Lesson Math circle level problem on average

Source code of 'Math circle level problem on average'

This Lesson (Math circle level problem on average) was created by by ikleyn(52906)

: View Source, Show
About ikleyn:

<H2>Math circle level problem on average</H2>

<H3>Problem 1</H3>The average of four three digit numbers is &nbsp;500. 
Two of the numbers are &nbsp;150 &nbsp;and &nbsp;230. 
What is the largest difference between the other two numbers?

<B>Solution</B>

<pre>
Let the four numbers be "a", "b", "c" and "d".

We are given that {{{(a+b+c+d)/4}}} = 500,  which implies

a + b + c + d = 4*500 = 2000.    (1)

We also are given that two of them are 150 and 230: let

a = 150  and  b = 230.

Then from (1)

c + d = 2000 - a - b = 2000 - 150 - 230 = 1620.

Thus we know about "c" and "d" that

c + d = 1620  and  c and  d are 3-digit numbers.

+---------------------------------------------------------------+
       |    The fact that  "c" and "d"  are 3-digit integer numbers    |
       |                   means that they are                         |
       | greater than or equal to 100 and lesser than or equal to 999. |
       +---------------------------------------------------------------+

So again, we know that c + d = 1620  and "c" and "d" are integer numbers 
greater than or equal to 100 and lesser than or equal to 999.

When the difference |c - d| is maximum ?

First answer,  which lies on the surface, is this:

when "c" is as great as it is possible; then d is minimal.

Second answer, which lies on the surface, is this:

when "d" is as small as it is possible; then c is maximal.

So, we should check, which of these two possibilities will really happen.

(1)  "c"s greatest possible value is 999; then "d"'s value is 1620-999 = 621 
          and it is in the allowable range between 100 and 999;
          then the difference |c-d| is |999-621| = 378.

(2)  "d"s least   possible value is 100; then "c"'s value is 1620-100 = 1520 
          and it is OUT of the allowable range between 100 and 999.

Therefore, the answer to the problem's question is THIS

the greatest possible difference  |c-d| is 378,  when c = 999 and d = 621.
</pre>

It can be derived in more formal fashion, but I consciously appeal to your intuition and common sense.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;To make it even &nbsp;MORE &nbsp;VISIBLE, &nbsp;you may think this way:

<pre>
    the sum c + d = 1620 is frozen;

hence, the average of numbers "c" and "d", {{{(c+d)/2}}} = {{{1620/2}}} = 810 is frozen, too,

and  100 <= c, d <= 999.

When the difference |c-d| is maximum ?

- It is maximum when the distance between "c" and "d" 
              on the number line is maximum, with their frozen average.

But it is obvious that it will happen 
         when "c"-value will reach its maximum, which is 999.
</pre>

My other lessons on &nbsp;<B>Average, &nbsp;Mean and Median</B>&nbsp; in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Geometric-Mean.lesson>WHAT IS Geometric mean</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Difference-between-Amean-and-Gmean.lesson>Difference between Arithmetic mean and Geometric mean</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/What-is-Median.lesson>WHAT IS Median</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Solved-problems-on-average-scores-weight-height-and-temperature.lesson>Solved problems on average scores, weight, height and temperature</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Solved-problems-on-average-scores.lesson>Solved problems on average scores</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Solved-problems-on-average-age.lesson>Solved problems on average age</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Miscellaneous-problems-on-average-values.lesson>Miscellaneous problems on average values</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Entertainment-problems-on-average.lesson>Entertainment problems on average</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/The-mean-and-the-total-value-problem-for-the-International-Fools-day-of-April-1.lesson>The mean and the total value problem for the International Fools' day of April, 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/OVERVIEW-of-lessons-on-Average-Mean-and-Median.lesson>OVERVIEW of lessons on Average</A>