Repeating Decimal: how do you write 0.0544 as a fraction? ('544' repeating)

[June]

Hey there,

Hope your day is going well. I have a question for you; how do you simplify 0.0544 as a fraction (the underlined part being the repeating portion)? It's something that wasn't covered in my notes and youtube videos and the web teaches me how to convert it if it were 0.544 but because of the extra decimal spot it doesn't work. Youtube references treating the number as x and creating a 10x value. You would then subtract x from 10x and solve for x resulting in a fraction that once inputed equals the decimal you're looking for. I tried to substitute 100x for 10x because of the extra decimal spot, however the fraction I ended up with once entered into the calculator gives me 0.0544, it does not repeat the value.

Thanks in advance!

June

 

[HallsofIvy]

Hey there,

Hope your day is going well. I have a question for you; how do you simplify 0.0544 as a fraction (the underlined part being the repeating portion)? It's something that wasn't covered in my notes and youtube videos and the web teaches me how to convert it if it were 0.544 but because of the extra decimal spot it doesn't work. Youtube references treating the number as x and creating a 10x value. You would then subtract x from 10x and solve for x resulting in a fraction that once inputed equals the decimal you're looking for. I tried to substitute 100x for 10x because of the extra decimal spot, however the fraction I ended up with once entered into the calculator gives me 0.0544, it does not repeat the value.

Thanks in advance!

June

x= 0.0544544544544... so multiplying by 10,
10x= 0.544544544... and then multiplying by 1000
10000x= 544.544544544....

Can you finish that?

 

[June]

x= 0.0544544544544... so multiplying by 10,
10x= 0.544544544... and then multiplying by 1000
10000x= 544.544544544....

Can you finish that?

Thank you for helping out. Does this look right?


Thanks
June

 

[Deleted member 4993]

x= 0.0544544544544... so multiplying by 10,
10x= 0.544544544... and then multiplying by 1000 .............................................(1)
10000x= 544.544544544 .............................................(2)

Can you finish that?

I'll just add one more step to help ..

subtract (1) from (2) to get
9990*x = 544
Now continue....

 

[JeffM]

Hey there,

Hope your day is going well. I have a question for you; how do you simplify 0.0544 as a fraction (the underlined part being the repeating portion)? It's something that wasn't covered in my notes and youtube videos and the web teaches me how to convert it if it were 0.544 but because of the extra decimal spot it doesn't work. Youtube references treating the number as x and creating a 10x value. You would then subtract x from 10x and solve for x resulting in a fraction that once inputed equals the decimal you're looking for. I tried to substitute 100x for 10x because of the extra decimal spot, however the fraction I ended up with once entered into the calculator gives me 0.0544, it does not repeat the value.

Thanks in advance!

June

The first step is to multiply your number so that the repeating pattern starts immediately to the right of the decimal point.

In your case you have \(\displaystyle x = 0.0544544544544....\)

So to get the repeating part immediately to the right of the decimal point, you just multiply by 10:

\(\displaystyle 10x = 0.544544544544....\)

The next step is to multiply your number so that the first occurrence of the repeating pattern starts immediately to the left of the decimal point.

\(\displaystyle 10,000x = 544.544544544544....\)

Now do you see that these two numbers are identical to the right of the decimal point so that if we subtract them there will be nothing to the right of the decimal point? So let's subtract.

\(\displaystyle 10,000x - 10x = 544 \implies 9,990x = 544 \implies\)

\(\displaystyle x = \dfrac{544}{9990} = \dfrac{2^5 * 17}{2 * 3^2 * 5 * 111} = \dfrac{2^4 * 17}{3^2 * 555} = \dfrac{272}{4995}.\)

The trick is to get a difference of multiples of your number so that the repeating decimal goes away with subtraction.

 

[HallsofIvy]

Thank you for helping out. Does this look right?
View attachment 9724

Thanks
June

It is odd that you would thank me for my reply and then not do anything I suggested!

 

[June]

The first step is to multiply your number so that the repeating pattern starts immediately to the right of the decimal point.

In your case you have \(\displaystyle x = 0.0544544544544....\)

So to get the repeating part immediately to the right of the decimal point, you just multiply by 10:

\(\displaystyle 10x = 0.544544544544....\)

The next step is to multiply your number so that the first occurrence of the repeating pattern starts immediately to the left of the decimal point.

\(\displaystyle 10,000x = 544.544544544544....\)

Now do you see that these two numbers are identical to the right of the decimal point so that if we subtract them there will be nothing to the right of the decimal point? So let's subtract.

\(\displaystyle 10,000x - 10x = 544 \implies 9,990x = 544 \implies\)

\(\displaystyle x = \dfrac{544}{9990} = \dfrac{2^5 * 17}{2 * 3^2 * 5 * 111} = \dfrac{2^4 * 17}{3^2 * 555} = \dfrac{272}{4995}.\)

The trick is to get a difference of multiples of your number so that the repeating decimal goes away with subtraction.

Thank you for your guidance. I've made notes of what you said so I can refer to it in the future.

 

[June]

It is odd that you would thank me for my reply and then not do anything I suggested!

I'm sorry, I clearly missed where you were going with the multiplying of numbers. I appreciate the time you took looking into this.

Thanks,
June