess) - 10 (off by) = 25 beans.
- Upper bound: 35 (Bryan's guess) + 10 (off by) = 45 beans.
The range of possible bean counts is the intersection of the lower and upper bounds for both Tanya and Bryan:
- Lower bound: maximum of Tanya's lower bound (45 beans) and Bryan's lower bound (25 beans) = 45 beans.
- Upper bound: minimum of Tanya's upper bound (85 beans) and Bryan's upper bound (45 beans) = 45 beans.
Therefore, there must be exactly 45 beans in the jar.
Now, let's discuss how to show your work:
1. Write down the given information: Tanya's guess (65 beans, off by 20) and Bryan's guess (35 beans, off by 10).
2. Calculate the lower and upper bounds separately for each guess.
3. Compare the lower and upper bounds for both Tanya and Bryan to find their intersection.
4. Determine the range of possible bean counts.
5. Conclude the final answer by identifying the specific number within the range.
6. Present your answer clearly, stating the number of beans in the jar is 45.
By following these steps, you demonstrate a clear understanding of the problem and show your work in a logical and organized manner.