One morning a big circus arrived in the Ramleela maidan at Delhi. The arrival of the circus was seen in the morning at 08:00 AM by Gopal. He passed this information on 08:15 to 2 other residents of the city. Each of these 2 people then informed the other 2 residents at 08:30, and again at 08:45, they reported the arrival of the circus every 2 to other uninformed residents This chain continued the same way till 12:00 PM. By 12:00 PM enough people were informed about the arrival of the circus.
Answer the following equations:
i. By 12:00 PM total how many people were informed about the arrival of the circus?
a. 131000 b. 131017 c.141000 d.65536
ii. By 10:00 AM total how many people were informed about the arrival of the circus?
a. 511 b. 256 c. 300 d.500
iii. From 10:00 AM to 11:00 AM how many people were informed about the circus?
a. 8000 b.7000 c.7680 d.7936
iv. What are the three terms between 16 and 256?
a. 64 , 32,128 b.16 , 32,256 c.32 , 64,128 d.16 ,32, 64
v. At 10:30 AM how many people were informed about the circus?
a. 512 b.1024 c.2048 d.2047
Answers
i. 131071 people were informed by 12:00 PM; ii. 511 people were informed by 10:00 AM; iii. 7936 people were informed from 10:00 AM to 11:00 AM; iv. 32, 64 and 128 are the three terms between 16 and 256; v. 1024 people were informed at 10:30 AM
Given:
Time at which Gopal sees the circus = 8:00 AM
Time at which Gopal passed this information to two other residents = 8:15 AM
Time at which these two residents further pass the information to two other residents = 8:30 AM
The chain continues till 12:00 PM
To Find:
i. how many people were informed by 12:00 PM?
ii. how many people were informed by 10:00 AM?
iii. how many people were informed from 10:00 AM to 11:00 AM ?
iv. What are the three terms between 16 and 256?
v. how many people were informed at 10:30 AM?
Solution:
The number of people informed increases exponentially every 15 minutes.
The number of people informed each hour will thus follow the given series-
1, 2, 2², 2³,... and so on
At 12:00 PM, 2¹⁶ new people will be informed
thus, the number of people informed by 12:00 PM will be-
2⁰ + 2¹ + 2² + ... + 2¹⁶
= 2¹⁷ -1
= 131071
Now, at 10 AM, 2⁸ new people will be informed
⇒ the number of people informed by 10:00 AM will be-
2⁰ + 2¹ + 2² + ... + 2⁸
= 2⁹ -1
= 511
Now, from 10:00 to 11:00 AM, the number of people informed will be-
2⁸ + 2⁹ + 2¹⁰ + 2¹¹ + 2¹²
= 2⁸(1 + 2 + 4 + 8 + 16)
= 7936
As we can see, the series goes on as follows-
1, 2, 4, 8, 16, 32, 64, 128, 256,...
Thus, the 3 terms between 16 and 256 are- 32, 64 and 128
Finally, at 10:30 AM, 2¹⁰ new people will be informed about the circus
= 1024
Hence, we get- i. 131071 people were informed by 12:00 PM; ii. 511 people were informed by 10:00 AM; iii. 7936 people were informed from 10:00 AM to 11:00 AM; iv. 32, 64 and 128 are the three terms between 16 and 256; v. 1024 people were informed at 10:30 AM.
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