Question 306388: Combined there are 202 Asians, African, Europeans, and Americans in a village. The number of Asians exceeds the number of Africans and Europeans by 71. The difference between the number of Europeans and Americans is 16. If the number of Africans is doubled, their population exceeds the number of Europeans and Americans by 18. Determine the number of Asians, Africans, Europeans, and Americans in this village.
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! Combined there are 202 Asians, African, Europeans, and Americans in a village. The number of Asians exceeds the number of Africans and Europeans by 71. The difference between the number of Europeans and Americans is 16. If the number of Africans is doubled, their population exceeds the number of Europeans and Americans by 18. Determine the number of Asians, Africans, Europeans, and Americans in this village.
A + F + E + M = 202
A - F - E + 0M = 71
0A + 0F + E - M = 16 --> E = M + 16 (substitute in)
0A + 2F - E - M = 18
to get
A + F + M + 16 + M = 202
A - F - M - 16 + 0M = 71
0A + 2F - M - 16 - M = 18
or
A + F + 2M = 186
A - F - M = 87
0A + 2F - 2M = 34 --> F - M = 17 --> F = M + 17 (substitute in)
to get
A + M + 17 + 2M = 186
A - M - 17 - M = 87
or
A + 3M = 169
A - 2M = 104 (picked this one of the 2 to multiply by -1 and add to other)
to get
A + 3M = 169
-A + 2M = -104 (add to one above)
to get
0A + 5M = 65 --> M = 13 (number of Americans is 13)(plug into 2nd)
- A + 2M = -104
to get
- A + 26 = -104
or
A - 26 = 104 --> A = 130 (number of Asians is 130)
now you got from A + F + E + M = 202
130 + F + E + 13 = 202
this
F + E + 143 = 202 --> F + E = 59
we know from above E = M + 16 so E = 13 + 16 = 29 (number of Europeans is 29)
so
F + 29 = 59 --> F = 30 (number of Africans is 30)
then
A + F + E + M = 202
130 + 30 + 29 + 13 = (A + F) + (E + M) = 160 + 42 = 202
Thus ends this difficult system of equations and substitution problem, hope this helps someone.
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