Question

How much $ 25\% $ antifreeze and $ 50\% $ antifreeze should be combined to give $ 40 $ gallons of $ 30\% $ antifreeze?

Answer 1

Hint: In this question, we need to determine the combination of gallons. Thus, we will construct the equations using the given conditions and consider that $ x $ of $ 25\% $ antifreeze and $ y $ of $ 50\% $ antifreeze should be combined to give $ 40 $ gallons of $ 30\% $ antifreeze. Then by solving it we will determine the required solution.

Complete step by step solution:
Now, let us consider that $ x $ of $ 25\% $ antifreeze and $ y $ of $ 50\% $ antifreeze should be combined to give $ 40 $ gallons of $ 30\% $ antifreeze.
Then, the equation can be given as,
 $ x + y = 40 $ $ \to \left( 1 \right) $
And, also it should give $ 30\% $ antifreeze.
Thus, the equation can be given as,
 $ 25\dfrac{x}{{40}} + 50\dfrac{y}{{40}} = 30 $
 $ \Rightarrow 25x + 50y = 30 \times 40 $
 $ \Rightarrow 25x + 50y = 1200 $
 $ \Rightarrow x + 2y = 48 $ $ \to \left( 2 \right) $
Now, let us rewrite the equation $ \left( 1 \right) $ as,
 $ x + y = 40 $
 $ x = 40 - y $
Now, let us substitute this in equation $ \left( 2 \right) $ , we have,
 $ 40 - y + 2y = 48 $
 $ 40 + y = 48 $
 $ y = 48 - 40 $
 $ y = 8\,gallons $
Now substituting the value of $ y $ in equation $ \left( 1 \right) $ , we have,
 $ x + 8 = 40 $
 $ x = 40 - 8 $
 $ x = 32\,gallons $
Hence, $ x = 32\,gallons $ and $ y = 8\,gallons $ .
So, the correct answer is “ $ x = 32\,gallons $ and $ y = 8\,gallons $ ”.

Note: Usually, to frame equations read the problem carefully and figure out what is asked to find. Then assign a variable to the quantity that we are finding. Also, mention what the variable represents. Again re-read the problem and write an equation for the quantities given in the problem. Then, finally by solving the equations, we can determine the required value.Students need to see the quantities and the relationship between them. They need to see the general quantities involved and how those are related to each other.