Faisal chooses four numbers. When each number is added to the mean (average) of the other three, the following sums are obtained: \(25\), \(37\), \(43\), and \(51\).
Determine the mean of the four numbers Faisal chose.
Extra Problem: Can you interpret the following picture puzzle? You may need to research the meanings of some mathematical symbols used in the puzzle.
Let \(a,b,c,\) and \(d\) represent the four numbers. It is possible to precisely determine the four numbers, but the problem asks for only their average, which is \(\frac{a+b+c+d}{4}\).
When the first number is added to the average of the other three numbers, the result is \(25\). Thus,
\[a+\frac{b+c+d}{3}=25\]
which can be rewritten as
\[3a+b+c+d=75 \tag{1}\]
When the second number is added to the average of the other three numbers, the result is \(37\). Thus,
\[b+\frac{a+c+d}{3}=37\]
which can be rewritten as
\[a+3b+c+d=111 \tag{2}\]
When the third number is added to the average of the other three numbers, the result is \(43\). Thus,
\[c+\frac{a+b+d}{3}=43\]
which can be rewritten as
\[a+b+3c+d=129 \tag{3}\]
When the fourth number is added to the average of the other three numbers, the result is \(51\). Thus,
\[d+\frac{a+b+c}{3}=51\]
which can be rewritten as
\[a+b+c+3d=153 \tag{4}\]
Adding equations \((1)\), \((2)\), \((3)\), and \((4)\), we obtain \(6a+6b+6c+6d=468\). Dividing this equation by \(6\) gives \(a+b+c+d=78\). It follows that \(\dfrac{a+b+c+d}{4}=19.5\).
Therefore, the average of the four numbers is \(19.5\).
Although it is not required, we could solve the system of equations to determine that the numbers are: \(-1.5\), \(16.5\), \(25.5\), and \(37.5\).
Extra Problem Solution:
The notation \(\dfrac{1}{n} \displaystyle \sum_{i=1}^{n} x_i\) is a mathematical short form which represents the average of the \(n\) numbers \(x_1,x_2,\ldots,x_n\). So the picture puzzle can be interpreted as “Be greater than average”.