SOLUTION: Please help me solve this: 90 principal accountants from selected shs attended a one week workshop at ajumako and the following shows their choice of meal they prefer at lunch.

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Question 1182254: Please help me solve this: 90 principal accountants from selected shs
attended a one week workshop at ajumako and the following shows their choice
of meal they prefer at lunch. 63 like Banku with stew (B) 55 like fried rice
with chicken (R) and 50 like fufu with mutton light soup. (F) 13 liked only
Banku 8 liked only fried rice 5 like only fufu and 22 liked all three meals.
Find the number of accountants who liked
a. Banku and fried rice
b. Only two of the meals
c. None of the three meals.
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

90 principal accountants from selected shs
attended ... 

s+t+u+v+w+x+y+z = 90

63 like Banku with stew (B)
 
s+t+v+w = 63

55 like fried rice with chicken (R)
 
t+u+w+x = 55

and 50 like fufu with mutton light soup. (F)
 
v+w+x+y = 50

13 liked only Banku
 
s = 13

8 liked only fried rice
 
u = 8

5 like only fufu
 
y = 5

and 22 liked all three meals.
 
w = 22



Substituting those in the last 3 equations above,

13+t+v+22 = 63
t+8+22+x = 55
v+22+x+5 = 50

They simplify to

t+v     = 28
t  +x   = 25
  v+x   = 23

Solve that by solving for a letter and substituting

Solution to that system is t=15, v=13, x=10 

Now if we substitute those into

s+t+u+v+w+x+y+z = 90

13+15+8+13+22+10+5+z = 90
                86+z = 90
                   z = 4



Now we can answer the questions:

Find the number of accountants who liked
a. Banku and fried rice
That's t+w = 15+22 = 37

b. Only two of the meals
That's t+v+x = 15+13+10 = 38

c. None of the three meals.
That's z = 4

Edwin

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The other tutor started the solution using 8 variables; that is fine, but I think unnecessarily complicated, since we are given the values of 4 of those variables.

So I'm going to pirate his nice diagram and solve the problem using far fewer variables, leading to a much easier solution.

His diagram....



A revised diagram, using given numbers where possible....



With this diagram, I have only three variables, with the following three equations:

B: a+c+22+13=63; a+c=28 [1]
R: a+b+22+8=55; a+b=25 [2]
F: b+c+22+5=50; b+c=23 [3]

With these three equations, we can easily find a+b+c...

Adding [1]+[2]+[3]...

2(a+b+c)=76
a+b+c=38 [4]

... then it is easy to solve for a, b, and c, by comparing [4] with [1], [2], and [3]:

a = 38-23=15
b=38-28=10
c=38-25=13

Now the diagram is this:



Now we can answer the questions....

a. How many like Banku and fried rice
22+a = 22+15 = 37

b. How many like only two of the meals
a+b+c = 15+10+13 = 38

c. How many like none of the three meals
90-(13+8+5+15+10+13+22) = 90-86 = 4