math problem

Question

Combined, there are 184 Asians, Africans, Europeans, and Americans in a village.

The number of Asians exceeds the number of Africans and Europeans by 54.

The difference between the number of Europeans and americans is 15.

If the number of Africans is doubled, their population exceeds the number of

Europeans and Americans by 35.

Determine the number of Asians, Affricans, Europeans, and Americans in the village.

Answer 1

This is a systems of equations problem.

First, assign letters to each race. These are your variables.

Asians = A

Africans = F

Europeans = E

Americans = M

 

 

You are told there are 184 total.

 

Therefore A + F + E + M = 184

 

 

 

You are told the number of Asians exceeds the number of Africans and Europeans by 54.

 

Therefore A - 54 = F + E

 

 

 

You are told that the difference between the number of Europeans and Americans is 15. You do not know if Europeans are more or Americans are more.

 

Therefore the absolute value of the difference = 15.

Abs(Americans-Europeans) = 15.

 

(Note the order of the subtraction doesn't matter due to the absolute value. Abs(Europeans-Americans)=15 is valid as well.)

 

 

 

You are told that if the number of Africans is doubled, their population exceeds the number of Europeans and Americans by 35. 2*Africans = Europeans + Americans +35.

 

Therefore 2*A = E + M + 35. 

 

 

You now have 4 equations and 4 variables which means there is are 1 or 2 unique solution to this set of equations:

 

1) A + F + E + M = 184

2) A - 54 = F + E

3) abs(M - E) = 15 - This can be broken down into 2 possibilities.

3a)M - E = 15 and 3b)E - M = 15

4) 2F = E + M + 35

 

 

This is the setup for the problem There are multiple ways to go about solving this. Just make sure you use both equations 3a and 3b separately. You should end up with 2 sets of answers.

 

I worked through it and got:

 

Using equation 3a) (Assuming Americans outnumber Europeans by 15)

 

M = 28

E = 13

F = 38

A = 105

 

Using equation 3b) (Assuming Europeans outnumber Americans by 15)

 

M = 13

E = 28

F = 38

A = 105