The Number Game
Let the three consecutive numbers be x, x+1, and x+2.
When we divide the product of these three numbers by the first number x, we get: (x)(x+1)(x+2)/x = (x+1)(x+2)
When we divide the product by the second number x+1, we get: (x)(x+1)(x+2)/(x+1) = (x)(x+2)/(x+1)
When we divide the product by the third number x+2, we get: (x)(x+1)(x+2)/(x+2) = (x)(x+1)
According to the problem, the sum of these three quotients is 74, so we can write the equation:
(x+1)(x+2) + (x)(x+2)/(x+1) + (x)(x+1) = 74
Expanding the terms and simplifying, we get:
3x^2 + 6x – 72 = 0
Dividing by 3 and factoring, we get: x^2 + 2x – 24 = 0
(x+6)(x-4) = 0
So x = -6 or x = 4. Since the numbers are consecutive, x must be positive, so x = 4.
Therefore, the three consecutive numbers are 4, 5, and 6.
Your Answer
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