"32 pages [per hour] " .
__________________________________________
Explanation:
_____________________________________________
Set up a ratio:
______________________________________________
2 ⅔ pages : 5 minute:: x pages: 1 hour ; Solve for "x" ;
Rewrite as:
_______________________________________
(2 ⅔ pages) / 5 minutes = "x" pages/ 1 hour.
Cross-multiply ; and write as:
(5 minutes) ("x" pages) = (2 ⅔ pages) / 1 hour. ;
________________________________________
Convert the "5 min" to its values in units of "hour" ;
________________________________________
5 min = ? hour ?
________________________________________
5 min * (
= (
hr ;
(
hr = [(5÷5)/(60÷5)] hr = (1/12) hr.
___________________________________________________
So we take our ratio, which is in the form of an equation, as follows:
___________________________________________________
(5 minutes) ("x" pages) = (2 ⅔ pages) * 1 hour.
___________________________________________________
and rewrite — substituting "(1/12 hr)" in lieu of "(5 min)" — as follows:
___________________________________________________
(1/12) * ("x" pages) = (2 ⅔ pages) * (1) ; Solve for "x" ;
___________________________________________________
Divide EACH SIDE of the equation by: "(1/12 hr)"; or, more efficiently, multiply EACH SIDE of the equation by "12 hr" ; since "dividing by "(1/12)" is the same as multiplying by "12" ;
(i.e. on the "left hand side" of the equation:
[ (1/12) x ] / (1/12) = x ;
AND: (1/12)x * 12 = (1x/12) * (12/1) = (12x/12) = x ; the same result.
{or: (1/12)x * 12 = (x/12) * (12/1) = ? ; The "12's" cancel out to "1's" ;
since: "12÷12 = 1" ; and we have:
" (x/1) * (1/1) = x * 1 = x ; which is the same value.
_____________________________________________________
to isolate "x" on one side of the equation:
_____________________________________________________
So, as explained above, we multiply EACH SIDE of the equation by "12" ; as follows:
_____________________________________________________
[ (1/12 hr) * ("x") ] * 12 = (2 ⅔ pages) * (1) * (12) ;
______________________________________________
Simplify:
______________________________________________
x = (2 ⅔ pages) * (12) ;
x = (8/3) * (12/1) = ?
The "3" cancels to a "1" ; and the "12" cancels to a "4" ;
_____________________________________________
since: "12÷3 = 4" ; and since: "3÷3 = 1 ".
_____________________________________________
So rewrite as:
_____________________________________________
x = (8/1) * (4/1) = 8 * 4 = 32 pages.
_____________________________________________
Alternate:
_____________________________________________
When we have: x = (2 ⅔ pages) * (12) ;
x = (8/3) * (12/1) = (8*12)/ (3*1) = 96/3 = 32 .
x = 32 pages.
_______________________________________________
Alternate:
_______________________________________________
When we have: x = (2 ⅔ pages) * (12) ;
x = (8/3) * (12/1) = (8*12)/ (3*1) ;
x = (8*12)/ (3*1) ;
__________________________________________________
We can cancel out the "12 in the numerator" and the "3" in the denominator, and replace with "4" and "1", respectively; and rewrite as:
x = (8*4) / (1*1) = (32/1) = 32.
x = 32 pages per hour.
__________________________________________________
