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34kurt
3 years ago
9

A list contains twenty integers, not necessarily distinct. Does the list contain at least two consecutive integers?

Mathematics
1 answer:
algol133 years ago
8 0

Answer:

Answer c is correct. Both statements arent sufficient separately, but together they are sufficient.

Step-by-step explanation:

Neither of the statments are sufficient alone. Lets first analyze why statment (1) alone is not sufficient.

If we suppose that our list is scattered, so our values are pretty distant one to the other, one value increasing by one wont change this. The list will still be scattered and our values will remain distant one to the other.

For a concrete example, consider a list with values multiples of a thousand. Our list can look this way {1000,2000,3000, ....., 19000, 20000}. So we have 20 distinct values. Adding one to any of this 20 values wont change the fact that we have 20 different values. However, our list doesnt have 2 consecutive integers.

Proving (2) not being sufficient is pretty straightfoward. If our list contains 20 equal values it wont have 2 consecutive values, because all values are equal. For example the list with values {0, 0, 0, 0, ....., 0} will have the value 0 occuring more than once but it doesnt have 2 consecutive values.

Now, lets assume both statements (1) and (2) are valid.

Since (2) is valid there is an integer, lets call it <em>k</em>, such that <em>k</em> appears on the list at least twice. Because (1) is True, then if we take one number with value <em>k </em>and we increase its value by one, then the number of distinct values shoudnt change. We can observe a few things:

  1. There are 19 numbers untouched
  2. The only touched value is <em>k</em>, so our list could only lost <em>k </em>as value after adding 1 to it.
  3. Since <em>k</em> appears at least twice on the list, modifying the value of one<em> number</em> with value <em>k</em> wont change the fact that the rest of the numbers with value <em>k</em> will <em>preserve</em> its value. This means that k is still on the list, because there still exist numbers with value k.
  4. The only number that <em>could</em> be new to the list is k+1, obtained from adding 1 to k

By combining points 2 and 3, we deduce that the lists doesnt lose values, because point 2 tells us that the only possible value to be lost is k, and point 3 says that the value k will be preserved!

Since the list doesnt lose values and the number of different values is the same, we can conclude that the list shoudnt gain values either, because the only possibility for the list to adquire a new value after adding one to a number is to lost a previous value because the number of distinct numbers does not vary!

Point 4 tells us that value k+1 was obtained on the new list after adding 1 to k. We reach the conclusion that the new list doesnt have new values from the original one, that means that k+1 was alredy on the original list.

Thus, the original list contains both the values k (at least twice) and k+1 (at least once), therefore, the list contains at least two consectutive values.

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Mis Meyer's paid $3600 all together for the equipment, furniture and decorations for her restaurant. The equipment cost $500 mor
Tju [1.3M]

Answer:

The equipment will cost $1740.

Step-by-step explanation:

Let the Cost of equipment, furniture and decoration be x, y and z.

Now, According to question,

x + y + z = 3600 ...... (1)  (cost of all items)

x = 500 + y (∵ equipment cost 500 more than furniture)

and y = 2z ( ∵ furniture twice as much as decoration)

so, z = y/2

Now substituting the value of x and z in eq (1)

x + y + z = 3600

500 + y + y + \frac{y}{2} = 3600

2y + \frac{y}{2} = 3600 - 500

\frac{5y}{2} = 3100

y = \frac{3100\times 2}{5} = 1240

So, the cost of furniture (y) = 1240

∴ Cost of equipment = y + 500 = 1240 + 500 = 1740

Therefore the cost of equipment was $1740.

3 0
3 years ago
What is the median of the following data set ? {3,4,2,8,5}
Varvara68 [4.7K]

<u>4</u>

<u></u>

Explanation:

3, 4, 2, 8, 5

<u>arrange in ascending order</u>

2, 3, 4, 5, 8

<u>Identify the integer at the middle</u>

median: 4

6 0
2 years ago
Two sides of a triangle are given as 28 inches and 42 inches. find all possible lengths for the third side of the triangle
timofeeve [1]
The longest side of  a triangle must be less than the sum of the other 2 sides

2 senarios

the 3rd side is the longest
the 3rd side is not the longest



for the 3rd side is the longest

3rdside<28+42
3rdside<70


for 3rd side isn't the longest
42 is longest
42<28+3rdside
14<3rdside

so we've got
3rdside<70 and 14<3rdside

so
14<3rdside<70

the 3rd side can be any number from 14in to 70in except 14in and 70in
6 0
3 years ago
a square dance set requires 4 couples 8 dancers with each couple standing on one side of a square. There are 250 people at a squ
sveta [45]

Given:

1 set requires 4 couples 8 dancers.

Total number of people at a square dance = 250.

To find:

The greatest number of sets possible at the dance.

Solution:

We have,

Total people = 250

1 set = 8 people.

\text{Number of possible sets}=\dfrac{\text{Total people}}{\text{People required for 1 set}}

\text{Number of possible sets}=\dfrac{250}{8}

\text{Number of possible sets}=31.25

Number of possible sets cannot be a decimal or fraction value. So, approx. the value to the preceding integer.

\text{Number of possible sets}\approx 31

Therefore, the number of possible sets at the dance is 31.

7 0
2 years ago
What number is 60 percent of 29
nevsk [136]
60% is 0.60 so you do 0.60 x 29 which = to 17.4
so your answer is 17.40 or 17.4 is 60% of 29
7 0
3 years ago
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