A rectangle becomes a square when its length and breadth are reduced by 10 m and 5 m, respectively. During this process, the rectangle loses 650 m2 of area. What is the area of the original rectangle in square meters?
2250
To find the area of the original rectangle, we need to determine its initial length and breadth. We are given information about how the rectangle transforms into a square and the area lost during this process.
Rectangle Dimensions and Transformation
Let the original length of the rectangle be \(L\) meters and the original breadth be \(B\) meters.
The area of the original rectangle is \(A_{\text{original}} = L \times B\) square meters.
- The length is reduced by 10 m, so the new length is \((L - 10)\) m.
- The breadth is reduced by 5 m, so the new breadth is \((B - 5)\) m.
After these reductions, the rectangle becomes a square. This means that the new length and the new breadth must be equal:
\[L - 10 = B - 5\]
We can rearrange this equation to express \(L\) in terms of \(B\):
\[L = B - 5 + 10\]
\[L = B + 5 \quad \text{(Equation 1)}\]
Area Loss Calculation
The area of the new square is \(A_{\text{new}} = (L - 10)(B - 5)\) square meters.
We are told that the rectangle loses 650 m\(^2\) of area during this process. This means the difference between the original area and the new area is 650 m\(^2\):
\[A_{\text{original}} - A_{\text{new}} = 650\]
Substitute the expressions for the areas:
\[L \times B - (L - 10)(B - 5) = 650\]
Now, expand the term \((L - 10)(B - 5)\):
\[(L - 10)(B - 5) = LB - 5L - 10B + 50\]
Substitute this back into the area loss equation:
\[LB - (LB - 5L - 10B + 50) = 650\]
Distribute the negative sign:
\[LB - LB + 5L + 10B - 50 = 650\]
The \(LB\) terms cancel out:
\[5L + 10B - 50 = 650\]
Add 50 to both sides:
\[5L + 10B = 650 + 50\]
\[5L + 10B = 700\]
Divide the entire equation by 5 to simplify:
\[L + 2B = 140 \quad \text{(Equation 2)}\]
Solving for Original Dimensions
Now we have a system of two linear equations with two variables:
- \(L = B + 5\)
- \(L + 2B = 140\)
Substitute the expression for \(L\) from Equation 1 into Equation 2:
\[(B + 5) + 2B = 140\]
Combine like terms:
\[3B + 5 = 140\]
Subtract 5 from both sides:
\[3B = 140 - 5\]
\[3B = 135\]
Divide by 3 to find the value of \(B\):
\[B = \frac{135}{3}\]
\[B = 45 \text{ meters}\]
Now that we have the breadth, substitute \(B = 45\) back into Equation 1 to find the length:
\[L = B + 5\]
\[L = 45 + 5\]
\[L = 50 \text{ meters}\]
Original Rectangle Area
Finally, we can calculate the area of the original rectangle using its length and breadth:
\[\text{Original Area} = L \times B\]
\[\text{Original Area} = 50 \text{ m} \times 45 \text{ m}\]
\[\text{Original Area} = 2250 \text{ m}^2\]
The area of the original rectangle is 2250 square meters.
| Parameter | Value |
|---|---|
| Original Length (\(L\)) | 50 m |
| Original Breadth (\(B\)) | 45 m |
| Original Area (\(L \times B\)) | \(50 \times 45 = 2250\) m\(^2\) |
| New Length (\(L-10\)) | \(50-10 = 40\) m |
| New Breadth (\(B-5\)) | \(45-5 = 40\) m |
| New Area (\((L-10)(B-5)\)) | \(40 \times 40 = 1600\) m\(^2\) |
| Area Lost | \(2250 - 1600 = 650\) m\(^2\) |
This confirms our calculations align with the problem statement.
Important Questions from Numerical Computation
Ratio of the angles of a triangle are 1 : 2 : 6. What is the smallest angle?
It would take one machine 4 hours to complete a production order and another machine 2 hour to complete the same order. If both machines work simultaneously at their respective constant rates, the time taken to complete the same order is ________ hours.
Two design consultants, P and Q, started working from 8 AM for a client. The client budgeted a total of USD 3000 for the consultants. P stopped working when the hour hand moved by 210 degrees on the clock. Q stopped working when the hour hand moved by 240 degrees. P took two tea breaks of 15 minutes each during her shift, but took no lunch break. Q took only one lunch break for 20 minutes, but no tea breaks. The market rate for consultants is USD 200 per hour and breaks are not paid. After paying the consultants, the client shall have USD_remaining in the budget.
What is the value of \(1 + \frac{1}{4} + \frac{1}{{16}} + \frac{1}{{64}} + \frac{1}{{256}} + \ldots ?\)
A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?
