### Instruction: The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number?### Response: I need to find three numbers that add up to 67 and satisfy the given conditions.
Let me call the smallest number x, then the middle number is x + 3, and the largest number is x + 3 + 7, or x + 10.
So I have x + (x + 3) + (x + 10) = 67.
I can simplify this equation by combining like terms.
I get 3x + 13 = 67.
I can subtract 13 from both sides to isolate 3x.
I get 3x = 54.
I can divide both sides by 3 to get x.
I get x = 18.
So the smallest number is 18, the middle number is 18 + 3, or 21, and the largest number is 18 + 10, or 28.
I can check that these numbers satisfy the given conditions.
The two larger numbers differ by 7, since 28 - 21 = 7.
The two smaller numbers differ by 3, since 21 - 18 = 3.
And they add up to 67, since 18 + 21 + 28 = 67.
So I have found the correct answer.