Bob plays a game at the school fair. he starts with 0 points. he gets 25 throws. he wins 12 points for hitting the target and loses 8 points for each miss. bob ends with a score of 0. how many hits and misses does bob have?

Hello,
Let's
     n the numbers of win throws
  25-n the number of lost

12*n+(-8)*(25-n)=0
==>12n-200+8n=0
==>20n=200
==>n=10 (win throws)
and 25-10=15 (lost)

Proof: 10*12+(-8)*15=120-120=0

For  a "11 year old nephew":

multiple of 12 ={0,12,24,36,48,60,72,84,96,108,120,132,144,}
multiple of 8  =[0,8,16,24,32,40,48,56,64,72,80,88,96,104,112,120,128,136,...}

Commun mutiples are {0,120,240,360...}

0=0*12-8*0    and (0-0 not equal to 25)
0=10*12-15*8 and 10+15=25 (get it)

0=20*12-8*30 and (20+8 not equal to 25)

Answer is thus win: 10 and lost 15.
This will be understandable for a kid of 11 years.