Four girls purchase a video game for $60. The first girl paid 1/2 the sum of the amounts paid by the other three girls. The second girl paid 1/3 the sum of the

Question

Four girls purchase a video game for $60. The first girl paid 1/2 the sum of the amounts paid by the other three girls. The second girl paid 1/3 the sum of the amounts paid by the other three. The third girl paid 1/4 the sum of the amounts paid by the other three girls. How much did the fourth girl pay?

Answer 1

Hi James

 

We have 4 girls, w,x,y,z

 

w = 1/2(x+y+z)

x = 1/3(w+y+z)  w = 3x-y-z

y = 1/4(w+x+z)  w = 4y-x-z

w+x+y+z = 60

w = 60-x-y-z

 

60-x-y-z = 3x-y-z

60-x = 3x

60=4x

x = 15

So the 2nd girl paid $15 and the other 3 paid a total of $45

 

Let's rewrite the equations above using this value of x

w = 1/2(15+y+z)    
15 = 1/3(w+y+z) w = 3(15)-y-z = 45-y-z
y = 1/4(w+15+z) w = 4y-15-z 
w+15+y+z = 60 
w = 60-15-y-z

 

45-y-z = 4y-15-z

45-y = 4y-15

60=5y

y = 12

The third girl paid $12 of the total

 

Let's rewrite the equations above using this value of y

w = 1/2(15+12+z) = 27/2 + z/2
15 = 1/3(w+y+z) w = 3(15)-12-z = 33-z
12 = 1/4(w+15+z) w = 4(12)-15-z 
w+15+12+z = 60
w = 60-15-12-z

 

27/2 + z/2 = 33-z

27 + z = 66-2z

3z = 39

z = 13

The 4th girl paid $13

 

and w, the first girl paid 60-15-12-13 = $20

 

Now let's check

20+15+12+13 = 60

 

20 = 1/2 (15+12+13) = 1/2 (40) = 20

15 = 1/3 (20+12+13) = 1/3 (45) = 15

12 = 1/4 (20+15+13) = 1/4 (48) = 12