How to calculate the ratio .724 : .276?

Indranil

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How to calculate the ratio .724: .276 in the simplest form? I have done below
(724/100) / (276/100) = 724/276 = 181 / 69, Now how to go to get simplest form like 7 : 3?
 
How to calculate the ratio .724: .276 in the simplest form? I have done below
(724/100) / (276/100) = 724/276 = 181 / 69, Now how to go to get simplest form like 7 : 3?
Do a prime factorization of numerator and denominator and cancel common prime factors.

\(\displaystyle \dfrac{724}{276} = \dfrac{2^2 * 181}{2^2 * 3 * 23} = \dfrac{181}{3 * 23}.\)

\(\displaystyle 181 \text { is composite } \implies \exists \text { a prime factor of } 181 < \sqrt{181} < 14.\)

\(\displaystyle \dfrac{181}{3},\ \dfrac{181}{5},\ \dfrac{181}{7},\ \dfrac{181}{11},\ \dfrac{181}{13} \not \in \mathbb Z^+ \implies\)

\(\displaystyle 181 \text { is a prime.}\)

\(\displaystyle \dfrac{724}{276} = \dfrac{181}{3 * 23} = \dfrac{181}{69}.\)
 
How to calculate the ratio .724: .276 in the simplest form? I have done below
(724/100) / (276/100) = 724/276 = 181 / 69, Now how to go to get simplest form like 7 : 3?

181/69 is the simplest form. there is nothing simpler that is actually equal to your ratio.

If you want a "simpler" form than 181/69, you have to approximate. Is that what you want? If so, what determines how much inaccuracy is allowed?

There are various ways to find close fractional approximations; for example, in Excel, I can enter either .724/.276 or 181/69 and get the decimal approximation 2.623188. I can then format that cell as "fraction - up to one digit", and it gives 2 5/8, or 21/8, which is the best approximation using one-digit numerator and denominator. (If you needed, I could show you how to find that; but most people don't have a good reason to go to all that effort.) If I allow two digits, I get 2 43/69 = 181/69, not surprisingly - the exact answer.

There are other possible goals. For example, you might just want 8ths. Or, you might want the simplest fraction that is equal to 2.623188 to three decimal places (since that is the precision of the given ratio). Since 2 5/8 is 2.625, that doesn't quite meet this requirement. By trial and error, I can find that using a denominator of 7 I find that 139/53 would do; but 181/69 uses numbers not much bigger, and is more accurate (i.e. exact).

Having said all this, I don't want to suggest you do any more than you have already done. If you are asked for simplest form, you have it; no one is going to ask for any more than that.
 
…724/276 = 181/69 … how … to get simplest form like 7 : 3?
If rounding, 8 : 3 is closer than 7 : 3

Compare the decimal forms. :cool:

724/276 = 2.6231…

00008/3 = 2.6666…

00007/3 = 2.3333…

 
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