Question Video: Converting Recurring Decimals to Fractions | Nagwa Question Video: Converting Recurring Decimals to Fractions | Nagwa

# Question Video: Converting Recurring Decimals to Fractions Mathematics

Answer the following questions for the recurring decimal 0.4 recurring, that is, 0.44444... Let π₯ = 0.4 recurring. Find an expression for 10π₯. Subtract π₯ from 10π₯ to find an expression for 9π₯. Find π₯.

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### Video Transcript

Answer the following questions for the recurring decimal 0.4 recurring, that is, 0.44444 and so on. Let π₯ be equal to 0.4 recurring. Find an expression for 10π₯. Subtract π₯ from 10π₯ to find an expression for nine π₯. Find π₯.

By breaking the question down into three individual steps, itβs showing us the process for converting a simple recurring decimal into a fraction. We have a simple recurring decimal. Itβs π₯ equals 0.4 recurring. That is, π₯ equals 0.444 and so on. It can be easier to write out a few digits of the recurring part so we can get an idea of the pattern. The question asks us to find an expression for 10π₯. Well, to get to 10π₯ from π₯, weβre clearly going to need to multiply it by 10. And since π₯ is equal to 0.4 recurring, it follows that weβll find an expression for 10π₯ by multiplying 0.4 recurring by 10.

π₯ multiplied by 10 is 10π₯ as required. And to multiply a decimal number by 10, we move the digits to the left exactly one space. So 10π₯ is equal to 4.444 and so on. The expression for 10π₯ is therefore 4.4 recurring. The second part of this question tells us to subtract π₯ from 10π₯. In doing so, weβre going to subtract the entire equation for π₯ from the entire equation from 10π₯. 10π₯ minus π₯ is nine π₯ as required.

Now letβs look at our recurring decimals. Notice how the digits after the decimal point are identical in both numbers. So that means when we subtract each four, we end up getting zero. So we see that 4.4 recurring minus 0.4 recurring is simply four. And so our expression for nine π₯ is simply four. The third part of this question asks us to find π₯. What itβs really saying is solve this equation for π₯. Solve the equation nine π₯ equals four.

Well, to solve the equation, weβll perform a series of inverse operations. Currently, nine π₯ means nine times π₯. So weβre going to divide both sides of our equation by nine. And so, when we do, we find π₯ is four divided by nine or four-ninths. π₯ is equal to four-ninths. Note that we originally defined π₯ to be equal to 0.4 recurring, but weβve just written that π₯ is equal to four-ninths. That must mean that 0.4 recurring is four-ninths.

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