Question

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

Answer 1

`16 2/7 - 2 1/2*1 1/7 + 9/14 = 14 1/14`

Explanation:

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 `5/2*8/7` first:

`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` by `2/2` to give it a denominator `14` like the other fractions:

`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` by `14` to give a quotient `14` and remainder `1`, so:

`197/14 = 14+1/14 = 14 1/14`

Answer 2

`14 1/14`

Explanation:

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

`:.=114/7-(5/2 xx 8/7)+9/14`

`:.=114/7-(40/14)+9/14`

`:.=(228-40+9)/14`

`:.=197/14`

`:.=14 1/14`

Answer 3

`=14 1/14`

Explanation:

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.

`16 2/7 color(blue)(-2 1/2 xx 1 1/7) +9/14`

`=16 2/7color(blue)(-5/2 xx8/7) +9/14`

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.

`16 4/14 color(blue)(-40/14) +9/14" "` all have the same denominator

`=16 4/14 color(blue)(-2 12/14) +9/14" "` change to a mixed number

`= 14 (4-12+9)/14`

`=14 1/14`

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.)