A box contains tiles, numbered 1, 2, 3,..., 49, 50. Only tiles which are marked with a number congruent to $2 \pmod{5}$ are blue. One tile is chosen randomly from the box. What is the probability that the tile is blue?
Positive integers which are congruent to $2\pmod{5}$ belong to the set $$\{2+5(0), 2+5(1), 2+5(2), ..., \}.$$To find the largest element of this set which is less than or equal to 50, we look for the largest possible integer $n$ such that $$2+5(n-1) \le 50.$$Solving this inequality, we find $n \le 53/5$, so the maximum integer solution is $n=\lfloor 53/5 \rfloor = 10$. Since there are 50 total tiles, the probability that the tile is marked with a number congruent to $2 \pmod{5}$ is $\dfrac{10 \; \text{blue tiles} }{50 \; \text{total tiles}} = \boxed{ \frac{1}{5} } .$
\frac{1}{5}
Level 3
Number Theory
Problem: A box contains tiles, numbered 1, 2, 3,..., 49, 50. Only tiles which are marked with a number congruent to $2 \pmod{5}$ are blue. One tile is chosen randomly from the box. What is the probability that the tile is blue?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability that a randomly chosen tile is blue, we first need to determine how many tiles are blue. A tile is blue if its number is congruent to 2 modulo 5. This means we are looking for numbers of the form 5k + 2, where k is a non-negative integer.

Next, we identify the range of k that keeps the tile number within the limits of 1 to 50. Setting up the inequality:

1 ≤ 5k + 2 ≤ 50

Subtracting 2 from all parts gives:

-1 ≤ 5k ≤ 48

Dividing the entire inequality by 5 results in:

-0.2 ≤ k ≤ 9.6

Since k must be a non-negative integer, the possible values for k are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This gives us 10 valid values for k.

Now, we calculate the corresponding blue tile numbers:

- For k = 0: 5(0) + 2 = 2
- For k = 1: 5(1) + 2 = 7
- For k = 2: 5(2) + 2 = 12
- For k = 3: 5(3) + 2 = 17
- For k = 4: 5(4) + 2 = 22
- For k = 5: 5(5) + 2 = 27
- For k = 6: 5(6) + 2 = 32
- For k = 7: 5(7) + 2 = 37
- For k = 8: 5(8) + 2 = 42
- For k = 9: 5(9) + 2 = 47

Thus, the blue tiles are: 2, 7, 12, 17, 22, 27, 32, 37, 42, and 47. This gives us a total of 10 blue tiles.

Next, we calculate the total number of tiles in the box, which is 50. The probability of selecting a blue tile is the ratio of the number of blue tiles to the total number of tiles:

Probability = (Number of blue tiles) / (Total number of tiles) = 10 / 50 = 1 / 5.

Therefore, the final answer is $\frac{1}{5}$.<|END_OF_LATENT|> The final answer is $\frac{1}{5}$.