Question #f7fe4

First, let's rearrange the order to that we can identify the smallest and largest numbers. Here are the symbols that I use and what it means. `>` is 'greater than' and `<` is 'less than'.

We know that the negative numbers should be smallest, as they are below 0. Remember the greater the number of the negative the smaller the number, so `-5 1/6 < -2 1/3` .

We also know that `8/17 < 3/5` because if we convert these fraction into decimals we get `8/17 = 0.42105263157` and `3/5 = 0.6`, and `0.42105263157` is smaller than `0.6` . Therefore `3/5 > 8/19`.

We know `4 1/7` has the greatest value since it is a positive number and it is greater than `3/5` because. To prove this lets rewrite `4 1/7` as `29/7`.

To get a common denominator for both fractions, we do `29/7 xx 5/5` and `3/5 xx 7/7`. So we get `29/7 = 135/35 and 3/5 = 21/35`.

This proves that `4 1/7 > 3/5` because `135/35 > 21/35`

Now that we know the order. Let's find the difference between the smallest and largest numbers.

Here's the order `-5 1/6, -2 1/3, 8/19, 3/5 and 4 1/7`

The largest number is `4 1/7` and the smallest is `-5 1/6`.
Let's find their difference `4 1/7 - (-5 1/16)`.
Firstly, we need to convert these mixed numbers into improper fractions so that we can subtract them more easily so `4 1/7 = color(green)(29/7` and `-5 1/6 = color(green)(-31/6`.

Now let's find the common denominator of these two fractions. Now the LCM of 6 and 7 is 42 so that will be our common denominator.
Let's do the operation to make the denominators of both of these fractions the same.
`29/7 xx 6/6 = color(green)(174/42) and -31/6 xx 7/7 = color(green)(-217/42`

Remember never add or subtract the denominators!

Finally, let's find the difference:
`174/42 -(-217/42) = 174/42 + 217/42 = color(green)(391/42`

Simplify the fraction so make it a Mixed number. We know that 42 goes into 391 nine times, therefore it will be 9 followed by a fraction. When we multiply 42 by 9 we get 378 and we know that the numerator of the fraction will be 13 because 391 - 378 is 13. Finally, we can write it as a fraction `9 13/42`.

`:.8/19=0.421052631`

`:.-2 1/3=-2.333333333`

`:.3/5=0.6`

`:.4 1/7=4.142857143`

`:.-5 1/6=-31/6=-5.166666667`

Largest number`=0.6=3/5`

Smallest number`=-5.166666667=-5 1/6=-31/6`

`:.3/5-(-31/6)`

`:.=3/5+31/6`

`:.=(18+155)/30`

`:.=173/30`

`:.color(magenta)(=5 23/30`