Question 1204434
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I'll convert each of the given mixed numbers into decimal form
78 & 2/5 = 78.4
12 & 1/2 = 12.5
2 & 3/5 = 2.6
4 & 1/2 = 4.5
Each result is exact.


Draw out a horizontal number line. Let's say the number line starts at x = 0 and ends at x = 78.4 to represent the width of the wall.
Don't worry about the diagram being to scale or not.


One picture takes up 12.5 cm of horizontal space along this number line.
Five pictures of the same width take up 5*12.5 = 62.5 cm
Add in the four gaps between neighboring pictures and we add another 4*2.6 = 10.4 cm
So far 62.5+10.4 = 72.9 cm is accounted for.


I'll assume that Peter is hanging the pictures so that they are symmetrically spaced between each opposite wall. 


If so then let k be the distance from the wall's edge to the first picture near that wall. It also represents the buffer distance from the other wall to the last picture. 


We add 2 copies of k onto the 72.9 and set it equal to 78.4 so we can solve for k.


k+72.9+k = 78.4
2k+72.9 = 78.4
2k = 78.4-72.9
2k = 5.5
k = 2.75


Let's say we started at the left wall.
Move 2.75 cm to the right to arrive at the left edge of the first picture. 
Move another (12.5)/2 = 6.25 cm to the right to arrive at the x coordinate of the center of the 1st picture. 
Move left (4.5)/2 = 2.25 cm so that we can locate the x coordinate of the first nail.
This places us at the x coordinate 2.75 + 6.25 - 2.25 = 6.75


Answer: <font color=red>6.75 cm</font>
This converts to the mixed number 6 & 3/4 or we can write it as {{{matrix(1,2,6,3/4)}}}


Edit: Tutor MathTherapy arrived at a different answer because he interpreted the problem differently than I did. 
Consider labeling the first four nails as A,B,C,D
My interpretation is that distance(A,B) = 4.5 cm while MathTherapy's interpretation is distance(B,C) = 4.5
Upon closer inspection, both interpretations seem valid. The wording "leaving 4.5 cm between each pair" is a bit vague.
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