The average of 15 numbers is 19. If the average of the first 10 numbers is 15 and the average of the last 6 numbers is 25, what is the 10th number?

Sure, let's solve this step by step.

1. Understand the problem:

   • We have 15 numbers.
   • The average of these 15 numbers is 19.
   • The average of the first 10 numbers is 15.
   • The average of the last 6 numbers is 25.
   • We need to find the 10th number.

2. Calculate the total sum of the 15 numbers:

   • The average of 15 numbers is 19.
   • Therefore, the total sum of these 15 numbers is $15 \times 19 = 285$.

3. Calculate the total sum of the first 10 numbers:

   • The average of the first 10 numbers is 15.
   • Therefore, the total sum of these 10 numbers is $10 \times 15 = 150$.

4. Calculate the total sum of the last 6 numbers:

   • The average of the last 6 numbers is 25.
   • Therefore, the total sum of these 6 numbers is $6 \times 25 = 150$.

5. Identify the overlap and solve for the 10th number:

   • Notice that the 10th number is included in both the first 10 numbers and the last 6 numbers.
   • The total sum of the 15 numbers is the sum of the first 10 numbers plus the sum of the last 6 numbers, but the 10th number is counted twice in this addition.
   • Therefore, we need to subtract the 10th number once to get the correct total sum.

   Let's denote the 10th number as $x$.

   The equation becomes: $150 + 150 - x = 285$

6. Solve for $x$: $300 - x = 285$ $x = 300 - 285$ $x = 15$

So, the 10th number is $15$.