Answer:
Emma started with 10
Jack with 8
and Ben with 12
(30 total)
Step-by-step explanation:
Emma started with 10
Jack with 8
and Ben with 12
(30 total)
after the swap, they each end up with 10:
Ben kept 8 and got 2 (a fifth of 10)
Jack kept 6 and got 4 (a third of 12)
Emma kept 8 and got 2 (a quarter of 8)
The constraints that the total number of counters is < 40, and that they each pass more than one counter makes this a unique solution. Without these constraints, any combination in the ratio of 4:5:6 (J:E:B) would work...
Now for the work:
If Ben passed 1/3, he kept 2/3
If Jack passed 1/4, he kept 3/4
If Emma passed 1/5, she kept 4/5
so we know:
2/3B + 1/5E = 3/4J + 1/3B = 4/5E + 1/4J
you can rearrange these in pairs to find the ratio between B,E, and J, the original number of counters each had.
I actually multiplied everything by 60 first to get rid of the fraction, even though they came back quickly:
40B + 12E = 20B + 45J = 48E + 15J
From the first two parts:
20B = 45J - 12E (**)
and from the last two:
20B = 48E - 30J (***)
giving: 45J - 12E = 48E - 30J
or 75J = 60E, or J = 4/5E
now plugging back into **
20B = 45(4/5 E) - 12E
20B = 36E - 12E = 24E
B = 6/5 E
so now J:E:B are in the ratio of (4/5E):E:(6/5)E, or
4:5:6
we can say J = 4x, E = 5x, and B = 6x, and find values of x that satisfy the given conditions
if x = 1, J = 4, E = 5, and B = 6, but Emma and Jack have to pass more than one, so this doesn't work
if x = 2, J = 8, E = 10, and B = 12 (the solution that works)
if x = 3, J = 12, E = 15, and B = 18, giving a total of 45 counters (too many)
