How Often Does This Happen?
A man was born on Jan 1st for some year in the first half of the nineteenth century. On Jan 1st of the year x 2 , it turns out that the man was x years old. When was he born?
The answer is 1806.
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5 solutions
Integer, not integral
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Ok changed
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How can an equation be both greater and less than 0? Wouldn't that violate the line rule for functions? I was under the belief that it would be 1842 since 7 squared is 49, and 49 minus 7 is 42...
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@Jeff Kavanaugh – See the constants of the 2 equations, these are different
43^2=1849. So,why the answer can't be 43????????
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You are like me Got half the answer right. What he is asking is the year he was born not his age. At 1849 he was 43 years old. so he was born 43 years before that 1849-43=1806
43 is not the answer. It is his age. Answer is 1849-43=1806
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sorry. i was wrong totally!!!!!!!!!!! i was just out of my mind.
The man is x years old in the year x 2
Therefore he was x − x = 0 years in the year x 2 − x .Hence he was born in the year x 2 − x .
Since it is given that he was born in the first half of the nineteenth century,hence 1 8 0 0 < x 2 − x < 1 8 5 0 .Solving,we get that the only integral value of his age is 4 3 and hence he was born in the year 4 3 ( 4 2 ) = 1 8 0 6
good one..+1
1805 is also valid, it is possible to have been born in 1805 and be 43 years old in 1849
Thanks for pointing out that if someone is born on June 1st 1805, then they are 43 years old on January 1st 1849, which satisfies the conditions of the problem.
I have since edited the statement for clarity.
In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the menu. This will notify the problem creator (and eventually staff) who can fix the issues. Also, if you include a more detailed explanation of your reasoning, that would help others easily understand what you are referring to.
The first half of the 19th century is from 1800 to 1850 inclusive. Since it is stated that the man is x years old at the year x 2 , and was born in the first half 19th century, let's first assume that 1 8 0 0 < x 2 − x < 1 8 5 0 . Using square root algorithm on 1850, the maximum value of x is 43 (basing on the assumption previously made). Now, assuming the man was 43 years old at the year 4 3 2 = 1 8 4 9 , let's get the year of when he was born based on this assumption: 1 8 4 9 − 4 3 = 1 8 0 6 . Seeing this, x can't get lower than 43 (since 1806 is already very near 1800 and if x = 4 2 , the year of which the man would be born is 1722, which isn't anymore in the 19th century, but in the 18th century) nor can it get higher than 43 (since 4 4 2 − 4 4 = 1 9 3 6 − 4 4 = 1 8 9 2 is in the second half of the 19th century, not in the first half). Thus, the year of when he was born is 1806 .
I'm not sure if my solution is quite right, but I hope it helps a lot! :)
I'm willing to accept critical remarks/comments or questions from anyone regarding my solution, thanks. :))
Moderator note:
The following statement is false:
Since it is stated that the man is x years old at the year x 2 , and was born in the first half 19th century, let's first assume that 1 8 0 0 < x 2 < 1 8 5 0 .
What is true, is that the man was born in x 2 − x , and thus we have 1 8 0 0 ≤ x 2 − x < 1 8 5 0 .
The following statement is false:
Since it is stated that the man is x years old at the year x 2 , and was born in the first half 19th century, let's first assume that 1 8 0 0 < x 2 < 1 8 5 0 .
What is true, is that the man was born in x 2 − x , and thus we have 1 8 0 0 ≤ x 2 − x < 1 8 5 0 .
@Calvin Lin , thanks for correcting me!:)
Let the year of birth be y .It is given that 1 8 0 0 ≤ y < 1 8 5 0 and x 2 = y + x .therefore 1 8 0 0 ≤ ( x − 1 ) x < 1 8 5 0 .A simple check gives x = 4 3 and hence y = 4 2 × 4 3 = 1 8 0 6 .
Let y be the year of birth. It is given 1 8 0 0 < y < 1 8 5 0 since he was born in first half of 19th century. And since he was x years old in the year x 2 , he was born in the year x 2 − x . Thus y = x 2 − x . Solving the inequality, x 2 − x − 1 8 0 0 > 0 a n d x 2 − x − 1 8 5 0 < 0 . Solve these two and the only integer value of x we get is 43.