Word Problem

Question

We mix 300 ml of brown paint with white paint. With that mix we obtain 1L of beige paint. If the brown paint costs 19.35/L and 4L of beige paint costs 36.80, how much does 1L of white paint cost.

Answer 1

I'm not sure whether there's an easier way to solve this than my method, I'm imagining it has to do with weight averages, but I'm not familiar with that concept and find it personally confusing believe it or not. So...let's do it my way. Pure logic.

 

In any word problem, you want to take the facts one piece at a time, starting with, what is the problem asking? Then, break down the facts. 

 

Question: cost of 1L of white paint?

 

Facts:

1. 300 mL of brown + white paint = 1 L of beige 
2. Brown paint costs 19.35/L
3. 4 L of beige costs 36.80

There are two facts we can derive. First of all, we know that 1 L = 1000 mL (because a milliliter is a 1000th of a Liter). Fact #1 is missing the quantity of white paint, but because we're so bright, we can say:

1000 mL total minus 300 mL brown = 700 mL white.

 

Secondly, fact #3 give us the cost of 4 L of beige, and I can only imagine they stated it this way to make the careless student excited and quickly give the answer for the price per gallon for beige paint by simply dividing by 4. This is why it's so important to make sure you keep in mind what the question is asking. Regardless, we want the cost of a single Liter, so let's divide by 4 and say:

1 L of beige costs 9.20

 

The new, derived facts are:

1. 300 mL brown + 700 mL white = 1 L of beige 
2. Brown paint costs 19.35/L
3. Beige paint costs 9.20/L

This problem is a bit tricky. The question is asking about a LITER of white paint's cost, but the solution for beige has 700 mL, so we have to start of by saying, how much does 700 mL white paint cost?

 

 

 

So here's the big picture, keeping in mind that a Liter of beige costs 9.20:

   Cost of 300mL BR    +     Cost of 700 mL W    =     9.20  (Cost of 1000mL beige)

 

 

Let's look at it in terms of percentages. If a liter of beige takes 300 mL brown and 700 mL white, then every beige mixture is 300/1000 brown, and 700/1000 white. Translation: 30% brown + 70% white = 100% beige

 

 

We don't know what the cost of 700 mL of white is yet, but we CAN figure out what the cost of 300 mL of brown is, by taking the cost per Liter of brown and multiplying by 30%. ⇒ 19.35 x 0.3 =  5.805 (Even though there's no such thing as half a cent, don't round to 5.81 yet. Rule of thumb, only round the final answer.)

 

Now let's take this number and plug it into our big picture equation.

 

Cost of 300mL BR + Cost of 700 mL W = 9.20 (Cost of 1000mL beige)

 

5.805 + Cost of 700 mL W = 9.20 (Cost of 1000mL beige)

 

(Subtract 5.805 from both sides and...)

 

Cost of 700 mL W = 3.395

 

So, we're done, right?....NOT EVEN! The question asks for the cost of a LITER of white paint, not 700 mL. So, let's set up a ratio. 700 mL to 1000 mL is a 7/10 ratio.

 

7 parts = 3.395

10 parts = ?

 

In fraction form:

 7   =   3.395

10           x 

 

Cross multiply:

 

7x = 3.395 * 10

 

Multiply the right side of the equation:

 

7x = 33.95

 

Divide both sides by 7:

 

x = 4.85

 

Ta-da! Now, make sure your answer makes sense. Does it make sense that 700 mL of the white paint costs about 3.40, and 1 L costs 4.85? Yup, sounds good to me.