M17-29

I solved it by approximating.

1/9 + 1/99 + 1/999

1/9 + 1/99 = 12/99

Therefore, 12/99 + 1/999 = x

approximate : 12/100 + 1/1000 = 120/1000 + 1/1000 = 121/1000 = 0.121

1/8 = 0.125
1/9 = 0.111

Therefore, 1/8.

An alternative approach: I see that a lot of people are opting for (B). But we do not need to get carried away with the details to see why (C) makes more sense. Why not estimate? It can help to know that a neat feature of ninths is that a single-digit integer in the numerator will be repeated infinitely after the decimal—1/9 is 0.11111..., 2/9 is 0.22222..., and so on. You can even solve something such as 29/9 by writing it as a mixed number and converting to a decimal appropriately: 29/9 = 3 2/9 or 3.22222...

In this problem, we can calculate or use the value of 1/9 to work the sum:

1/9 = 0.11111

1/99 ~ 1/100 or 0.01 (1/99 would be a tad greater)

1/999 ~ 1/1000 or 0.001 (again, 1/999 would be a tad greater)

\(0.111+0.01+0.001 = 0.122\)

We can deduce that our estimate of 0.122 will be slightly less than the actual value. Now, between (B) and (C), we should know that (B) is 0.11111... It is beneficial to have eighths memorized to a whole—each 1/8 is worth 0.125, half of a quarter. Thus, we need to place our figure between 0.11111... and 0.125. Test each difference (whether on your noteboard or in your head):

(B) 0.122 - 0.111... ~ 0.011

(C) 0.125 - 0.122 = 0.003

(C) would be even closer, remember, than 3/1000, far better than 1/100 (or, for comparative purposes, 10/1000). The answer must be (C).

I hope this approach may help others who come across the thread. As always, good luck with your studies.

- Andrew

I think this is a high-quality question and I agree with explanation.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

­Used intuition for this question. Since working with 10s (scientific notation) is easier, if you approximate 1/10 + 1/100 + 1/1000 to be closer to 1/9, 1/9 + 1/99 + 1/999 should be closer to 1/8.

The answer must be higher than 1/9. So, A is not correct.

Now B, 1/8. The difference between 1/9 and 1/8 is 1/72. To know to which one the result of the sum is closer, we can draw a line between 1/9 and 1/8, they are separated by 1/72. To the middle point we have 1/144. If 1/99 + 1/999 is higher than the middle point, it will be closer to 1/8.

1/99 + 1/999 is higher to 1/144, so it is closer to 1/8.­

\(\frac{1}{9} + \frac{1}{99} + \frac{1}{999}\)

Recall that \(\frac{1}{9} = \frac{11.11}{100}\) (1/9 = 11.11%)

1/99 is approximately 1/100

1/999 is approximately 1/1000 = .1/100

\(\frac{11.11}{100} + \frac{1}{100} + \frac{.1}{100}= 12.21/100\)

Also recall that 1/8 = 12.5%

Hence with the addition of the two terms, the sum has gone closer to 1/8.

Answer (C)

Fraction percent equivalents are discussed here: https://youtu.be/HxnsYI1Rws8

Could we also use geometric progression formula? since r is smaller than 1 we could estimate the sum ( I know its infinite but still good to estimate): b/1-r => (1/9)/(1-1/9) = 1/8

The point is that 1/9, 1/99, and 1/99 are not in geometric progression because the ratio of consecutive terms is not the same.