Question

Tony and Cleo are donating some of their books to a hospital. If Tony contributes half of his books and Cleo gives one third of hers, they will donate a total of 30 books. If tony gives two fifths of his books and Cleo contributes half of hers, they will donate a total of 31 books. How many books does each have?

Answer 1

Short Answer: Tony 40 Cleo 30
Givens
T = Tony
C = Cleo

Equations
1/2 T + 1/3 C = 30 (1)
2/5 T + 1/2 C = 31 (2)

Adjustments
Multiply (1) by 3
Multiply (2) by 2

New Equations
3/2 T + C = 90 (1a)
4/5 T + C = 62 (2a) Subtract (2a) from (1a)
(3/2 - 4/5)T = 28 the common multiple of 2 and 5 = 10.
(15/10 - 8/10)T = 28
7/10 T = 28 Multiply by 10
7T = 28*10
7T = 280 Divide by 7
T = 280/7 = 40

Put T = 40 into (1)
1/2 T + 1/3 C = 30
1/2 (40) + 1/3 C = 30
20 + 1/3 C = 30 Subtract 20 from both sides.
1/3 C = 30 - 20
1/3 C = 10 Multiply through by 3
C = 10 * 3
C = 30

Answer:
Cleo has 30 books.
Tony has 40 books.

Check
Use Equation 2
2/5 T + 1/2 C = 31
2/5*40 + 1/2 * 30 = ? 31
16 + 15 =?3`
31 = 31 They are equal.