terminating decimal, pls help

[tanviet]

which of the following has a decimal equivalent that is a terminating decimal

10/189
15/196
16/225
25/144
39/128

this is an official question and should be studied.

[mals24]

this is an official question and should be studied.

Yes Sir!!!

Answer should be E.

A fraction is a terminating decimal if the denominator is of the form: 2^n or 5^m or 2^n*5^m.

A, C, D are out straight away because their denominators have 3 as one of the factors.

Between B and E

128 = 2^7

So E is a terminating decimal.

[Ian Stewart]

A fraction is a terminating decimal if the denominator is of the form: 2^n or 5^m or 2^n*5^m.

Absolutely correct, with one important note: your fraction must be *reduced* before you apply this test. For example, if you try to apply the above to the fraction 27/48, you might mistakenly think that this is not a terminating decimal (because the denominator is divisible by a prime different from 2 or 5), but you need to reduce the fraction first: 27/48 = 9/16 = 9/(2^4), which must terminate.

[tanviet]

how do you know that if the denominator is in form 2^n, 5^m, or 2^n*5^m , the fractiom is terminating.

pls. explain this

[Ian Stewart]

how do you know that if the denominator is in form 2^n, 5^m, or 2^n*5^m , the fractiom is terminating.

If you have, say, the fraction a/2^n, you can multiply on the top and bottom by 5^n to get

(a*5^n)/(2^n * 5^n) = (a*5^n)/(10^n)

Since you have a power of 10 (so a number like 10, or 100, or 1000, etc) in the denominator, this must be a terminating decimal. Similarly, if you have a fraction like

21/125 = 21/5^3

we can calculate the decimal of this very quickly by multiplying the top and bottom by 2^3

21/5^3 = (2^3 * 21)/(2^3 * 5^3) = 162/10^3 = 162/1000 = 0.162

As long as the only primes in the denominator are 2, 5, or both, you can get 10^x in the denominator by multiplying on the top and bottom by a well-chosen power of 2 or 5. Since you can get a denominator that is a power of 10, the decimal must terminate.

[mals24]

If you have, say, the fraction a/2^n, you can multiply on the top and bottom by 5^n to get

(a*5^n)/(2^n * 5^n) = (a*5^n)/(10^n)

Since you have a power of 10 (so a number like 10, or 100, or 1000, etc) in the denominator, this must be a terminating decimal. Similarly, if you have a fraction like

21/125 = 21/5^3

we can calculate the decimal of this very quickly by multiplying the top and bottom by 2^3

21/5^3 = (2^3 * 21)/(2^3 * 5^3) = 162/10^3 = 162/1000 = 0.162

As long as the only primes in the denominator are 2, 5, or both, you can get 10^x in the denominator by multiplying on the top and bottom by a well-chosen power of 2 or 5. Since you can get a denominator that is a power of 10, the decimal must terminate.

Thanks a lot Ian for taking the time out and posting such great explanations.