Card distribution puzzle (simultaneous linear equations): Find the exact number of cards with C. Statements: If B gives A 3 cards, then B will have as many as A has now. If D takes 5 cards from B, D will have as many as E has now. A + C together equal twice E. B + D together equal A + C. Total cards with A, B, C, D, E is 150. Compute how many cards C has. Choose the correct value.

Let counts
A = a, B = b, C = c, D = d, E = e.

Translate statements
1) B gives A 3 ⇒ b − 3 = a. 2) D takes 5 from B ⇒ d + 5 = e. 3) a + c = 2e. 4) b + d = a + c. 5) a + b + c + d + e = 150.

Express in one variable
From 1: b = a + 3. From 2: e = d + 5. From 3: c = 2e − a = 2(d + 5) − a = 2d + 10 − a. From 4: (a + 3) + d = a + c ⇒ c = d + 3. Equate c: d + 3 = 2d + 10 − a ⇒ a = d + 7. Then b = d + 10, c = d + 3, e = d + 5.

Use total
a + b + c + d + e = (d + 7) + (d + 10) + (d + 3) + d + (d + 5) = 5d + 25 = 150. Solve: 5d = 125 ⇒ d = 25 ⇒ c = d + 3 = 28.