# How do you evaluate #16\frac { 2} { 7} - 2\frac { 1} { 2} \cdot 1\frac { 1} { 7} + \frac { 9} { 14}#?

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Given:

#16 2/7 - 2 1/2*1 1/7 + 9/14#

First let us convert all of the mixed numbers to improper fractions:

#16 2/7 = 16 + 2/7 = (16*7)/7+2/7 = 112/7+2/7 = 114/7#

#2 1/2 = 2 + 1/2 = (2*2)/2 + 1/2 = 4/2+1/2 = 5/2#

#1 1/7 = 1 + 1/7 = 7/7+1/7 = 8/7#

So our original expression can be rewritten as:

#114/7 - 5/2*8/7 + 9/14#

Next note that multiplication has higher precedence than addition or multiplication, so we need to perform the multiplication

#114/7 - color(blue)(5/2*8/7)+9/14 = 114/7 - (5*8)/(2*7)+9/14 = 114/7-40/14+9/14#

In order to add or subtract these fractions, they need to have the same denominators, so we multiply

#114/7 = (114*2)/(7*2) = 228/14#

So our express can be rewritten:

#228/14-40/14+9/14 = (228-40+9)/14#

Then note that addition and subtraction have the same priority, so we need to evaluate them from left to right:

#(color(blue)(228-40)+9)/14 = (188+9)/14 = 197/14#

To express this as a mixed number, we divide

#197/14 = 14+1/14 = 14 1/14#

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It is possible to work with the whole numbers separately when adding or subtracting.

Only for multiplication do we have to use improper fractions.

There are 3 terms, do the multiplication first.

We would usually cancel in the middle term. However we will need a common denominator in the next step so leave as it is. Simplify by multiplying straight across.

Working with mixed numbers gives us smaller numbers to work with. The numbers in improper fractions are often uncomfortably big.

(Answer in the same form as the numbers were given.)

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