WORD PROBLEM

Question

A 100% concentrate is to be mixed with a mixture having a concentrate of 42% to obtain 53 gallons with a concentrate of 71%. How much of the 100% concentrate will be needed? 

Answer 1

Let's say we need x gallons of the 100% concentrate. Then we will need 53-x gallons of the 42% concentrate. If the outcome is 53 gallons of 71% concentrate we get a linear equation to solve.

 

100x + 42(53 - x) = 71·53

 

100x + 2226 - 42x = 3763

 

58x = 1537

 

x = 1537/58 = 26.5

 

This means we need 26.5 gallons of the 100% concentrate.

 

Incidently we also need 26.5 gallons of the 42% concentrate.

 

This makes sense since 71% is exactly half way between 42% and 100% so we expect to need equal amounts of both mixtures.

 

Answer 2

100%(x)+42%(53-x)=71%(53)

1(x)+0.42(53-x)=0.71(53)

x+22.26-0.42x=37.63

0.58x+22.26=37.63

0.58x=37.63-22.26

0.58x=15.37

58x=1537

x=1537/58

x=26.5 gallons which is half of 53

you are mixing 26.5 gallons of the 100% solution with 26.5 gallons of the 42% solution

why the same number of gallons ?

the concentrate is 71% which is (100%+42%)/2=142%/2=71%