Question

How do you insert a pair of brackets to make each statement true `1/2+ 2/3 - 1/4 * 2/5 ÷ 1/6 = 6 2/5`?

Answer 1

`(1/2+ 2/3 - 1/4 * 2/5) ÷ 1/6 = 6 2/5`

Explanation:

`1/2+ 2/3 - 1/4 * 2/5 ÷ 1/6 = 6 2/5`

I first rewrote `-:1/6` as `*6/1`

`1/2+ 2/3 - 1/4 * 2/5 * 6/1 = 6 2/5`

Now that mixed number on the right confuses me, so write:

`1/2+ 2/3 - 1/4 * 2/5 * 6/1 = 32/5`

That looks better but there are too many denominators especially in the addition and subtraction. So let's write:

`30/60 + 40/60 - 15/60 * 2/5 *6 = 384/60`

Keeping the factors `15/60 * 2/5 *6` together means we'll need to do something with
`-15/60*2/5*6 = -3/60 * 2/3*6 = -36/60`

But the first two only sum to `70/60`, so that won't work.

Next, I divided `384` by `6` to see what we'd need `1/2+ 2/3 - 1/4 * 2/5` to be, if multiply by `6` was the last step.

`384 -: 6 = 64`

And, (Bonus)

`30/60 + 40/60 - 15/60 * 2/5 = 30/60 + 40/60 - (3/60 * 2/1)`

`= 30/60 + 40/60 - 6/60 = 64/60`

Last check:

`(1/2+ 2/3 - 1/4 * 2/5) ÷ 1/6`

`= (1/2 + 2/3 -(1/4*2/5)) -: 1/6`

`= (1/2 + 2/3 -1/10) -: 1/6`

`= (15/30 + 20/30 -3/30) -: 1/6`

`= (35/30 - 3/30)-: 1/6`

`= 32/30 -: 1/6`

`= 32/30 * 6/1 = 32/5 = 6 2/5`

It works!