need to describe 3 different problem solving approaches to this problem, one I think is algebra, but don't know how to solve, and i need to say how to solve each one in each different way, please help with this.
aaron is thinking of a number, if its doubled, increased by 10, cut into thirds, and reduced by 7, new number is 5, whats original number?
n = original number
( 2 * n + 10 ) / 3 - 7 = 5 Add 7 to both sides
( 2 n + 10 ) / 3 - 7 + 7 = 5 + 7
( 2 * n + 10 ) / 3 = 12 Multiply both sides by 3
2 n + 10 = 12 * 3
2 n + 10 = 36 Subtract 10 to both sides
2 n + 10 - 10 = 36 - 10
2 n = 26 Divide both sides by 2
n = 13
( 2 * n + 10 ) / 3 - 7 =
( 2 * 13 + 10 ) / 3 - 7 =
( 26 + 10 ) / 3 - 7 =
36 / 3 - 7 = 12 - 7 = 5
Sure! To solve the problem, we can apply three different problem-solving approaches: algebraic, numerical, and logical. Let's explain each of these approaches step by step:
1. Algebraic Approach:
To solve this problem algebraically, let's assume the original number is represented by 'x.' We can apply the given operations to 'x' and set up an equation to find its value. Here's how:
- Double the number: 2x
- Increase it by 10: 2x + 10
- Cut it into thirds: (2x + 10)/3
- Reduce it by 7: (2x + 10)/3 - 7
According to the problem, this new number is 5. Therefore, we can set up the equation:
(2x + 10)/3 - 7 = 5
To solve this equation and find the original number, we can isolate 'x' and perform algebraic operations:
(2x + 10)/3 = 12
2x + 10 = 36
2x = 26
x = 13
So, the original number is 13.
2. Numerical Approach:
In this approach, we can use trial and error by substituting different numbers into the given operations until we find a number that satisfies the condition. Here's how:
- Start with a number (e.g., 1).
- Double it: 1 * 2 = 2
- Increase it by 10: 2 + 10 = 12
- Cut it into thirds: 12/3 = 4
- Reduce it by 7: 4 - 7 = -3
Since -3 is not equal to 5, we need to try another number and repeat the steps until the result is 5. Continuing this process, you will find that when the original number is 13, the final result is indeed 5.
3. Logical Approach:
The logical approach involves reasoning and deduction. Using this approach, we can analyze the given operations and infer the possible original number. Here's how:
- Doubled and increased by 10: This suggests that the original number should be a positive even number because doubling any even number would result in an even number. Adding 10 to an even number would keep it even.
- Cut into thirds and reduced by 7: This suggests that the original number should be a multiple of 3 since cutting it into thirds would result in a whole number. Reducing a multiple of 3 by 7 might yield a result of 5.
Considering these deductions, if we try the number 12, it satisfies the requirements:
- Double 12: 12 * 2 = 24
- Increase it by 10: 24 + 10 = 34
- Cut it into thirds: 34/3 = 11.333 (not equal to 5)
Therefore, the logical approach indicates that the original number should be greater than 12. By trying a few more numbers, we find that 13 satisfies all the conditions, resulting in the new number of 5.