The answer is they sold 127 cold sandwiches, 254 hot dogs, and 181 hamburgers.

s – the number of sandwiches

d – the number of hot dogs

h – the number of hamburgers

The price for s number of sandwiches is: $2.50s

The price for d number of hot dogs is: $1.50d

The price for h number of hamburgers is: $2h

The students have collected $1060.50: $2.50s + $1.50d + $2h = $1060.50

The students have sold 562 items: s + d + h = 562

The students sold twice as many hot dogs as cold sandwiches: d = 2s

Let’s use substitution method.

First, let’s substitute d from the third equation into the second equation and solve it for h in the term of s:

s + d + h = 562

d = 2s

s + 2s + h = 562

3s + h = 562

h = 562 – 3s

Further, substitute h and d from the second and the third equations, respectively, into the first equation and solve it for s:

2.50s + 1.50d + 2h = 1060.50

d = 2s

h = 562 – 3s

2.50s + 1.50 * 2s + 2(562 – 3s) = 1060.50

2.50s + 3s + 1124 – 6s = 1060.50

-0.5s + 1124 = 1060.50

1124 – 1060.50 = 0.5s

63.5 = 0.5s

s = 63.5/0.5 = 127

d = 2s = 2 * 127 = 254

h = 562 – 3s = 562 – 3 * 127 = 562 – 381 = 181