Maths in a minute: Adding fractions (the easy way)

Ruler

Adding fractions is probably the first difficult bit of maths we come across at school. For example, to work out $\frac{5}{6} + \frac{7}{10}$ you first need to figure out that the lowest common multiple of 6 and 10 is 30, and that in order to get 30 in the denominator of both fractions you need to multiply the numerator 5 by 5 and the numerator 7 by 3. This gives

$\frac{5}{6} + \frac{7}{10} = \frac{(5 \times 5)}{30} + \frac{(7 \times 3)}{30} = \frac{25}{30} + \frac{21}{30} = \frac{(25+21)}{30} = \frac{46}{30}.$

You then need to get rid of the common factors of 46 and 30, giving the final result $\frac{23}{15},$ which bears no resemblance whatsoever to the original two fractions. Doing this as a ten-year-old who has never seen it before is pretty tough.

Here is an alternative recipe that always works and doesn't involve faffing around with lowest common denominators. Writing "top" for numerator and "bottom" for denominator, the idea is to do:

(top left x bottom right + top right x bottom left) / (bottom left x bottom right).

Applied to our example this gives:

  \[  \frac{(5 \times 10 + 7 \times 6)}{(6 \times 10)} = \frac{(50 + 42)}{60} = \frac{92}{60} = \frac{23}{15}.  \]    

The difference to the standard way of adding fractions is that you are not bothered with finding the lowest common denominator. You simply use the product of the two denominators as a common denominator. Then, in order to bring both fractions on that common denominator you only need to multiply the numerator of each by the denominator of the other. Easy!

Apparently this is how Vedic mathematicians in ancient India added up fractions. If you happen to speak German, you can also explore this method in musical form in this maths rap by DorFuchs. And even if you don't speak German, it's cute!

Comments

I was taught this method at school (London, late 60s). Why on earth anyone would think that making it more complicated by chucking in this unnecessary LCD business is entirely beyond me. Now I'm thinking I should check that my kids haven't been polluted by this nonsense!

...its also the way I was taught to do it 52 years ago in a junior school in Swindon!

Remember doing it this way at school - 50+ years ago.

ditto

abundantly obvious.

same process as lcm

just figuring the lcm the way i always did it.

but then i am incapable of learning and incapable of remembering. without 90 days practice lol.

so i figure what i deal with all over again one second at a time; boring things i practice so i no longer notice them they become habits.

Try adding
1/2 + 1/3 + 5/6 + 4/9
without looking for the lowest common denominator. The numbers get large fairly fast using the crude method touted here.

Since when are kids too stupid to comprehend lowest common denominator?

Method I was taught in Florida in the late 80s. Next year moved to Atlanta and was "corrected" and forced to use LCD method. Yeck.

Nothing new.

Same way I was taught in 1975, this does not deserve an article in 2014.

Mathematical short-cuts are often traps.

The student who learns to work fractions this way and never learns LCD will struggle in algebra and fail at calculus.

The goal IS NOT to teach easy computation--we have computers and calculators for that.

The goal IS to teach mathematical theory that can be generalized and used to learn more complex ideas.

exactly
this method works for here and now BUT
without the skills to do it properly youll run into trouble later in high school maths

Okay, so adding fractions works by using any common multiple of the denominators, not just the lowest... Doesn't seem ground-breaking and, frankly, what's the point?
Firstly students will also learn how to find the LCM of two numbers anyway.
Secondly, they need to simplify the fraction, which requires the notion common factors and it's not something that comes easy to them
Thirdly, if you try doing that to algebraic fractions or higher number fractions using this method, you'll get in trouble quickly...
Basically, uninteresting, a very particular case and not useful...
Sorry...

There really is nothing special to this method, at all. I came up with the thought in school once, I guess any student might come up with that idea, sooner or later, because finding LCDs was a pain in the ass when you could just do it this way, obviously...

What a shame that +Plus magazine is suggesting that confidence-building and efficient computation techniques are 'faffing'.

Students who know their times facts (times tables) have no difficulty finding LCM, those with less confidence can try the ' 2x bigger denom, does it work for smaller? 3xbigger denom, does it work for smaller' method. Those who find LCM an issue would be better if given activities and time to understand the LCM concept and build instant recall skills in times facts. If LCM is beyond them, using bigger CMs just gives more opportunity to make computation errors with the bigger numbers - not a confidence-building approach.

Fractions techniques are brilliant tools to simplify otherwise complex calculations. Fractions are already short-cuts for multiply/division. I teach them as a step into using 'professional tools' as you would teach an apprentice the techniques of the master craftsman.