In Exercises 45–50, express each repeating decimal as a fraction ...

Question

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms.

0.5 ̅ (repeating 5) = 5/10 + 5/100 + 5/1000 + 5/10,000 + ...

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to rewrite and simplify the reoccurring decimal 0. as a whole number and in this case we're given the fractions associated or that represents 0.9 repeating where we have nine divided by 10 plus nine, divided by 100 plus nine, divided by 2009 plus nine, divided by 10,000 and so on. So in order to represent this as a whole number, we're going to want to determine what's happening to the terms here. So the difference between the first and the second term is that we have this extra zero in the denominator and once again from the second to the third we have an extra zero. So this is giving me an indication that we are multiplying by some ratio. So I have a geometric sequence here. And when we have a geometric sequence we want to try to find that common ratio and to find the common ratio I can divide the second term nine over 100 By the first term. 9/10. And because I'm dividing two fractions, I'm going to make this easier by turning this into a multiplication. So I need to switch the denominator and numerator of my second fraction. So now this becomes 10-9. And from here you can see that the nines cancel out to one and I can cancel a zero in the denominator of my first fraction and a zero And the numerator of my second fraction, which means that my common ratio is 1/10. This means that when I multiply my first term, 9/10 times 1/10 I should get my second term. So nine times one is nine and 10 times 10 is 100. And this is in fact my second term and I can do that again multiplying my second term nine over 100 by the common ratio 1/10. I should get the third term. So nine times one is nine and 100 times 10 is and that is indeed my third term. So it seems like this common ratio is good now that we have the common ratio in order to represent this decimal as a whole number, we need to figure out the infinite sum of the sequence. So we can follow the infinite sum formula for geometric sequences which states that the infinite sum is equal to the first term. A one divided by one minus the common ratio are so we have all of these values so we can go ahead and plug in. So a one is equal to nine divided by And that is being divided by 1 -2 Which is 1/10. And I'm going to rewrite this one As 10/10 to make it easier to simplify. So when I do that I know how of nine divided by 10 Divided by 10 -1 is nine And the same denominator 10. And from here I'm going to do the same thing I did before and switch my denominator and numerator in the second fraction to make this a multiplication. So now I have 10 divided by nine for my second fraction. And here you can see that these tens cancel out to one and these nines cancel out to one, which means that this whole expression becomes one, so my infinite sum is equal to one. So when I write to 0.9 recurring decimal as a whole number, I write it as one. And you can see that this aligns with answer choice C. So hopefully this video helped and see you next time.

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.5 ̅ (repeating 5) = 5/10 + 5/100 + 5/1000 + 5/10,000 + ...

Key Concepts

Repeating Decimals

Geometric Series

Fraction Simplification

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