Mr. Dhoni does plantation work on Mr. Tendulkar’s square farm. He is paid in kind for the work he does on the basis of the square feet area he covers. He is paid 1 potato per square feet of area he covers every month. In the last one month, Mr. Tendulkar has expanded his farm, while maintaining its square shape, and this has meant that Mr. Dhoni has received 101 more potatoes this month than in the previous month. How many potatoes did Mr. Dhoni receive as his salary last month, and what is the current size of the farm?
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2500 potatoes in the prev month, and the current dimension of the farm is 51*51 feet
Dhoni’s last months salary : 2500 Potatoes
Current Farm Size : 2601 Sq unit
Answer =
dhoni recieved 50 potatoes
size of current farm : 25921 sqft
shape of farm is presently square
add 101 sq ft to it and should remain a square
basically we are looking at a figure which is currently perfect square of a number ( area of square= L^2) and remains a perfect square if 101 is added to it
(50)^2 = 2500
2500 +101 = 2601 (2601 is square of 51)
the farm used to be 2500 sq ft before expansion, hence Mr dhoni recd 2500 potatoes in previous month. the current size of farm is 51×51 Ft i.e- 2601 sq ft
Potatoes-2500
Current size-2601
2500 potatoes last month
2601 potatoes this month
current area =51 sq feet area
50
51
for the puzzle :
difference betn two squares should be 101
x original side
x+y new side
sq(x+y) – sq(x) =101
y(y+2x)=1*101
y=1 and x=50
Hence, new side 51 and old side 50.
Before expansion
sizz of square farm is 50×50 = 2500
After expansion
size of square farm is 51×51 = 2601
previous months salary is 2500
The difference between the square of two natural numbers is always equal to sum of the natural numbers 51^2 -50^2= (51+50)=101
As mr doni recieves salary on per square feet basis his current salary will be suppose x and after increase in field it is y..
then
Y^2=x2+101
Y^2= 2500 + 101
Y^2= 2601
Y= 51
Where x=50
Mr doni previous salary was 2500…
a^2-b^2=101
Simple way:
101/2=50.5
50.5 b/w 50 & 51
so a=51, b=50
Initial wage=50^2=2500 potatoes.
Mr Dhoni received 2500 potatoes last month and the current sieze of the farm is 2601 square feet
Size in the beginning = x^2
Size after increasing = y^2
y^2 = x^2 +101
ie (y+x)(y-x)=101
=>(y+x)(y-x) = 101.1 (Since product of 2 numbers is a prime number, one number is the number itself and the other is 1).
=> y = 51, x = 50.
So Dhoni received 2500 potatoes last month (50*50 = 2500) and current size of the farm is 51*51 = 2601.
Ans.: 2500 potatos as last salary and atpreseant area is 2601.
Soln: 101 is a prime no. so the possible factor are 1 & 101 so if i assume inital length be “a” and final length be “b” so the final equaltion : (b+a)(b-a)=101 so b+a=101 & b-a=1 so a=50 & b=51
Lets say initial width of the farm was x feet, and it was expanded by y feet.
Therefore last mongth Mr. Dhoni received x*x potatos, and this ,mongth he received (x+y)*(x+y) potatos.
Therefore ((x+y)*(x+y))-(x*x)=101
ie (2*x*y)+(y*y)=101
assuming x and y are whole numbers, y is a factor of 101.
But 101 is prime.
therefore either y=101, or y=1
if y=101
(2*x*y)+(y*y)=(2*x*101)+(101*101)
so
101=101*((2*x)+101)
which results into x=-50.
Since x is assumed a whole number y cannot be 101.
Therefore y=1.
here x=50.
So last mongth Mr. Dhoni received 2500 potatos as salory, and the current size of the farm is 2601 sq feet.
ASSUMING WE ARE TALKING OF ONLY WHOLE NOS., AND NO FRACTIONS, ETC.
Lets say the farm size was x square feet initially, and after the increase it is y square feet.
y-x = 101
both y and x should be perfect squares. So let y= a^2, x= b^2.
a^2 – b^2 = 101
(a+b) (a-b) = 101
Lets try to split 101 into factors.
As 101 is a prime no, 101 = 101 * 1 (only two factors)
Therefore, a-b = 1 and a+b = 101
=> Solve for a, b, we get a=51, b=50.
Therefore, Last month salary was b^2=2500 potatoes.
This month it is a^2 =2601 potatoes.