Bob's Favorite Number
You and your 3 friends are out to dinner. You want to know Bob's favorite number. None of them lie, but they are not necessarily correct in their statements.
You: (to Bob) What is your favorite number?
Bob: 1488 is my favorite number.
Barry: I'm sorry. Bob multiplies every number he says by 2.
Bill: I'm sorry, Barry divides every number he says by 4 to make up for Bob's exaggerations.
Bob: I'm sorry. Bill is not very bright. He adds 8 to every number he mentions.
What is Bob's favorite number?
This is not an original problem .
The answer is 372.
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1 solution
Let's rephrase and change back the confusing numbers into 3 unknowns. Then, we'll have the new conversation to go like this :
You : (to Bob) What is your favorite number?
Bob : (1488/x) is my favorite number.
Barry : I'm sorry. Bob multiplies every number he says by x (formerly 2).
Bill : I'm sorry, Barry divides every number he says by y (formerly 4) to make up for Bob's exaggerations.
Bob : I'm sorry. Bill is not very bright. He adds z (formerly 8) to every number he mentions.
From Barry's & Bill's statements, we get 2 = x (originally intended number) ÷ y (adjustment),
AND
From Bill's & Bob's statements, we get 4 = y (originayl intended number) + z (adjustment),
AND
From Bob's & Barry's statements, we get 8 = z (originally intended number) × x (adjustment),
8 = zx & 4 = y + z & 2 = x / y
y/x = 1/2 to be multiplied with zx = 8
we get
(y/x) × (zx) = (1/2) × (8)
yz = 4 that can be use together with y + z = 4
yz + y + z = 4 + 4 = 8
yz + y + z + 1 = 8 + 1
9 = (y + 1)(z + 1)
(y,z) = (2,2) or (0,8) or (8,0) or (-4,-4) or (-2,-10) or (-10,-2)
Rechecking with yz = 4 = y + z, only y = z = 2 works. Then x is just x = 8/z = 4, so answer must be 1488/4 = 372.