The average of 6 consecutive even numbers is 37. What is the largest of these numbers?Options

Sure, let's solve this step by step.

1. Understanding the problem:

   • We are given that the average of 6 consecutive even numbers is 37.
   • We need to find the largest of these 6 numbers.

2. Setting up the problem:

   • Let the 6 consecutive even numbers be $x, x+2, x+4, x+6, x+8, x+10$.
   • These numbers are consecutive even numbers, so they increase by 2 each time.

3. Finding the average:

   • The average of these numbers is given by the formula: $\text{Average} = \frac{\text{Sum of the numbers}}{\text{Number of numbers}}$
   • The sum of the numbers is: $x + (x+2) + (x+4) + (x+6) + (x+8) + (x+10)$
   • Simplifying the sum: $x + x + 2 + x + 4 + x + 6 + x + 8 + x + 10 = 6x + 30$
   • The average is given as 37, so: $\frac{6x + 30}{6} = 37$

4. Solving for $x$:

   • Multiply both sides by 6 to clear the fraction: $6x + 30 = 37 \times 6$
   • Calculate $37 \times 6$: $37 \times 6 = 222$
   • So, we have: $6x + 30 = 222$
   • Subtract 30 from both sides: $6x = 192$
   • Divide both sides by 6: $x = 32$

5. Finding the largest number:

   • The largest number in the sequence $x, x+2, x+4, x+6, x+8, x+10$ is $x+10$.
   • Substitute $x = 32$: $x + 10 = 32 + 10 = 42$

So, the largest of these 6 consecutive even numbers is 42.