Question

A math teacher has two 15 question tests from previous years. No one question appears on both tests. She wants to make two new 20 question tests. She decides to add the first five questions of each 15 question test to the bottom of the other 15 question test. How many questions will these two new tests have in common,

Answer 1

Let

`T_1 = a_1,\ a_2,\ \ldots, a_(15)`

be the 15 questions of the first test, and let

`T_2 = b_1,\ b_2,\ \ldots, b_(15)`

be the 15 questions of the second test.

If we append the first 5 questions of each test to the bottom of the other, the two tests becomes

`T_1 = a_1,\ a_2,\ \ldots, a_(15),\ b_1,\ b_2,\ b_3,\ b_4,\ b_5`

`T_2 = b_1,\ b_2,\ \ldots, b_(15),\ a_1,\ a_2,\ a_3,\ a_4,\ a_5`

So, the tests have 10 questions in common:

`a_1,\ a_2,\ a_3,\ a_4,\ a_5,\ b_1,\ b_2,\ b_3,\ b_4,\ b_5`