E. Using Fractions with the Imperial System

Parts of units in the imperial system are often listed as fractions, instead of decimals. Often when fractions are given in a problem, it is expected that they will also be used in the solution to the problem.

 

Example 1

Covert «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«/math» mi into yards.

When performing an operation on a mixed fraction such as «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«/math», it is often best to convert it to an improper fraction. A mixed fraction is really a whole number added to a fraction:

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mn»4«/mn»«mn»4«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mrow»«mn»4«/mn»«mo»+«/mo»«mn»3«/mn»«/mrow»«mn»4«/mn»«/mfrac»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mn»7«/mn»«mn»4«/mn»«/mfrac»«/math»

Recall that a shorter way of converting is to simply multiply the denominator by the whole number and add the product to the numerator to get the new numerator:

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»1«/mn»«mfrac»«mn mathcolor=¨#FF00FF¨»3«/mn»«mn mathcolor=¨#1C82BA¨»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mo»(«/mo»«mn mathcolor=¨#1C82BA¨»4«/mn»«mo»§#8729;«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo»)«/mo»«mo»+«/mo»«mn mathcolor=¨#FF00FF¨»3«/mn»«/mrow»«mn»4«/mn»«/mfrac»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mn»7«/mn»«mn»4«/mn»«/mfrac»«/math»

 

 

Example 2

Darwin needs two pieces of plastic border edging for a garden that is in the corner of his yard. If one side is of the garden is  «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»12«/mn»«mo»`«/mo»«mn»7«/mn»«mfrac»«mn»5«/mn»«mn»8«/mn»«/mfrac»«mo»`«/mo»«mo»`«/mo»«/math» and the other is «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mo»`«/mo»«mn»9«/mn»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mo»`«/mo»«mo»`«/mo»«/math», how much border edging is required?


 Adding fractions requires a common denominator. You can change the denominator in a fraction by multiplying by a second fraction that has the same value in the numerator and the denominator.

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»5«/mn»«mn»8«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»5«/mn»«mn»8«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mo»§#8729;«/mo»«mfrac mathcolor=¨#B94A48¨»«mn»2«/mn»«mn»2«/mn»«/mfrac»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mn»5«/mn»«mn»8«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mn»3«/mn»«mo»§#8729;«/mo»«mn»2«/mn»«/mrow»«mrow»«mn»4«/mn»«mo»§#8729;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mn»5«/mn»«/mstyle»«mstyle displaystyle=¨true¨»«mn»8«/mn»«/mstyle»«/mfrac»«mo»+«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mn»6«/mn»«/mstyle»«mstyle displaystyle=¨true¨»«mn»8«/mn»«/mstyle»«/mfrac»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mstyle displaystyle=¨true¨»«mn»11«/mn»«/mstyle»«mstyle displaystyle=¨true¨»«mn»8«/mn»«/mstyle»«/mfrac»«/math»

To determine the total length, the individual measurements will need to be added. This addition can be completed by converting everything to inches and then adding them or by adding the different units separately. This example shows adding the units separately.

Start by adding the smallest unit, the inches.

There are 12 inches in a foot, so this measurement can be written as

Now add this measurement to the feet.

So of border edging is required.