M15-25

Question

In the decimal notation of the number  (\frac{2}{23})^3 , what is the third digit after the decimal point?

A. 0
B. 1
C. 2
D. 4
E. 8

Bunuel · Accepted Answer

Official Solution:

In the decimal notation of the number  (\frac{2}{23})^3 , what is the third digit after the decimal point?

A. 0
B. 1
C. 2
D. 4
E. 8

Observe that  (\frac{2}{23})^3 < (\frac{2}{20})^3 .
 
Now,  (\frac{2}{20})^3 = (\frac{1}{10})^3 = 0.001 .
 
Finally, since  (\frac{2}{23})^3 < 0.001 , the third digit after the decimal point of  (\frac{2}{23})^3  must be less than 1, which is the third digit after the decimal point of 0.001. Therefore, it must be 0.

Answer: A

akshatcy · Answer

Using approximation,
(2/23)^3
 \frac{2}{23}  ~ 0.08
 (0.08)^3 
After decimal there are two places, squaring it will make it 4 places,
 (0.0064) * (0.08) 
Now multiplying it by 0.08 will make it 6 places after decimal,
0.000512
Therefore, third place after decimal will be 0

dp3791 · Answer

I thought the digit 1 is the third from the right of the decimal point. So, if 1 is not the third, in which position is it? Are not the decimal points supposed to be the ones after the comma, in which case 1 would be the third and the 2 zeros would be first and second. I don't understand this concept at all.... Can someone pplease help?!

Bunuel · Answer

(2/23)^3 = 0.00 0 657516232431988164707816224213035259307964165365332456... 0 is the third digit to the right of the decimal point.

1 is the third digit to the right of the decimal point of  (\frac{2}{20})^3= 0.001 . Because  (\frac{2}{23})^3 \lt (\frac{2}{20})^3 , then the third digit to the right of the decimal point of (2/23)^3 must be 0.

Bunuel · Answer

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

Dbrunik · Answer

Hahahaha I was sitting here trying to do long division. Big mistake, great explanation and good question on estimate and number properties

Posted from my mobile device

Tommy3603 · Answer

I like the solution - it’s helpful.

Fisayofalana · Answer

I like the solution - it’s helpful. Three exponents or fewer, I usually just do the math. But estimation here is the best path.

M15-25

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