I have a question, how do you reduce large fraction fast? If you don't remember more multiplication.

- Faman39

I have a question, how do you reduce large fraction fast? If you don't remember more multiplication.

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- welshfella

find their prime factors

- Faman39

for example 456/678

- welshfella

456 = 2 * 2 * 2* 3 * 19
678 = 2 * 3 * 113
cancel out the duplicates
in this case they are 2 and 3
so we have 2* 2* 19 / 113 = 76/113

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## More answers

- welshfella

that took me a while - then I'm an old guy! lol

- Faman39

Do we usually need to reduce all large fraction with 2 or 3 ? yeah, I was thinking that how we can reduce large fraction like that faster if we don't remember longest or bigger multiplications

- welshfella

I don't think there's a faster way of doing this for large numbers All the division you do are pretty simple.
You can use a calculator of course.

- mathmate

@Faman39
It depends on whether you can divide mentally by one-digit numbers.

- Faman39

Actually, I need to take tabe test within few month. They dont allow us to use calculator and they give really short time to solve those type of problem.

- welshfella

|dw:1443102044415:dw|

- welshfella

- thats how i do it by hand

- mathmate

Then practise dividing by single digit numbers.
For the example,
try 2
456/678=228/339
then try 3
228/339=76/113
and that's the end of it.
Also, learn dvisibilities so you can spot easy factors, like 2,3,5,11.
Example of a difficult one:
529/253
If you know divisibilities, you know 253=23*11 (note 2+3=5, therefore divisible by 23 and 11)
Then try divide 529 by 23, or 529=23^2
so
529/253=23/11
Also, memorizing the squares of numbers from 1-20 (or even 30) helps you factorize the difficult ones (like 529).

- Faman39

Thank you so much for helping :)

- welshfella

Practicing as much as possible is I believe, the key here.

- Faman39

okay. Thanks so much @welshfella @mathmate

- welshfella

one tip If you have a large odd number , to find if its divisible by 3, add up all the digits until you get a single number - if you get 3,6 or 9 then its divisible by 3.

- welshfella

yw

- mathmate

you're welcome! ;)

- Faman39

@DanJS @ganeshie8, Do you guys have any ideas for my this post?

- DanJS

reducing fractions, i just use a calculator, but nah just keep taking common factors from top and bottom

- DanJS

i use a calculator to do addition, :-)

- DanJS

even numbers, keep taking out 2's till you cant anymore

- ganeshie8

try reducing
\(\dfrac{1963}{2567}\)

- ganeshie8

actually i have just figured out a nice method
do you know how to find the gcd of two numbers using euclid gcd algorithm ?

- phi

Here is info you should learn
1) divisibility rules
https://en.wikipedia.org/wiki/Divisibility_rule#Divisibility_rules_for_numbers_1.E2.80.9320
(at least the easy ones up to 10)
2) prime numbers
https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_500_prime_numbers
It is good to learn the primes up to 100. The more you remember of these, the easier to simplify big fractions

- phi

In general, people are not fast at reducing fractions, and with calculators and computer so common, there probably isn't any need to be.

- ganeshie8

Let \(d=\gcd(a,b)\).
Then it is easy to find the integers satisfying \(a=a'd\) and \(b=b'd\).
\[\dfrac{a}{b} = \dfrac{a'd}{b'd} = \dfrac{a'}{b'}\]
It cannot reduce further since \(\gcd\) is the max factor that is common.

- Faman39

Thank you so much @phi @ganeshie8 @DanJS for coming here and helping me. I missed you all, wonderful openstudy moderators and friends.

- ganeshie8

np
look up euclid gcd algorithm incase if you haven't before, pretty sure you will find it interesting :)

- Faman39

gcd, okay. Thing is, I can't use calculator on tabe test for solving large fraction like this. So I was confused and don't know what to do. And they give really short time to solve those problem

- ganeshie8

as the name says, it is an algorithm from "euclid" days: 300BC
he had been able to invent that simple yet powerful algorithm probably because there were no calculators at that time

- Faman39

okay, I will try it. Thanks so much for amazing tricks! :)

- Faman39

Can you show me one more how do you solve large fraction with euclid? Cuz I am dumb enough :D

- ganeshie8

you'r not dumb
do you have teamviewer ?

- Faman39

No

- ganeshie8

so do you see how gcd is related to our problem ?

- Faman39

yes.

- ganeshie8

explain quick
so that i know that you know :)

- DanJS

is there a way to show a ridiculous ratio is in reduced form. one that would take time longer than the universe to calculate by hand or machine?

- Faman39

You set it up large fraction as cross multiply breaking in order reduce them easily

- DanJS

are there infinite primes?

- ganeshie8

lol yes and yes
@DanJS

- Faman39

I know, Im wrong @ganeshie8

- ganeshie8

You're not wrong, I did not phrase the question properly.
If you understand the question, you will answer it easily. Let me rephrase the question..

- DanJS

I think there is a numberphile video on that actually on the youtube

- ganeshie8

maybe lets see an example

- ganeshie8

\(\dfrac{15}{6}\)
is that in reduced form ?

- Faman39

5/2

- ganeshie8

how did u get 5/2 ?

- ganeshie8

you have cancelled 3
whats special about 3 ?

- ganeshie8

gcd(15, 6) = ?

- Faman39

okay common factor is 3 for 15 and 6

- ganeshie8

3 is not just a common factor
it is the GREATEST common factor
i want you see that first

- mathmate

Try 437/851 to see the power of Euclid!

- Faman39

well, this is large one and I can't figure out what would be common factor of large number like this cuz I often forgot large multiplication

- ganeshie8

Exactly! that is the kind of problem which makes you love euclid dude.

- ganeshie8

so you do see that if you figure out the "greatest common factor" of 437 and 851 "somehow", then you can reduce the fraction easily

- Faman39

5?

- ganeshie8

what 5

- Faman39

common factor. I am guessing but not sure

- ganeshie8

Look at the numbers : 437 and 851
Is any one of them "appear" like divisible by 5 ?

- ganeshie8

both the numbers must be divisible by "common factor"
that is the definition of "common factor"

- Faman39

I think, two?

- Faman39

that's also wrong

- ganeshie8

Are you saying 437 and 851 are divisible by 2 ?

- ganeshie8

doesn't matter if it is right/wrong... i appreciate that you're asking nice questions :)

- Faman39

okay. But I still don't get, how do I reduce large fraction quick if I can't figure out common factor or use euclid

- Faman39

okay. But I still don't get, how do I reduce large fraction quick if I can't figure out common factor or use euclid?

- ganeshie8

first you need to practice few simple things

- Faman39

okay

- ganeshie8

what do you know about even and odd ?

- Faman39

okay.

- ganeshie8

how do you say an integer is even or odd ?

- ganeshie8

Is 6 even/odd ?

- Faman39

even

- ganeshie8

why

- Faman39

I don't know why, but I know it's even

- ganeshie8

good, is 31 even ?

- Faman39

odd

- ganeshie8

how about 437 ?

- Faman39

odd?

- ganeshie8

Yes!
here is the trick :
An integer is even if it is divisible by 2.

- ganeshie8

6 is even because it is divisible by 2 :
6/2 = 3
leaves no remainder.

- ganeshie8

31 is not even because it is not divisible by 2 :
31/2 = 15, R=1
remainder is not 0, so not divisible by 2.

- ganeshie8

so when somebody asks you the definition of an "even" number, you can say this :
A number is even if it is divisible by 2.

- Faman39

okay. Thanks so much for explaining me thoroughly :)

- ganeshie8

437 and 851
so do you see why "2" cannot be a common factor of above numbers ?

- Faman39

cuz they are odd number?

- ganeshie8

Yes, neither of them is divisible by 2; they are odd.
So 2 is not a factor of any of them.

- ganeshie8

do you know how to tell if a number is divisible by 5 just by looking at the number ?

- Faman39

no

- ganeshie8

Is 10 divisible by 5 ?

- Faman39

yes

- ganeshie8

why

- Faman39

cuz 5x2 is 10, so yes

- ganeshie8

good, Is 13 divisible by 5 ?

- Faman39

no

- ganeshie8

why

- Faman39

13 is not divisible with any number

- ganeshie8

right, but why is it not divisible by 5

- ganeshie8

just divide and see

- ganeshie8

13/5 = 5*2, R=3
remainder is not 0,
so we say that 13 is not divisible by 5

- ganeshie8

to say that a 13 is divisible by 5, the remainder when 13 is divided by 5 must be 0.
but here the remainder is 3, which is not 0.
so 13 is not divisible by 5.

- Faman39

got it now

- ganeshie8

lets do couple more quick examples

- ganeshie8

Is 18 divisible by 5 ?

- Faman39

no, cuz it will also leave remainder of 3

- ganeshie8

Good!
Is 22 divisible by 5 ?

- Faman39

No, it will also give us remainder of 2 not 0.

- ganeshie8

Exactly!
Now, can you write down first few positive integers that are divisible by 5 ?

- Faman39

11, 2

- ganeshie8

how

- Faman39

11x2 = 22 and 2x11 = 22 as well

- ganeshie8

Ahh ok. My question was different question.
I was asking for the first few integers that are divisible by 5.

- Faman39

20, 10, 25, 30, 35, 40, 45

- ganeshie8

good, see anything special in them ?

- Faman39

they are increasing my 5

- Faman39

they are increasing by 5

- ganeshie8

thats true, more importantly, notice that the units digit in each of those numbers is either 0 or 5

- ganeshie8

so can we say this :
if the units digit is 0 or 5, then the number is divisible by 5.
?

- Faman39

hmm, got it :) thanks

- ganeshie8

so, Is 3434345 divisible by 5 ?

- Faman39

well, I think so but not sure about it, cuz it has two 34 and one 45

- ganeshie8

It doesn't matter. The units digit is 5, so it is divisible by 5.

- Faman39

okay. so how do I divide with 5 this such long number?

- ganeshie8

we don't need to.
if the units digit is 5 or 0, then the number is divisible by 5.

- Faman39

okay, I understood what you're saying. But if I need to reduce the largest like that digit, in that I often get blank, because I can't find common factor and how to reduce to them. Because I often forget bigger multiplication

- ganeshie8

one thing at a time

- Faman39

okay.

- ganeshie8

by now, you know how to tell if a number is divisible by 5, yes ?

- Faman39

yes

- ganeshie8

Is 898985 divisible by 5 ?

- Faman39

yes, it is divisible by 5

- ganeshie8

Yes, notice that, we don't need to divide the number and look at remainder to tell if a number is divisible by 5.
That is really powerful.

- Faman39

ok

- ganeshie8

Is 894852 divisible by 5 ?

- Faman39

It is not divisible by 5

- ganeshie8

good, I see you got it!

- ganeshie8

Now throw them away, we don't really need all those divisibility rules because we're going to use euclid algorithm to find the gcd directly.

- Faman39

okay

- ganeshie8

Here it is
https://www.youtube.com/watch?v=AJn843kplDw
it is just 2 minutes long, but it explains very nicely... let me know if something doens't make sense..

- Faman39

okay, it makes sense in video but how do I apply it in fraction?

- ganeshie8

suppose we want to reduce \(\dfrac{12}{30}\)
step1)
find \(\gcd(12, 30)\)

- Faman39

so, we just need to divide denominator with numerator

- Faman39

is that correct?

- ganeshie8

yes, that video explains it better

- Faman39

I am sorry, I must be made you really tired. Extremely apologize you had to stuck here quite long for me. Thanks soooooo much for taking time for explaining everything thoroughly. You really helped me to understand everything very clearly.

- Faman39

You made everything so easy to understand

- ganeshie8

hey, no im not tired, i want you master euclid algorithm and use it to reduce fractions
but just so you know... when i was first learning, it took me more than a week to understand euclid algorithm

- Faman39

But, really you did a lot to explain, which really helped. It mean a lot to me that you stayed with me quite long until I don't understand everything well. Thanks so much! I really appreciate what you did for me today. I was quite struggling with this kind of problem

- Faman39

So thank you sooooooooooooooooooooooooo much for help!

- Faman39

I can't believe you made everything so clear and easy for me, now I feel quite confident about doing the large fractions

- ganeshie8

np
good to see that that you took time to find efficient methods in reducign fractions, that's exactly the kind of attitude is necessary to get good at math. keep it up :)

- ganeshie8

there is always more than one way to do a particular thing in math
but often there is exactly one best way to do a thing.. if you spend time, you will find it...

- Faman39

Thanks for encouragement :) You're really genius math magician and amazing instructor :) I had really great time learning from you. Truly really enjoyed.

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