Question

If the length of the canal to be laid each year is proportional to the estimated cost for materials and labour, what fraction of the total length is intended to be completed by the third year?

• A. 0.72
• B. 0.68
• C. 0.70
• D. 0.74
• A. 23.53
• B. 22.53
• C. 24.53
• D. 21.53
• A. 3.47%
• B. 3.07%
• C. 2.87%
• D. 3.27%
• A. 52.78%
• B. 37.78%
• C. 42.78%
• D. 47.78%

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem states that the length of the canal built each year is directly proportional to the cost of materials and labour for that year. We need to find the fraction of the total length completed by the end of the third year (i.e., by the end of 2022). This fraction will be the ratio of the cumulative cost of materials and labour up to 2022 to the total cost of materials and labour over all four years.
Step 2: Key Formula or Approach:
The categories for materials are Bolster, Brace, Slabs, and Other material.
\[ \text{Fraction of length} = \frac{\text{Sum of (Materials + Labour) cost for 2020, 2021, and 2022}}{\text{Total (Materials + Labour) cost for all four years (2020-2023)}} \] Step 3: Detailed Explanation:
First, we calculate the combined cost of materials and labour for each year.
Cost for 2020:
Materials Cost = 0.0 (Bolster) + 0.0 (Brace) + 0.0 (Slabs) + 0.0 (Other material) = 0.0
Labour Cost = 3.2
Total Cost (2020) = 0.0 + 3.2 = 3.2 lakh.
Cost for 2021:
Materials Cost = 142.5 + 105.0 + 22.5 + 37.5 = 307.5
Labour Cost = 37.5
Total Cost (2021) = 307.5 + 37.5 = 345.0 lakh.
Cost for 2022:
Materials Cost = 120.0 + 67.5 + 18.0 + 27.0 = 232.5
Labour Cost = 30.0
Total Cost (2022) = 232.5 + 30.0 = 262.5 lakh.
Cost for 2023:
Materials Cost = 112.5 + 90.0 + 24.0 + 31.5 = 258.0
Labour Cost = 27.0
Total Cost (2023) = 258.0 + 27.0 = 285.0 lakh.
Now, we calculate the required cumulative and total costs.
Cumulative cost by the end of the third year (2022):
\[ \text{Cumulative Cost} = 3.2 + 345.0 + 262.5 = 610.7 \text{ lakh} \] Total cost over four years:
\[ \text{Total Cost} = 3.2 + 345.0 + 262.5 + 285.0 = 895.7 \text{ lakh} \] Finally, we find the fraction:
\[ \text{Fraction} = \frac{610.7}{895.7} \approx 0.6817 \] Step 4: Final Answer:
The fraction of the total length completed by the third year is approximately 0.68.

Answer 2

Step 1: Understanding the Question:
We need to calculate the total increase in the estimated cost due to a 5% rise in the cost of all materials for the years 2022 and 2023.
Step 2: Key Formula or Approach:
The total rise in cost is the sum of the increase in material costs for 2022 and 2023.
Increase for a year = 5% of (Total material cost for that year).
Total Rise = (Increase for 2022) + (Increase for 2023).
Step 3: Detailed Explanation:
First, we find the total cost of materials for the years 2022 and 2023.
Total Material Cost for 2022:
\[ \text{Cost}_{2022} = \text{Bolster} + \text{Brace} + \text{Slabs} + \text{Other material} \] \[ \text{Cost}_{2022} = 120.0 + 67.5 + 18.0 + 27.0 = 232.5 \text{ lakh} \] Increase in cost for 2022:
\[ \text{Increase}_{2022} = 5% \text{ of } 232.5 = 0.05 \times 232.5 = 11.625 \text{ lakh} \] Total Material Cost for 2023:
\[ \text{Cost}_{2023} = \text{Bolster} + \text{Brace} + \text{Slabs} + \text{Other material} \] \[ \text{Cost}_{2023} = 112.5 + 90.0 + 24.0 + 31.5 = 258.0 \text{ lakh} \] Increase in cost for 2023:
\[ \text{Increase}_{2023} = 5% \text{ of } 258.0 = 0.05 \times 258.0 = 12.90 \text{ lakh} \] Total Rise in Estimated Cost:
\[ \text{Total Rise} = \text{Increase}_{2022} + \text{Increase}_{2023} \] \[ \text{Total Rise} = 11.625 + 12.90 = 24.525 \text{ lakh} \] Step 4: Final Answer:
The estimated cost will rise by approximately 24.53 lakh.

Answer 3

Step 1: Understanding the Question:
The question asks for the percentage increase in the total estimated cost if the cost for "contingencies" (Exigencies) is doubled.
Step 2: Key Formula or Approach:
Percentage Increase = \( \frac{\text{Increase in Cost}}{\text{Original Total Cost}} \times 100% \).
The increase in cost is equal to the original total cost of Exigencies, as it is being doubled.
Step 3: Detailed Explanation:
First, we need to calculate the original total estimated cost by summing all the values in the table.
Sum of costs for each year:
2020: 62.3 + 0.0 + 0.0 + 0.0 + 0.0 + 3.2 + 11.3 + 1.5 = 78.3
2021: 11.3 + 142.5 + 105.0 + 22.5 + 37.5 + 37.5 + 22.5 + 22.5 = 401.3
2022: 3.3 + 120.0 + 67.5 + 18.0 + 27.0 + 30.0 + 22.5 + 6.3 = 294.6
2023: 0.8 + 112.5 + 90.0 + 24.0 + 31.5 + 27.0 + 21.0 + 7.5 = 314.3
\[ \text{Original Total Cost} = 78.3 + 401.3 + 294.6 + 314.3 = 1088.5 \text{ lakh} \] Next, calculate the total cost for Exigencies over four years. This amount will be the increase in the total estimate.
\[ \text{Total Exigencies Cost} = 1.5 (2020) + 22.5 (2021) + 6.3 (2022) + 7.5 (2023) = 37.8 \text{ lakh} \] Since this cost is doubled, the increase is equal to the original amount, i.e., 37.8 lakh.
Now, calculate the percentage increase:
\[ \text{Percentage Increase} = \frac{37.8}{1088.5} \times 100% \] \[ \text{Percentage Increase} \approx 0.03472 \times 100% \approx 3.47% \] Step 4: Final Answer:
The total estimate increases by approximately 3.47%.

Answer 4

Step 1: Understanding the Question:
The total expenditure needs to be brought down to Rupees1,050 lakh from its original value. This reduction is achieved by cutting the management expenditure equally across all four years. We need to find the percentage reduction in the management cost for the year 2021.
Step 2: Key Formula or Approach:
1. Calculate the total reduction required.
2. Calculate the reduction per year (since it's an equal reduction).
3. Calculate the percentage reduction for 2021 using the formula:
\( \text{Percentage Reduction} = \frac{\text{Reduction Amount for 2021}}{\text{Original Management Cost for 2021}} \times 100% \)
Step 3: Detailed Explanation:
From the previous question, the original total expenditure is 1088.5 lakh.
The target expenditure is 1050 lakh.
Total reduction required:
\[ \text{Total Reduction} = 1088.5 - 1050 = 38.5 \text{ lakh} \] This reduction is to be applied equally over four years from the management expenditure.
Reduction in management expenditure per year:
\[ \text{Reduction per year} = \frac{38.5}{4} = 9.625 \text{ lakh} \] The original management expenditure for the year 2021 is given in the table as 22.5 lakh.
Now, we can calculate the percentage reduction for 2021:
\[ \text{Percentage Reduction for 2021} = \frac{\text{Reduction per year}}{\text{Original Management Cost for 2021}} \times 100% \] \[ \text{Percentage Reduction for 2021} = \frac{9.625}{22.5} \times 100% \] \[ \text{Percentage Reduction for 2021} \approx 0.42777 \times 100% \approx 42.78% \] Step 4: Final Answer:
The percentage reduction for the year 2021 is 42.78%.