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?
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. 