Question 174071: a total of 15 delegates from Asia, Africa, America, and Europe meet at an international confrence. Each continent sends a different number of delegates and each has at least one delegate.
America and asia send a toal of 6
asia and Europe send a total of 7
Which continent has sent 4 delegates?
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! this question has multiple possible answers.
the answer is asia.
here's why:
let:
a = asia
b = africa
c = america
d = europe
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a total of 15 delegates were sent:
a + b + c + d = 15
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asia and america sent a total of 6:
a + c = 6
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asia and europe sent a total of 7:
a + d = 7
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from a + c = 6, we get c = 6-a
from a + d = 7, we get d = 7-a
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substituting in a + b + c + d = 15, we get
a + b + 6 - a + 7 - a = 15
combine like terms we get:
-a + b + 13 = 15
subtracting 13 from both sides we get:
-a + b = 2
adding a to both sides we get":
b = a + 2
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substituting for b in -a + b + 13 = 15 we get:
-a + a + 2 + 13 = 15
combining like terms we get:
15 = 15
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since this is an identity, it appears that any value of any of the variables will get us the same answer.
for example:
let b = 20
that makes a = 18
that make c = 6 - 18 = -12
tht makes d = 7 - 18 = -11
that makes a + b + c + d =:
18 + 20 - 12 - 11 = 38 - 23 = 15
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this was an extreme example that is not realistic because nobody can send negative delegates but it just proves the point that any value assigned to any variable will provide the correct solution.
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there are limits, however, and those limits are:
each continent has a different number of delegates.
each continent sends at least one delegate.
each continent sends a different number of delegates.
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the additional information to help narrow the options down are:
b = a + 2
c = 6 - a
d = 7 - a
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the first equation tells us that b can't be less than or equal to 2, since if b were less than or equal to 2, then a would be less than or equal to 0.
this means that b has to be greater than 2.
this also means that the lower limit of a is 1.
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the second equation tells us that a can't be greater than or equal to 6, since if a is greater than or equal to 6, then b is less than or equal to 0.
we now know that a can be as low as 1 but no greater than 5.
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the third equation tells us that a can't be greater than or equal to 7 which is unnecessary since if a can't be greater than or equal to 6, then it can never be equal to or greater than 7.
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since a can be as low as 1 and as high as 5, i'll use a as my control variable.
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what i found was:
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the option: a = 1, b = 3, c = 5, d = 6, is not valid because no continent sent 4 delegates.
the option: a = 2, b = 4, c = 4, d = 5, is not valid because two continents sent the same number of delegates (4).
the option: a = 3, b = 5, c = 3, d = 4, is not valid because two continents sent the same number of delegates (3).
the option: a = 4, b = 6, c = 2, d = 3, is valid since all requirements are met.
the option: a = 5, b = 7, c = 1, d = 2, is not valid because no continent sent 4 delegates.
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you answer is:
a = 4
this means that asia is the continent that sent 4 delegates.
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we have:
asia sent 4
africa sent 6
america sent 2
europe sent 3
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total is 15.
asia and america sent a total of 6 (4+2=6)
asia and europe sent a total of 7 (4+3=7)
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