Step-by-step explanation:
I never learned this (for good reasons, I say). for me this is the strangest thing in math. math is precise. we are not guessing around when we can actually calculate things.
this is definitely ok, when just trying to decide, if one fraction is larger than the other, but to use that for an actual operation among fractions, is the biggest nonsense I have ever seen in math lessons (and not simpler than doing the precise calculation). again, math is a precise thing. the only valid level of imprecise answers is rounding, but only for the end result, not for intermediate results, because the rounding errors are rarely compensating for each other, they mostly pile up making the end result even less precise.
enough with the venting ...
benchmark fractions are the simple fractions (and their multiples)
1 (=1/1), 1/2, 1/3, 1/4, 1/5, 1/6
and then in steps of 2 :
1/8, 1/10, 1/12, ...
we try to find the closest simple fractions to our actual fractions and use them to estimate the actual result.
the simplest (but also normally the least precise) benchmark approach is using 1/2 as benchmark unit and 0, 1/2, 1 as benchmarks.
so, is 10/12 closest to 0, to 1/2 or to 1 ?
we know that 0 in 12ths would be 0/12, 1/2 would be 6/12, and 1 would be 12/12.
so, out of these 3 values 10/12 is closest to 12/12 (=1), so that is what you are choosing as representation.
is 3/8 closest to 0, 1/2 or to 1 ?
we would have 0/8, 4/8, 8/8.
and 3/8 is closest to 4/8 (= 1/2), and that is what we are choosing here as representation.
and the estimated result of 10/12 - 3/8 = 1 - 1/2 = 1/2 or 4/8 or 6/12.
let's pick 1/4 as benchmark unit.
so, are the numbers closest to
0, 1/4, 2/4, 3/4 or 1 (4/4) ?
for 10/12 we get then
0/12, 3/12, 6/12, 9/12, 12/12
and 10/12 is closest to 9/12 (3/4), which we pick as representation.
for 3/8 we get
0/8, 2/8, 4/8, 6/8, 8/8
if equidistant, we have to round up, so, we get 4/8 (2/4), which we pick as representation.
and the estimated calculation is
10/12 - 3/8 = 3/4 - 2/4 = 1/4
which is really horrible ...
2 and 4 are factors of 12 and 8. all smaller benchmark fractions (<1/4; so, e.g. 1/5, 1/6, 1/8. ...) are not meeting that criteria, making the benchmark finding much harder. so, I would not try that.
the real result is
the LCM of 12 and 8 is 24 (2×12 and 3×8).
so, we have
20/24 - 9/24 = 11/24 (almost 1/2).
so, actually, the benchmark unit of 1/2 is giving the best result here.
but to just do the exercise of benchmark units that are not factors of both denominators, let's pick 1/6
so, are the numbers closest to
0, 1/6, 2/6, 3/6, 4/6, 5/6 or 1 (6/6) ?
for 10/12 we get (all nicely)
0/12, 2/12, 4/12, 6/12, 8/12, 10/12, 12/12
so, 10/12 is closest to 10/12 (of course) = 5/6, which we pick as representation.
for 3/8 we get (ugh, we have to go for 24ths here too as 4×6 and 3×8)
0/24, 4/24, 8/24, 12/24, 16/24, 20/24. 24/24
3/8 = 9/24
so, the closest number to 9/24 is 8/24 (= 2/6), which we pick as representation.
and the estimated calculation is
10/12 - 3/8 = 5/6 - 2/6 = 3/6 = 1/2
did I say already that I don't like this ? that I don't see the advantage to the actual calculation ?
so, just in case : I don't like this (ugh) !
