Answer is 11
METHOD 1: Strategy: Find the sum of the five numbers.
If the average of the five numbers is 8, then the sum of those five numbers is 5 × 8, or 40. Subtract the sum of the two known numbers, 7, from 40. The sum of the other three numbers is 33. The value of any one of the three equal numbers is 33 ÷ 3, or 11.
METHOD 2: Strategy: Start with all equal.
The simplest set of 5 numbers whose average is 8 is 8, 8, 8, 8, and 8. Replacing one of the 8's with 2 drops the total by 6, and replacing another with 5 drops the total still further by 3. The total of the five numbers must be increased by 9 (since 6 + 3 = 9), so each of the three remaining 8's must be increased by 3 (three 3's = 9). That means the numbers are 2, 5, and three 11's. The value of any one of the three equal numbers is 11.
FOLLOW-UPS: The average of five different natural numbers is 8. Consider the largest of these numbers.(1)What is its largest possible value? [30]
(2) What is its smallest possible value? [10]
METHOD 1: Strategy: Find the sum of the five numbers.
If the average of the five numbers is 8, then the sum of those five numbers is 5 × 8, or 40. Subtract the sum of the two known numbers, 7, from 40. The sum of the other three numbers is 33. The value of any one of the three equal numbers is 33 ÷ 3, or 11.
METHOD 2: Strategy: Start with all equal.
The simplest set of 5 numbers whose average is 8 is 8, 8, 8, 8, and 8. Replacing one of the 8's with 2 drops the total by 6, and replacing another with 5 drops the total still further by 3. The total of the five numbers must be increased by 9 (since 6 + 3 = 9), so each of the three remaining 8's must be increased by 3 (three 3's = 9). That means the numbers are 2, 5, and three 11's. The value of any one of the three equal numbers is 11.
FOLLOW-UPS: The average of five different natural numbers is 8. Consider the largest of these numbers.(1)What is its largest possible value? [30]
(2) What is its smallest possible value? [10]
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