The scores on a certain exam were 60, 75, 100, 70 and x

Question

The scores on a certain exam were 60, 75, 100, 70 and x. If the median score was 5 points lower than the average (arithmetic mean) score on the exam, which of the following could be x?

A. 75
B. 80
C. 85
D. 90
E. 95

OE
 Rearranging the scores that we know: 60, 70, 75, 100. We are told that the median score is 5 points lower than the average score; the median score represents the number in the middle, or in this case the third highest score.
Assume for now that the number will be 75.
That would make the average score on the test 80, so the total for all 5 scores would be 80 x 5 = 400.
If 60 + 75 + 100 + 70 + x = 400, then x = 400 - (60 + 75 + 100 + 70) = 95.
This in fact would make 75 the median, x = 95

Hi, I want to know if we can use algebra to solve this, please.

PareshGmat · Accepted Answer

Look at the options available

Value of x seems to 75 < = x < = 95

A. 75
B. 80
C. 85
D. 90
E. 95

So position of x is fourth

60 , 70 , 75 , x , 100

Median = 75

Median score was 5 points lower than the average, so average = 80

60 + 70 + 75 + x + 100 = 400

x = 95 = Answer = E

GMarc · Answer

what do you mean by using algebra ?

goodyear2013 · Answer

I mean to solve it without picking numbers

GMarc · Answer

I don’t think it is possible for the simple reason that this problem involves the « median » value for which it is almost impossible to get a algebraic formula. When you encounter the median in a problem, you should always try to pick numbers because it is unlikely you will solve the problem without doing so.

catalysis · Answer

I think it is relatively easy to solve using algebra. We can see from the answer choices that the median will have to be 75, because all answer choices make x equal to 75 or larger. Therefore, we know we are aiming for x to be a number such that the average of all 5 scores is 80 (because the prompt states that the median is 5 points lower than the average).

Thus, we can set up the simple algebraic equation:
60 + 70 + 75 + 100 + x = 80*5
305 + x = 400
x = 95

Thus, we can set up the simple algebraic equation: 
60 + 70 + 75 + 100 + x = 80*5
305 + x = 400
x = 95

veergmat · Answer

Average= (60+70+75+100+X)/5 = (305+x)/5

Median= Avg.-5
 = (280+x)/5
 = 56+ x/5

Now, put each option in the place of x to match the value of Median with the given numbers

For, x=90 
Median comes out to be 75.

Average of the given 4 nos. is 76.25 and median is 72.5. Difference between Average and median is 3.75
Now, on adding one more no. that is X the difference increases from 3.75 to 5, this means average has increased so,
 x>76.25.

chenm93 · Answer

1.  Mean = \frac{(60 + 70 + 75 + 100 + X)}{5}

2.  Mean = Median + 5

Mean = \frac{(305 + X)}{5} = \frac{305}{5} + \frac{X}{5} = 61 + \frac{X}{5} 
 Median = 61 + \frac{X}{5} - 5 
 Median = 56 + \frac{X}{5}

The Median value must either be 60, 70, 75, 100 or X.

60 = 56 + \frac{X}{5}; X = 20 
 70 = 56 + \frac{X}{5}; X = 70 
  75 = 56 + \frac{X}{5}; X = 95  
 100 = 56 + \frac{X}{5}; X = 220

X = 56 + \frac{X}{5} 
 \frac{4}{5}*X = 56; X = 70

You can see that the only answer that is also an answer choice is when X = 95, and the median is 75.

SVaidyaraman · Answer

The average has to end in 0 or 5 because we can see all the numbers end in 5 or 0 and the average is 5 more than the median.

The total of the 4 numbers is 305. Since this is a "could be" question, we can answer it immediately by seeing adding 95 to 305 , makes the average 80. We see 75 is indeed the median. Hence E.

ScottTargetTestPrep · Answer

The average is:

(60 + 75 + 100 + 70 + x)/5 = (305 + x)/5

Let’s plug in answer choices to see which could be x, given that the median is 5 less than the average:

A) average = 380/5 = 76, and scores are 60, 70, 75, 75, 100; mean is 76 and median is 75 doesn’t work.

B) average = 385/5 = 77, and scores are 60, 70, 75, 77, 100; mean is 77 and median is 75 doesn’t work.

C) average = 390/5 = 78, and scores are 60, 70, 75, 85, 100; mean is 78 and median is 75 doesn’t work.

D) average = 395/5 = 79, and scores are 60, 70, 75, 90, 100; mean is 79 and median is 75 doesn’t work.

E) average = 400/5 = 80, and scores are 60, 70, 75, 95, 100; mean is 80 and median is 75, which is exactly 5 points less; this is possible.

Answer: E

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The scores on a certain exam were 60, 75, 100, 70 and x

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