The perimeter of a rectangular field is 140m. If the length of the field is increased by 2m and its breadth decreased by 3m,the area is decreased by 66sq.m. find the length & breadth of the field.
Well, well, well, it seems like we have a rectangle with a bit of a size-changing problem! Let's solve this puzzler, shall we?
Let's call the original length of the field L and the original breadth B. According to the problem, we know that the perimeter of the rectangle is 140m:
Perimeter = 2(L + B) = 140
Now, let's focus on the second part of the problem. The length is increased by 2m, so the new length is L + 2. The breadth is decreased by 3m, so the new breadth is B - 3. And what do we get? The area decreases by 66 square meters. So we have the following equation:
(L + 2)(B - 3) = LB - 66
Now, let's simplify things a bit, like peeling a banana. Expand the left side of the equation to get:
LB + 2B - 3L - 6 = LB - 66
The LB cancels out, and we are left with:
2B - 3L - 6 = -66
Now, let's move some things around, like rearranging furniture in a clown car:
2B - 3L = -66 + 6
2B - 3L = -60
Alright, now let's go back to that first equation. The perimeter of the rectangle is 140m, or in expanded form:
2L + 2B = 140
But we can simplify that further, like trying to squeeze into a pair of pants two sizes too small. Divide both sides of the equation by 2:
L + B = 70
Now we have a system of two lovely equations:
L + B = 70
2B - 3L = -60
Now we can solve this using some clever clown calculations! Let's multiply the first equation by 3:
3L + 3B = 210
Now, let's add the two equations together, like juggling oranges:
2B - 3L + 3L + 3B = -60 + 210
Simplifying a bit:
5B = 150
Divide both sides by 5:
B = 30
Now, substitute this value back into the first equation:
L + 30 = 70
Subtract 30 from both sides:
L = 40
So, the length of the field is 40m and the breadth is 30m. Ta-da! The clown team has solved it!
Let's denote the length of the field as L and the breadth as B.
We are given two pieces of information:
1. The perimeter of the rectangular field is 140m, which can be expressed as:
Perimeter = 2(L + B)
140 = 2(L + B)
2. If the length of the field is increased by 2m and the breadth is decreased by 3m, the area is decreased by 66 sq.m. This can be expressed as:
(L + 2)(B - 3) = Area - 66
(L + 2)(B - 3) = LB - 66
Now, we can solve these two equations simultaneously to find the values of L and B.
Step 1: Simplify the first equation:
140 = 2L + 2B
70 = L + B
Step 2: Substitute the value of (L + B) from the first equation into the second equation:
(L + B + 2)(B - 3) = LB - 66
(70 + 2)(B - 3) = LB - 66
72(B - 3) = LB - 66
72B - 216 = LB - 66
Step 3: Rearrange the equation:
LB - 72B + 150 = 0
Step 4: Factor the equation:
(B - 10)(L - 15) = 0
Step 5: Solve for possible values of B and L:
Case 1: B - 10 = 0
B = 10
Substitute B into the first equation to find L:
70 = L + 10
L = 60
Case 2: L - 15 = 0
L = 15
Substitute L into the first equation to find B:
70 = B + 15
B = 55
Therefore, the length of the field can be either 60m and the breadth can be 10m, or the length can be 15m and the breadth can be 55m.
To find the length and breadth of the rectangular field, we can set up a system of equations based on the given information.
Let's assume the length of the field is "L" and the breadth is "B".
1. From the first sentence, we know that the perimeter of the rectangular field is 140m. The perimeter of a rectangle is given by the formula: Perimeter = 2(length + breadth). So we can write the first equation as: 2(L + B) = 140.
2. From the second sentence, we know that if the length of the field is increased by 2m and the breadth is decreased by 3m, the area is decreased by 66 sq.m. The area of a rectangle is given by the formula: Area = length x breadth. So the second equation can be written as: (L + 2)(B - 3) = (L)(B) - 66.
Now we have a system of two equations:
1. 2(L + B) = 140
2. (L + 2)(B - 3) = (L)(B) - 66
We can solve this system of equations to find the values of L and B.
Let's solve the first equation for L:
2L + 2B = 140
2L = 140 - 2B
L = (140 - 2B) / 2
L = 70 - B
Now substitute this value of L in the second equation:
[(70 - B) + 2](B - 3) = (70 - B)(B) - 66
(72 - B)(B - 3) = 70B - B^2 - 66
72B - 3B^2 - 216 + 9B = 70B - B^2 - 66
9B - 70B = -66 + 216
-61B = 150
B = 150 / -61
Substitute the value of B back into the first equation to find L:
L = 70 - B
Therefore, the values of L and B are given by L = 70 - B and B = 150 / -61.