A department of motor vehicles asks visitors to draw numbered tickets from a dispenser so that they can be served in order by number. Six friends have graduated from truck-driving school and go to the department to get commercial driving licenses. They draw tickets and find that their numbers are a set of evenly spaced integers with a range of 10. Which of the following could NOT be the sum of their numbers?a)1,254b)1,428c)3,972d)4,316e)8,010Correct answer is option 'D'. Can you explain this answer?

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A department of motor vehicles asks visitors to draw numbered tickets from a dispenser so that they can be served in order by number. Six friends have graduated from truck-driving school and go to the department to get commercial driving licenses. They draw tickets and find that their numbers are a set of evenly spaced integers with a range of 10. Which of the following could NOT be the sum of their numbers?

a)
1,254
b)
1,428
c)
3,972
d)
4,316
e)
8,010

Correct answer is option 'D'. Can you explain this answer?

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Most Upvoted Answer

A department of motor vehicles asks visitors to draw numbered tickets ...

Explanation:

Given Information:
The numbers drawn by the friends are a set of evenly spaced integers with a range of 10.

Let's analyze the options:

a) 1,254:
- The sum of evenly spaced integers can be calculated using the formula: Sum = (Number of terms / 2) * (First term + Last term)
- In this case, the first term can be calculated as (Sum - (Number of terms - 1) / 2) / Number of terms
- Substituting the values, we find that 1,254 can be the sum of their numbers.

b) 1,428:
- Similarly, using the formula, we find that 1,428 can be the sum of their numbers.

c) 3,972:
- By applying the formula, we find that 3,972 can be the sum of their numbers.

d) 4,316:
- Calculating the first term using the formula, we find that it does not result in evenly spaced integers with a range of 10.
- Therefore, 4,316 could NOT be the sum of their numbers.

e) 8,010:
- Applying the formula, we find that 8,010 can be the sum of their numbers.

Conclusion:
Option D, 4,316, could NOT be the sum of their numbers as it does not align with evenly spaced integers with a range of 10.

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Community Answer

A department of motor vehicles asks visitors to draw numbered tickets ...

Let's assume the smallest number drawn by one of the friends is x. Since the range is 10, the largest number drawn would be x + 10.

The sum of evenly spaced integers can be found using the formula:

Sum = (Number of terms / 2) * (First term + Last term)

In this case, the number of terms is 6 (since there are 6 friends) and the first term is x, while the last term is x + 10.

So, the sum of the numbers would be:

Sum = (6 / 2) * (x + (x + 10))
= 3 * (2x + 10)
= 6x + 30

Now, let's check each option to see if it can be obtained as the sum of the numbers:

A) 1,254:
6x + 30 = 1,254
6x = 1,224
x = 204
The sum can be obtained.

B) 1,428:
6x + 30 = 1,428
6x = 1,398
x = 233
The sum can be obtained.

C) 3,972:
6x + 30 = 3,972
6x = 3,942
x = 657
The sum can be obtained.

D) 4,316:
6x + 30 = 4,316
6x = 4,286
x ≈ 714.333
The sum cannot be obtained with evenly spaced integers since x is not an integer.

E) 8,010:
6x + 30 = 8,010
6x = 7,980
x = 1,330
The sum can be obtained.

Therefore, the sum that could NOT be the sum of their numbers is D) 4,316.

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