|
Question 1003608: During dinner the guests were told to share dishes with one another, every 2 of them share a dish of rice, every 4 of them a dish of broth, every 4 of them a dish of meat. If there are 65 dishes in all, how many guests were there?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! as far as i can tell, you have 2 equations in 4 unknowns which which can be reduced to 2 equations in 3 unknowns, which means you can't have a single solution common to both equations, but you can have a solution that is relative to the value of one of the reduced variables.
what this means is that, if you assume the value of one of the variables, then you can solve for the other variables.
i base my conclusion on the following:
let a = numbe of rice dishes.
let b = number of broth dishes.
let c = number of meat dishes.
you have 2 equations.
a + b + c = 65
2a + 4b + 4c = g
that's 2 equations with 4 unknowns.
the unknowns are a,b,c,g.
g is the number of guests.
since the same number of guests eat broth dishes and meat dishes, you can combine these into one variable which is called d.
d is the number of broth and meat dishes combined.
this means that d = b + c.
you get b + c = d
you get 4b + 4c = 4 * (b + c) = 4 * d = 4d
your equations become:
a + d = 65
2a + 4d = g
now you have 2 equations in 3 unknowns.
multiply both sides of the first equation by 4 and bring down the second equation to get:
4a + 4d = 260
2a + 4d = g
subtract the second equation from the first to get:
2a = 260 - g
solve for g to get g = 260 - 2a
what this says is that the number of guests is dependent on the number of rice dishes that were served.
if you know the number of rice dishes that were served, then you can figure out the total number of guests.
if not, then you can't.
you could also have solved this for g in terms of d.
you start with:
a + d = 65
2a + 4d = g
multiply the first equation by 2 and bring down the second equation to get:
2a + 2d = 130
2a + 4d = g
subtract the first equation from the second to get 2d = g - 130
solve for g to get g = 2d + 130
so you have 2 ways you can solve this.
the first way is to get g in terms of a.
the second way is to get g in terms of d.
your 2 equations are:
g = 260 - 2a
g = 2d + 130
using the first equation, you assume a value of a and then solve for d and then compute g.
using the second equation, you assume a value of d and then solve for a and then compute g.
these 2 solutions will complement each other as they should.
i think the only restriction is that a or d can't be greater than 65, becuse if they are, then one of them will have to be negative which is not allowed.
for example:
assume a = 45
this means d = 65 - 45 = 20
you have a = 45 and d = 20
assuming a = 45, you will have calculated d = 20
assuming d = 20, you will have calculated a = 45
now you can solve for g using either a or d
when a = 45, g = 260 - 2*45 = 260 - 90 = 170.
when d = 20, g = 2d + 130 = 40 + 130 = 170.
you derive the same value of g either way.
you do not necessarily know that b and c are equal in value, but you do know that the sum of a + b is equal to d, and you also do know that the sum of 4b + 4c is equal to the sum of 4 * (b + c) which is equal to the sum of 4 * d.
best i can do,
hopefully is helps.
|
|
|
| |
