Question

Regina rode her bike `2 1/2` miles from her house to school and then `1 1/9` miles to her friend's house. How many miles did Regina ride in total?

Answer 1

Explanation takes longer than doing the actual maths with shortcuts.

`3 color(white)(.)11/18`

Explanation:

A fraction's structure is `("count")/("size indicator")->("numerator")/("denominator")`

You can not `ul("directly")` add or subtract the counts (numerators) unless the size indicators (denominators) are the same

`color(blue)("Converting these into 'improper fraction'")`

Consider: `2 1/2" "` Write as #2+1/2" "->" "color(green)([2color(red)(xx1)] color(white)(.)+1/2)#
`" = "color(green)([2color(red)(xx2/2)]+1/2)`
#" = "color(green)([4/2] color(white)(...)+1/2#

`" "=" "(4+1)/2 = 5/2`

Consider `1 1/9`: by the same method we have:`" " 10/9`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting this all together")`

`" "2 1/2 + 1 1/9" "=" "5/2" "+" "10/9`

`" "color(green)(=[5/2color(red)(xx1)]+[10/9color(red)(xx1)])`

`" "color(green)(=[5/2color(red)(xx9/9)]+[10/9color(red)(xx2/2)])`

`" "color(green)(=" "[45/18]color(white)(.)+" "[20/18])`

Now that the size indicators (denominators) are the same we can directly add the counts (numerators

`" "=(45+20)/18=65/18`

`" "=3 11/18`

Answer 2

Alternative approach

`3 11/18`

Explanation:

`2 1/2+1 1/9`

`[2+1]+[1/2+1/9]`

`3+9/18+2/18`

`3+11/18`

`3 11/18`