Question

The average of 7 consecutive whole numbers is 7. Then 1 is added to the first number, 2 is added to the second number, 3 is added to the third number, and so on. What is the average of the seven new numbers? Use the variable n to solve. Let n be the first number.

Answer 1

Let's use the variable n to represent the first number.

The seven consecutive whole numbers can be represented as: n, n+1, n+2, n+3, n+4, n+5, n+6.

We know that the average of these numbers is 7, so:

(n + n+1 + n+2 + n+3 + n+4 + n+5 + n+6) / 7 = 7

Multiplying both sides of the equation by 7 gives us:

7n + 7(n+1) + 7(n+2) + 7(n+3) + 7(n+4) + 7(n+5) + 7(n+6) = 7 * 7

Simplifying both sides of the equation gives us:

7n + 7n + 7 + 14n + 21 + 21n + 42 + 28n + 49 + 35n + 70 + 42n + 84 = 49

Combine like terms:

147n + 273 = 49

Subtract 273 from both sides of the equation:

147n = 49 - 273

147n = -224

Now, divide both sides of the equation by 147:

n = -224 / 147

Simplifying the fraction gives us:

n ≈ -1.52

Therefore, the first number is approximately -1.52.

Adding 1, 2, 3, 4, 5, 6, and 7 respectively to each of the numbers n, n+1, n+2, n+3, n+4, n+5, n+6, we get:

n = -1.52
n+1 = -0.52
n+2 = 0.48
n+3 = 1.48
n+4 = 2.48
n+5 = 3.48
n+6 = 4.48

The average of these numbers can be calculated by adding them together and dividing by 7:

(-1.52 + -0.52 + 0.48 + 1.48 + 2.48 + 3.48 + 4.48) / 7

= (10.88) / 7

= 1.55 (rounded to two decimal places)

Therefore, the average of the seven new numbers is approximately 1.55.