if you had 15 pieces of plywood and they averaged 2.3 pieces thick per inch, how thick would the 15 pieces be if you stacked them together?

Question

Any help would be greatly appreciated 

Answer 1

To change units without changing the value, we use the Multiplicative Identity (that is, 1).  In this problem:

   there are :

        1 = (2.3 pieces) per inch

        1 = (2.3 pieces) / (1 inch)

        1 = (1 inch) / (2.3 pieces)

 

We have:

      15 pieces

       (15 pieces)*1

       (15 pieces)*((1 inch)/(2.3 pieces))

       (15)/(2.3) inches          [note: pieces cancel out]

       6.5 inches        [note: if you are working with plywood, then 6.52 can not be obtained]

Answer 2

You get the average by adding all of the measured values and dividing by the number of measurements.  To get the average pieces per inch, you had to measure the total stacked thickness of all 15 (probably).  But instead of an average measurement of thickness per piece, we are given average pieces per inch.   However, we can turn this into average thickness by inverting the number.  Instead of 2.3 pieces (thickness) per inch, change it to 1/2.3 to convert it to inches (thick) per piece.   Then multiplying by 15 pieces will give the total thickness. 

 

You can see how this works out by careful attention to the units of measurement:

 

# pieces / inches thick inverted = inches thick / # pieces 

# pieces x inches thick / # pieces = inches thick. 

 

Your answer works out to be 15/2.3, or 6.52173913 inches total thickness.   This makes sense: if there are 2.3 pieces per inch, average, then the average thickness is less than an inch and close to 0.4 inches.