Problem Solving

Hi LUANDATO,

We're told that Leon, Billy, and Jim ate a total of 78 hot dogs in a hot dog eating contest, Jim ate FOUR TIMES as many hot dogs as Leon but HALF as many hot dogs as Billy. We're asked how many MORE hot dogs Billy ate than Leon. This question relates the number of hotdogs eaten by each person in terms of MULTIPLES of the number of hotdogs that the others ate, so we can set this up as a RATIO.

First, we'll relate Jim to Leon....
Jim ate FOUR TIMES as many hot dogs as Leon
J:L = 4:1

Next, we'll relate Jim to Billy....
J:B = 1:2

We can now combine ratios (we just have to 'equalize' the value of J):
J:L:B
4:1:8

This means that for ever 13 hotdogs eaten, Jim ate 4, Leon ate 1 and Billy ate 8.
With a total of 78 hotdogs, there are 78/13 = six 'groups of 13'

Thus....
Jim ate 6(4) = 24 hotdogs
Leon ate 6(1) = 6 hotdogs
Billy ate 6(8) = 48 hotdogs

Billy ate 48 - 6 = 42 more hotdogs than Leon.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

LUANDATO wrote:Leon, Billy, and Jim ate an astounding total of 78 hot dogs in a hot dog eating contest. If Jim ate four times as many hot dogs as Leon but half as many hot dogs as Billy, how many more hot dogs did Billy eat than Leon?

A. 13
B. 20
C. 39
D. 42
E. 60

Let x = # of hot dogs that Jim ate

Jim ate four times as many hot dogs as Leon
In other words, Leon ate 1/4 as many hot dogs as Jim ate
So, x/4 = # of hot dogs that Leon ate

Jim ate half as many hot dogs as Billy
In other words, Billy ate twice as many hot dogs as Jim ate
So, 2x = # of hot dogs that Billy ate

In TOTAL they ate 78 hot dogs
So, x + x/4 + 2x = 78
Solve to get: x = 24

So...
Jim ate 24 hot dogs
Leon ate 6 hot dogs
Billy ate 48 hot dogs

How many more hot dogs did Billy eat than Leon?
48 - 6 = 42 = D

Cheers,
Brent

LUANDATO wrote:Leon, Billy, and Jim ate an astounding total of 78 hot dogs in a hot dog eating contest. If Jim ate four times as many hot dogs as Leon but half as many hot dogs as Billy, how many more hot dogs did Billy eat than Leon?

A. 13
B. 20
C. 39
D. 42
E. 60

We can let the number of hot dogs eaten by Leon, Billy, and Jim = L, B, and J, respectively, and create the equations:

L + B + J = 78

and

J = 4L

J/4 = L

and

J = (1/2)B

2J = B

Substituting, we have:

J/4 + 2J + J = 78

J/4 + 3J = 78

Multiplying by 4 we have:

J + 12J = 312

13J = 312

J = 24

Thus, B = 2(24) = 48 and L = 24/4 = 6.

So Bill ate 48 - 6 = 42 more hot dogs than Leon.

Answer: D