Three friends Amy, Brad, and Chadwick have a certain number of oranges in their possession.

It is known that:

(1) The number of oranges possessed by each of the three friends are two digit positive integers.

(2) The number of oranges possessed by Brad is obtained by adding the two digits of the number of oranges possessed by Amy and summing the result with the original number of oranges in Amy's possession.

(3) The number of oranges possessed by Chadwick is obtained by reversing the digits corresponding to the number of oranges in Brad's possession.

(4) The total number of oranges possessed by the three friends is 272.

Determine the number of oranges possessed by each of the three friends.


Solution to the Problem:


Amy has 85 oranges, Brad has 98 oranges, and Chadwick has 89 oranges.

Let the number of oranges possessed respectively by Amy and Brad be (10x+y) and (10a+b).
Then the total number of oranges in Todd's possession is (10b+a)

By the problem: (i) 10x+ y + 11(a+b) = 272; (ii) 10a+ b = 11x + 2y

Reducing both sides of (i) in modulo 11, we have:
y-x = - 3(Mod 11). Since x and y must be less than 10, this is possible if y- x = -3, 8.


If y -x = 8, then:
Either (y,x) = (9,1); giving a+b=23 in terms of (i). This is not feasible as max(a+b) < 9+9 = 18.
Or, (y,x) = (8,0); giving a+b = 24 in terms of (i), which is not feasible for similar reason.

If y-x = -3; then x = y+3, so that:
a+ b+ y = 22 and 10a + b -13y = 33 in terms of (i) and (ii).
Accordingly, 9a - 14y = 11.
Since, both a and y must be less than 10, this is feasible only if (a,y) = (9,5)
Hence, x =5+3 = 8, so that b = 22 - 8 - 5 = 9.

Consequently, the respective number of oranges in the possession of Amy, Brad and Chadwick are 85, 98 and 89.

Correctly solved by:

1. Colin (Yowie) Bowey Beechworth, Victoria, Australia
2. Ritwik Chaudhuri Santiniketan, West Bengal, India
3. Brijesh Dave Mumbai City, Maharashtra, India
4. Dr. Hari Kishan D.N. College,
Meerut, Uttar Pradesh, India
5. Davit Banana Istanbul, Turkey
6. Seth Cohen Concord, New Hampshire
7. Ivy Joseph Pune, Maharashtra, India
8. Kelly Stubblefield Mobile, Alabama