Hard word problem

Question

A fertilizer for grass contains 24% nitrogen and another that contains 12%. What quantity of each one do you need to combine to get 100kg of fertilizer that contains 21% nitrogen

Answer 1

Hello, Jordann.

 

The first sentence tells us that we have two solutions of fertilizer with different concentrations of nitrogen. The second sentence is asking us how much we need of each to make 100 kg with a particular concentration.

 

There are two unknowns here in this problem: the amount of solution with 24% N -- call it x -- and the amount of solution with 12% N -- call it y. We know that the total mass is 100 kg. Therefore, it must be true that x + y = 100. We have two unknowns and only one equation, so we'll need to use more information in the problem to get a second equation.

 

The second equation comes from the fact that we're mixing different solutions together to get a different concentration. A solution that is 24% of nitrogen by mass means that if we have 100 kg of this solution, there will be 24 kg of nitrogen in it. Likewise, 100 kg of the 12% solution will have 12 kg of nitrogen in it. The final solution we're trying to create has 21% by mass, so there will be 21 kg of nitrogen in it. Thus, our second equation is

 

(24/100) * x + (12/100) * y = (21/100) * 100 = 21 kg

 

This equation expresses the mass balance of nitrogen. The two terms on the left are products of concentration (mass/quantity) and quantity, which give units of mass. If you take chemistry, this will become more intuitive. The first equation, on the other hand, expresses the mass balance of the whole solution.

 

Solving these equations is a piece of cake by substitution. Writing them side by side, we have

 

x + y = 100

0.24x + 0.12y = 21

 

The first equation can be written as x = 100 - y. Substituting this into the second equation gives

 

0.24(100 - y) + 0.12y = 21

24 - 0.24y + 0.12y = 21

-0.12y = -3

y = 25 kg

 

Now we can solve for x. x = 100 - 25 = 75 kg.

 

Therefore, we need 75 kg of the 24% solution and 25 kg of the 12% solution to make 100 kg of the 21% solution. This answer makes sense as well. Since 21% is closer to 24%, we should need more of the 24% than the 12% solution. Indeed, this is the case.

 

I hope you found this useful. Take care.

 

Edward S.

Answer 2

Let x = amount of 24% fertilizer, then 100-x = amount of 12% fertilizer.  Set up the following equation:

 

x(.24) + (100-x)(.12) = 100(.21)

 

.24x + 12 - .12x = 21

 

.12x = 9

 

x = 75

 

You need 75 kg of 24% fertilizer and 25 kg of 12% fertilizer