Ethan had a sum of money. If he bought 5 muffins and 6 tarts, he would be short of $5. If he bought 3 muffins and 4 tarts, he would have $19 left. Given that he had $61, how many tarts could he buy at most?
Cost of 1 muffin = 1 M
Cost of 1 tart = 1 T

5 M + 6 T - 5 = 61
5 M + 6 T = 61 + 5 
5 M + 6 T = 66 --- (1)

3 M + 4 T + 19 = 61 
3 M + 4 T = 61 - 19 
3 M + 4 T = 42 --- (2)

Make M the same.

(1) _{x 3}
15 M + 18 T = 198 
15 M = 198 - 18 T ---- (3)

(2) _{x 5}
15 M + 20 T = 210 
15 M = 210 - 20 T ---- (4)

(3) = (4)
198 - 18 T = 210 - 20 T 
20 T - 18 T = 210 - 198 
2 T = 12

1 T = 12 ÷ 2 = 6 

Cost of 1 tart = $6 

Number of tarts that Ethan can buy at most
= 61 ÷ 6
= 10 r 1

Most number of tarts that Ethan can buy = 10

Answer(s): 10