* ⭕TMF * Telangana Mathematical Forum * Let's do this calculation * There will be four friends. The four friends go to Narendra's garden and get some banana pound. Because the garden is too far away from home, it is dark. Well, they are lying in someone's room to share those bananas the next day. 1. After a while a man gets up and takes my portion now and makes four pieces of banana. One is surplus. Throwing it outside for a massacre, he takes his part. 2. If the second person rises and repeats the four parts, one is left. He throws one of them and takes his part. 3. The third person gets up and makes the remaining four parts again, leaving one more. He throws it away and takes his part. 4. The fourth person gets up and makes the remaining four parts again, leaving one to the other. All come early. They will never tell anyone what they have done. The Silent is made up of four parts and those whose parts are captured. This time the massacre had nothing left. But how many bananas were brought from Narendra's garden? Let's say *
Answers
765 bananas were brought from Narendra's garden
Step-by-step explanation:
Bananas were brought from Narendra's garden = B
after 1st person
one part =(B-1)/4
remaining = B - (B-1)/4 - 1 ( 1 massacre)
= ( 4B - B + 1 -4)/4
= (3B -3)/4
= 3(B - 1)/4
after2nd person
= (3/16)(3B - 7)
after 3rd person
= (3/64) (9B - 37)
after 4th person
= (3/256)(27B - 175)
(3/256)(27LB- 175) = 4n ( equallu distributed nothing left)
=> (3/256)(27B - 175) = 4
=> 81B - 525 = 1024n
=> 81B = 1024n + 525
=> we need to find n such that (1024n + 525)/81 is an integer
(81 * 12n + 52n + 81*6 + 39)/81
= (52n + 39) / 81 is integer
=> 13(4n+3)/81
=> 4n + 3 = 81k
4n = 3(27k -1)
k = 3
n = 60
=> 81B = 1024 * 60 + 525
=> B = 765
Bananas mascare one part remaining
765 1 191 573
573 1 143 429
429 1 107 321
321 1 80 240
240 0 = 4×60 0
765 Bananas were brought from Narendra's garden
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The total number of bananas they brought from Narendra's garden are 765 in number.
Explanation :
Let the number of bananas were brought by them from Narendra's garden be 'X'
From 'X', total number of bananas the first one took = (X -1)/4
We need to eliminate the banana he threw away.
Bananas remained are = X - ((X -1)/4 ) - 1
= ( 4X - X + 1 - 4) / 4
= 3(X - 1) / 4
Similarly, the no.of second one had took with him = (3/16)(3X - 7)
the third one = (3/64) (9X - 37)
the fourth one = (3/256)(27X - 175)
Then, on talking 4 equal parts from the bananas left on the next morning, the share happened exactly. As in :
(3/256)(27X - 175) = 4n
81X - 525 = 1024n
81X = 1024n + 525 ----------------- (1)
Here, n is the integer which we need to find.
(81 * 12n + 13 * 4n + 81 * 6 + 13 * 3)/81
As per this statement, (4n+3)/81 is an integer. On solving,
4n + 3 = 81c ; where c is a constant.
4n = 3 ( 27c -1 )
Let us check, on substituting which integer in c, the total value becomes a multiple of 4.
c = 0; 3(27(0)-1) = -3 (Incorrect)
c = 1; 3(27(1)-1) = 3(26) (Incorrect)
c = 2; 3(27(2)-1) = 3(53) (Incorrect)
c = 3; 3(27(3)-1) = 3(80) (Suits!) As 80 is a mutiple of 4.
So, at c=3, we get
4n = 3(80)
n = 3(20)
n = 60
As n=60, substitute this value in eqn (1), As in :
81X = 1024 * 60 + 525
81X = 61440+ 525
X = 61965 / 81
X = 765
Therefore, the total number of bananas they brought from Narendra's garden are 765.
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