Math, asked by riyayadav23ya, 2 months ago

In a group of 10 elves and trolls, each were given a token with a different number from 1 to 10 written upon it. They were each asked whar number was on their token and all answered with a number from 1 to 10 . The sum of the answers was 36. Each troll told a lie and each elf told the truth.WHAT IS THE SMALLEST NUMBER OF TROLLS THERE COULD BE IN THE GROUP?​

Answers

Answered by irisychisholm
0

Answer:

B: 3

Step-by-step explanation:

Hope this answer helps! Also I tried it out so I know it's the right answer

Answered by qwstoke
0

Let's denote the number on the token of the i-th person by xi, for i = 1, 2, ..., 10. Since each person gave an answer from 1 to 10, we have:

x1 + x2 + ... + x10 = 1 + 2 + ... + 10 = 55

However, we are told that the sum of the answers was 36. Since the elves tell the truth, the sum of the numbers on their tokens must be 1 + 2 + ... + 5 = 15. Therefore, the sum of the numbers on the tokens of the trolls is 36 - 15 = 21.

Let's assume that there are t trolls in the group. Then, the sum of the numbers on their tokens is equal to the sum of the first t positive integers minus the sum of the next 10 - t positive integers. That is,

x6 + x7 + ... + x10 + (1 - y1) + (2 - y2) + ... + (10 - yt) = 1 + 2 + ... + t - (t + 1) - ... - 10

where yi = 0 or 1 depending on whether the i-th troll lies or tells the truth. Simplifying, we get:

x6 + x7 + ... + x10 - (y1 + y2 + ... + yt) = (t/2)(t + 1) - 55

We know that the sum of the numbers on the tokens of the trolls is 21, so we can write:

x6 + x7 + ... + x10 = 21 + y1 + y2 + ... + yt

Substituting this into the previous equation, we get:

21 + y1 + y2 + ... + yt - (y1 + y2 + ... + yt) = (t/2)(t + 1) - 55

21 = (t/2)(t + 1) - 55

t^2 + t - 152 = 0

Solving for t using the quadratic formula, we get:

t = 7 or t = -22

Since the number of trolls must be a positive integer, we conclude that there are 7 trolls in the group.

#SPJ3

Similar questions