Solving for the Difference Between the First and Third Number

This problem involves finding the difference between two numbers when their relationships to each other and their average are provided. We need to use algebra to solve for the unknown numbers.

Setting Up the Number Relationships and Average

Let the three numbers be represented by $N_1$, $N_2$, and $N_3$. According to the question:

• The first number ($N_1$) is twice the second number ($N_2$): $N_1 = 2 \times N_2$
• The first number ($N_1$) is thrice the third number ($N_3$): $N_1 = 3 \times N_3$
• The average of the three numbers is $121$: $ \frac{N_1 + N_2 + N_3}{3} = 121 $

Step-by-Step Calculation

1. Express all numbers in terms of one variable:

   It's easiest to express $N_2$ and $N_3$ in terms of $N_1$, as $N_1$ is related to both.

   • From $N_1 = 2 \times N_2$, we get $ N_2 = \frac{N_1}{2} $.
   • From $N_1 = 3 \times N_3$, we get $ N_3 = \frac{N_1}{3} $.
2. Substitute into the average equation:

   Now, substitute the expressions for $N_2$ and $N_3$ into the average formula:

   $ \frac{N_1 + \frac{N_1}{2} + \frac{N_1}{3}}{3} = 121 $
3. Simplify the numerator:

   To add the fractions in the numerator, find a common denominator, which is $6$.

   $ N_1 + \frac{N_1}{2} + \frac{N_1}{3} = \frac{6N_1}{6} + \frac{3N_1}{6} + \frac{2N_1}{6} = \frac{6N_1 + 3N_1 + 2N_1}{6} = \frac{11N_1}{6} $
4. Solve for the first number ($N_1$):

   Substitute the simplified numerator back into the average equation:

   $ \frac{\frac{11N_1}{6}}{3} = 121 $

   This simplifies to:

   $ \frac{11N_1}{18} = 121 $

   Now, solve for $N_1$:

   $ 11N_1 = 121 \times 18 $ $ N_1 = \frac{121 \times 18}{11} $

   Since $121 \div 11 = 11$, we have:

   $ N_1 = 11 \times 18 $ $ N_1 = 198 $
5. Calculate the third number ($N_3$):

   Using the relationship $N_3 = \frac{N_1}{3}$:

   $ N_3 = \frac{198}{3} $ $ N_3 = 66 $
6. Find the difference between the first and third numbers:

   The question asks for the difference between $N_1$ and $N_3$.

   $ \text{Difference} = N_1 - N_3 $ $ \text{Difference} = 198 - 66 $ $ \text{Difference} = 132 $

Final Answer Summary

The first number is $198$, the second number is $99$, and the third number is $66$. The difference between the first number ($198$) and the third number ($66$) is $132$.