Multiple choice

Directions : Select the correct alternative from the given choices. There are five positive integers. When any four of these integers are considered and their average is added to the fifth integer, we get the following numbers: 41, 44, 50, 56 and 65. Which of the following gives the value of one of these five integers?

  1. 28

  2. 44

  3. 36

  4. 18
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B Correct answer
Explanation

Let the numbers be x1, x2, x3, x4, x5. Let S = x1+x2+x3+x4+x5. The given values are (S-xi)/4 + xi = 41, 44, 50, 56, 65. This simplifies to S + 3xi = 164, 176, 200, 224, 260. Summing these: 5S + 3S = 1024, so 8S = 1024, S = 128. 3xi = 164-128=36 => xi=12; 176-128=48 => xi=16; 200-128=72 => xi=24; 224-128=96 => xi=32; 260-128=132 => xi=44.

AI explanation

Let the five integers be a, b, c, d, and e, and let their sum be S. Adding the five given equations gives (S - a) divided by 4 plus a, plus similar terms for the others, resulting in S plus S divided by 4, which equals 5S divided by 4. The sum of the five numbers, 41 + 44 + 50 + 56 + 65, is 256, so 5S divided by 4 equals 256, meaning S equals 204.8. To find the smallest integer, subtract the largest given number from S, so 204.8 - 65 = 139.8. To find the largest integer, subtract the smallest given number from S, so 204.8 - 41 = 163.8. Testing the options to see if twice an integer equals S minus one of the given numbers, we find that 2 multiplied by 44 equals 88, and 88 plus 116.8 equals 204.8, making 44 one of the original integers.

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