Auditing a pond-desilting bill before payment
You are assisting the three-member works committee of a Gram Panchayat in Sitapur district, auditing a pond-renovation bill before payment. The pond is a straight-sided battered tank: top surface 24 m × 18 m, bottom 16 m × 10 m, and the uniform measured depth is 3.00 m. Contractor A's bill reads: volume = top area × depth = 24 × 18 × 3 = 1,296 cubic metres at ₹450 per cubic metre = ₹5,83,200, and this figure is already entered in the measurement book. Contractor B, quoting the same ₹450 rate, has billed ₹3,85,200. The Junior Engineer says both figures are 'approximately right' and wants to release the average of the two bills, ₹4,84,200, to avoid delaying payment to labour. The committee may pass the bill only on a technically defensible measurement, and the MGNREGA social audit is due next month.
1. Compute the correct volume of the excavated tank using the prismoidal (mid-section) formula, show why the top-area × depth method is wrong, and quantify Contractor A's over-billing.
Read the worked answer
Step 1 - Three parallel sections.
- Top: 24 × 18 = 432 m²
- Bottom: 16 × 10 = 160 m²
- Middle (average of top and bottom dimensions, since the sides batter uniformly): 20 × 14 = 280 m²
Step 2 - Prismoidal formula V = (h/6)(A₁ + 4Aₘ + A₂) V = (3/6)(432 + 4×280 + 160) = 0.5 × (432 + 1120 + 160) = 0.5 × 1712 = 856 m³.
Step 3 - Why A is wrong. Top area × depth treats the tank as a straight-sided box of full top area all the way down; it ignores the batter, which removes material on all four sides. It is a systematic over-estimate, not a rounding difference. (Even the cruder average-end formula, (h/2)(A₁+A₂) = 1.5 × 592 = 888 m³, overstates by 32 m³ - so 856 m³ is the defensible figure.)
Step 4 - Over-billing. Correct value = 856 × ₹450 = ₹3,85,200. A's bill = ₹5,83,200. Excess = ₹1,98,000, i.e. 440 m³ too much - an overstatement of 51.4% of the true quantity.
Step 5 - Verdict on the JE's suggestion. Averaging two figures, one of which is demonstrably wrong, is not a measurement; it would pass on ₹98,000–₹99,000 of unjustified public expenditure and would fail the social audit.
2. Back-compute Contractor B's volume from his bill. Does it match your computed quantity, and what should the committee actually pay?
Read the worked answer
Back-computation: volume = bill ÷ rate = ₹3,85,200 ÷ ₹450 = 856 m³.
Comparison: this exactly equals the prismoidal volume computed from the section dimensions. So B's bill is consistent with the correct method - but note that consistency alone does not certify the work; the committee must still check that the measured depth of 3.00 m in the measurement book matches the field measurement and that the muster rolls support the person-days claimed.
What to pay: ₹3,85,200 at the quoted rate, subject to the depth verification below and to standard deductions (labour cess, royalty/earth-sale if applicable, and any material component). The figure entered in the measurement book must be corrected from 1,296 m³ to 856 m³ and the correction signed by the JE, with the reason recorded as "top-area method used; prismoidal recomputation".
What not to do: do not release the average ₹4,84,200, and do not simply reduce A's bill without correcting the measurement book - an uncorrected book will regenerate the same inflated figure in the next bill or in the audit para.
3. Field re-measurement at nine grid points gives an average depth of 2.60 m instead of 3.00 m. Recompute the payable amount and state what action follows for the measurement book and the contractor's bill.
Read the worked answer
Step 1 - Recompute with h = 2.60 m (areas unchanged): V = (2.60/6)(432 + 4×280 + 160) = (2.60/6) × 1712 = 0.4333 × 1712 ≈ 741.87 m³.
Step 2 - Payable amount: 741.87 × ₹450 = ₹3,33,840 (exact: 1712 × 2.60 = 4451.2; ÷6 = 741.8667; ×450 = ₹3,33,840).
Step 3 - Effect of the two corrections together. The original bill of ₹5,83,200 must fall to ₹3,33,840 - a reduction of ₹2,49,360 (42.8%). Contractor B's bill also falls short by ₹51,360 and must be revised, not paid as submitted.
Step 4 - Actions that follow:
- Record the nine grid readings in the measurement book with a sketch, get the JE's and the Gram Pradhan's signatures, and strike out the 3.00 m entry with a dated remark - never overwrite it.
- Issue a written show-cause to Contractor A for claiming an unsupported depth and an inflated method; hold the payment until the revised bill is submitted.
- Report the case to the social audit committee, since the same JE may have certified other bills using the top-area method.
Reasoning principle: the lower of the two defensible quantities governs, and the method and the measurement must both be correct before any rupee moves.