Video Transcript
By finding the sum of an infinite geometric sequence, express 0.43 recurring as a common fraction.
First, we observe that this notation of the line over only the three means it is the three that is recurring. So this is the decimal 0.4 followed by an infinitely long string of threes. We want to express this decimal as a fraction. And we’re told to approach this problem by finding the sum of an infinite geometric sequence. We need to consider then how we can split this decimal up into a sum. Well, we can separate the values in each decimal place. So we can write this decimal as 0.4 plus 0.03 plus 0.003 plus 0.0003 and so on. The terms 0.03, 0.003, 0.0003, and so on are each 10 times smaller than the previous term. And so they form a geometric sequence with a common ratio of 0.1, or one-tenth.
The first term in the sequence 𝑎 sub one is 0.03. As the absolute value of the common ratio is less than one, this series is convergent. And so we can find its sum. The sum of an infinite geometric sequence with first term 𝑎 sub one and common ratio 𝑟, whose absolute value must be strictly less than one, is 𝑎 sub one over one minus 𝑟. Substituting 0.03 for 𝑎 sub one and 0.1 for 𝑟, we have that the sum of this infinite geometric sequence is 0.03 over one minus 0.1. That’s 0.03 over 0.9. And we can then multiply both the numerator and denominator by 100 so that they’re both integers. And this gives three over 90. Three over 90 can of course be simplified to one over 30 by dividing both the numerator and denominator by three.
We found then that the recurring decimal, 0.03 recurring, is equal to one over 30. But we also need to add on the 0.4. Well, 0.4 as a fraction is four-tenths, or two-fifths. So 0.43 recurring is equal to four-tenths plus one over 30. That’s 12 over 30 plus one over 30, which is equal to 13 and over 30. And this fraction can’t be simplified any further, as the numerator and denominator have no common factors other than one. So we have our answer to the problem.
We can check this using short division. To divide 13 by 30, we can first divide by 10, giving 1.3, and then divide 1.3 by three. First, threes into one don’t go, so we put a zero and carry the one. Threes into 13 go four times with a remainder of one. Threes into 10 go three times with a remainder of one. Threes into 10 go three times with a remainder of one. Threes into 10 go three times with a remainder of one. And we can see that this is going to continue infinitely. We see then that 13 divided by 30 is indeed equal to the recurring decimal 0.43 recurring. And so our answer to the problem is that 0.43 recurring as a common fraction is 13 over 30.
