Question

how do you decide by which power of 10 to multiply an equation when writing a decimal with repeating digits as a fraction?

Answer 1

Answer: it depends on the denominator as which power of 10 is the nearest factor of denominator

Step-by-step explanation:

Thank you

Answer 2

Answer:

The repeating digits in the decimal number decides the which power of 10 is to be multiplied.

Step-by-step explanation:

• A non-terminating recurring decimal is a decimal that will never end.
• One or more numbers after the decimal point will be repeated.
• Identify the repeating digits in the decimal number.
• To convert the given decimal to a fraction, multiplying with 10 or powers of 10 on both sides.
• The choice of 10 or which power of 10, will always be determined by the digits that repeat in the decimal component.
• Choose 10 or power of 10 in such a way that, all the repeating digits must come to the LEFT of the decimal point.
• For the next equation, see whether the repeating digits are immediately to the RIGHT of the decimal point.
• Then choose again 10 or power of 10, to achieve this.
• The intention of multiplying with 10 or power of 10 is to get two equations which has only the recurring part on the right side of the decimal.

Let x = 1.04242424242 . . . is the repeating decimal number.

Two digits 42 are the repeating digits.

To get the first equation, up to 42 must be on left side of the decimal.

3 digits must be shifted LEFT, so multiply with 1000, that gives you

1000x = 1042.42424242 ...                        ...(1)

For next equation, 42 must be immediate RIGHT of the decimal, so the digit should be taken to left, therefore multiply with 10. This gives

10x = 10.4242424242 ...                            ...(2)

See that in both the equations, the right hand side of the decimal is same.

Now subtract both, which gives us

$990x = 1032\implies x=\dfrac{ 1032}{990}$

is the required number.

So, the repeating digits in the decimal number decides the which power of 10 is to be multiplied.

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