Question

im pretty supitd so plzz answer and i got the 2nd and 4th one and it said it was wrong and i only have 1 try left so help me plzzzzzzz

Answer 1

Answer: A) Yes, because the number can be written as the quotient of two integers

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Step-by-step explanation:

Any repeating decimal can be expressed as a fraction of two integers.

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Optional section:

Here's a proof showing how the given number can be expressed as a fraction

Let x = 0.020020200202002...

The block of digits "02002" repeats forever.

Multiply both sides by 10^5 = 100,000 to move the decimal over 5 spots to the right.

The equation

x = 0.020020200202002...

will turn into

100000x = 2002.0200202002...

We have this system of equations

`\begin{cases} x = 0.020020200202002\ldots\\100000x = 2002.0200202002\ldots\end{cases}`

Subtracting said equations leads to

100,000x - x = 99,999x on the left

2002.0200202002... - 0.0200202002... = 2002 on the left

Note how the decimal portions line up perfectly to cancel out when we subtract

After those two subtractions, we end up with the equation

99,999x = 2002

Divide both sides by 99,999 and we isolate x

x = 2002/(99,999)

Use your calculator to find that,

2002/(99,999) = 0.02002020020202...

which helps confirm our answer.