How many liters of a 32 % alcohol solution must be mixed with 15 liters of a 81% alcohol solution to get a 46% alcohol in the entire solution?

37.5 liters of 32% solution

Step-by-step explanation:

Assign variables

let "a" be the number of liters to add from the 32% alcohol solution

let "c" be the number of liters to have a 46% alcohol solution

Create equations using the information given in the problem.

Equation for total liters

a + 15 = c

Equation for percentage

0.32a + 0.81(15) = 0.46c

0.32a + 12.15 = 0.46c

Substitute the expression for total liters into the equation for percentage. Both equations have "c".

0.32a + 12.15 = 0.46c                   Equation for percentage

0.32a + 12.15 = 0.46(a + 15)                 Substituted expression

0.32a + 12.15 = 0.46a + 6.9

Start isolating "a"

0.32a + 12.15 - 12.15 = 0.46a + 6.9 - 12.15     Subtract 12.15 on both sides

0.32a = 0.46a - 5.25

0.32a - 0.46a = 0.46a - 0.46a - 5.25             Subtract 0.46a on both sides

-0.14a = -5.25

-0.14a/-0.14 = -5.25/-0.14                    Divide both sides by -0.14

a = 37.5                      Liters to add from 32% alcohol solution

If you need to know how many liters your final solution is:

Substitute a = 37.5 into the equation for total liters

a + 15 = c  

37.5 + 15 = c                     Add

c = 52.5                 Liters of final solution

Check your answer using the equation for percentage

0.32a + 0.81(15) = 0.46c             Substitute a=37.5 and c=52.5

0.32(37.5) + 0.81(15) = 0.46(52.5)              Multiply each term

12 + 12.15 = 24.15               Add to simplify

24.15 = 24.15            Same answer

LS = RS                   Left side equals right side

Therefore 37.5 liters of the 32% alcohol solution must be added with 15 liters of an 81% alcohol solution.

 37.5 liters

Step-by-step explanation:

Note: Create a table. Multiply across and add down. The bottom row creates the equation.

                                   0.46(x + 15) = 0.32x + 12.15

                                   0.46x + 6.9 = 0.32x + 12.15

                                   0.14x  + 6.9 =              12.15

                                   0.14x           =               5.25

                                          x           =               37.5