Determine the number of Asians, Africans, Europeans, and Americans in this village

Question

Combined, there are 178 Asians, Africans, Europeans, and Americans in a village. The number of Asians exceeds the number of Africans and Europeans by 57. The differenc 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 38. Deter the number of Asians, Africans, Europeans, and Americans in this village.

Answer 1

This type of problem aims to "trick" you by providing you with all the necessary equations to solve the problem - but disguises them in a word problem. The way to solve these is to figure out the equations from the word problem then work through them to solve for the missing values. Let's walk through the text to find the equations:

Combined, there are 178 Asians, Africans, Europeans, and Americans in a village.

In this first sentence, they've provided us with our first equation. I'm going to use the following variables in the equations, but you could use any variable to represent these numbers:

s (Asian)

f (Africans)

e (Europeans)

m (Americans)

So in this first sentence, we know that if we add the number of all four villager types, we get 178. Thus:

s + f + e + m = 178

The number of Asians exceeds the number of Africans and Europeans by 57.

Here we have our 2nd equation; we know that the total number of Asian villagers is equal to the African villagers plus the European villagers plus 57. Thus:

s = f + e + 57

The difference between the number of Europeans and Americans is 16.

Our third equation says that if you subtract the number of American villagers from the number of European villagers, we'd have 16 left over. Thus:

e - m = 16

If the number of Africans is doubled, their population exceeds the number of Europeans and Americans by 38.

The 4th and final equation says that if we had twice as many African villagers, it would be the same as if we added all European and American villagers - then added 38 more. Thus:

f * 2 = e + m + 38

All in all, we have the following four equations (rearranged slightly):

s + f + e + m = 178

f + e + 57 = s

e - m = 16

e + m + 38 = 2f

Now, we need to determine the number of Asians, Africans, Europeans, and Americans in this village. We have all the information we need. There are a variety of methods to solve this problem, but I'll just cover the method in which we solve for single variables to plug into other equations. Well work through solving for the first value, a single one of these values, before using that number to find all the other values.

First, we need to find an equation that we can quickly solve for a single variable (in other words, get a variable on one side of the equals sign by itself). If you notice, we already have one equation in that format:

s = f + e + 57

So we already know what s equals (even if we don't have the exact number yet). We can quickly get another one of these equations into this same format. We can rearrange e - m = 16 by adding m to both sides. This leaves us with:

e = m + 16

Let's plug these values back into the first equation:

s + f + e + m = 178 (plug in the value for s we have above)

(f + e + 57) + f + e + m =178 (Let's also plug in the value for e for each time we have e in the equation)

f + (m+16) + 57 + f + (m+16) + m = 178 (Now, let's combine terms)

2f + 3m + 89 = 178

We could go through and try to isolate either f or m like we did with s and e, but notice something here. We have the term 2f in our simplified equation above. We already know what 2f equals from our 4th equation, 2f=e + m + 38. Let's just plug in the value for 2f (instead of trying to divide everything by 2 then plugging that into the equation).

2f + 3m + 89 = 178 (plug in the equation for 2f)

(e + m + 38) + 3m + 89 = 178 (remember, we already know what e equals, let's plug that value in here)

(m + 16) + m + 38 + 3m + 89 = 178 (Combine terms)

5m + 143 = 178 (subtract 143 from both sides)

5m = 35 (divide both sides by 5)

m = 7

There! You've solve for the number of American villagers (7). We can now use this value to solve for the rest of the number of villagers. Find an equation that, if you plug in the number of American villagers, will allow you to solve for another villager type number. In this case...

e - m = 16, or when rearranged, e = m+16

Thus, the number of European villagers is equal to 7 + 16, or 23.

We're already halfway done. Let's try to solve for another missing variable using the two known variables. We can't use the first equation, as there are still two unknown variables in that equation. We want to find the equation that only has 1 unknown variable in it. So here, we'd use:

e + m + 38 = 2f

Let's plug in the known numbers:

23 + 7 + 38 = 2f

68 = 2f (divide both sides by 2)

f = 34

And we have 3 of the four values. Now, we can use the f + e + 57 = s equation to solve for the final missing variable (s). We could use the first equation, but I want to save that to the end because I want to use it to test and make sure I got all the right values. So let's plug in the known values:

34 + 23 + 57 = s

114 = s

Now we have all four values. But we need to check to make sure we got the right numbers. Let's plug the four known values into the first equation:

s + f + e + m = 178

114 + 34 + 23 + 7 = 178

178 = 178

It works! That's how you confirm you got the correct values.

Good luck!