What fraction is between 1/313 and 1/212 ?

5 Answers
Dec 17, 2017

2/525

Explanation:

1/2=0.512=0.5

1/3=0.33333313=0.333333

So

2/5=0.425=0.4

Dec 17, 2017

There are infinitely many possibilities, but the midpoint between 1/313 and 1/212 is 5/12512

Explanation:

For any two numbers aa and bb, the average 1/2(a+b)12(a+b) lies between them.

So to find a fraction between 1/313 and 1/212 we can form the average:

1/2(1/3+1/2) = 1/2(2/6+3/6) = 1/2(5/6) = 5/1212(13+12)=12(26+36)=12(56)=512

Dec 17, 2017

5/12512

Explanation:

First, make the denominators equal:

(1/2) * (3/3) = 3/6(12)(33)=36

(1/3) * (2/2) = 2/6(13)(22)=26

Now figure out the middle number between the numerators:

2/6 < x < 3/626<x<36

x = 2.5/6x=2.56

Since 2.52.5 is a decimal, multiply both the numerator and denominator to make the 2.52.5 into a whole number:

(2.5/6) * (2/2) = 5/12(2.56)(22)=512

Check the answer:

(1/3) < (5/12) < (1/2) " "?(13)<(512)<(12) ?

1/212 is the same thing as 6/12612

1/313 is the same thing as 4/12412

so

(4/12) < (5/12) < (6/12)(412)<(512)<(612) is true!

Dec 20, 2017

There are many ... but 5/12512 is exactly half-way between them.

Explanation:

There are many fractions between 1/2 and 1/312and13

If you use the LCD you end up with 3/6 and 2/636and26.
The values between 2 and 32and3 are all fractions.

However, use a larger value in the denominator:

6/12 and 4/12612and412

Now it is easy so see that a fraction exactly between them is 5/12512

Consider an even bigger value in the denominator:

12/24 and 8/241224and824

Now we have the fractions: 9/24, 10/24, 11/24924,1024,1124 lying between them.

You can continue in this using larger and larger values and each time you will find more and more fractions between 1/2 and 1/312and13

There are infinitely many equivalent fractions.

Dec 29, 2017

2/525 is one of them.

Explanation:

The fraction (a+c)/(b+d)a+cb+d is between color(red)(a/b)ab and color(blue)(c/d)cd for any integers a,ca,c and positive integers b,db,d.

Proof:
Assuming that a/b< c/dab<cd

color(red)(a/b)=color(red)(a/b)*(b+d)/(b+d)ab=abb+db+d

=color(red)(a/b)*b/(b+d)+color(red)(a/b)*d/(b+d)<=abbb+d+abdb+d<

< a/b*b/(b+d)+c/d*d/(b+d)<<abbb+d+cddb+d<

< color(blue)(c/d)*b/(b+d)+color(blue)(c/d)*d/(b+d)<cdbb+d+cddb+d

=color(blue)(c/d)*(b+d)/(b+d)=color(blue)(c/d)=cdb+db+d=cd

The middle term
a/cancel(b)*cancel(b)/(b+d)+c/cancel(d)*cancel(d)/(b+d)=a/(b+d)+c/(b+d)=(a+c)/(b+d)

That means
a/b< (a+c)/(b+d)< c/d

In our case the fraction between 1/3 and 1/2 is (1+1)/(3+2)=2/5

It's a nice trick if you need a fraction between fractions quick.

Ford's circles are closely related to that.