Utility Billing mathematically models the consumption and cost accounting of essential public services, primarily electricity, municipal water, and piped natural gas (PNG). In CBSE Class 11 Applied Mathematics, utility bill computations apply piecewise step functions, progressive tiered pricing, standing fixed demand charges, regulatory fuel surcharges, and municipal duties.
Key Utility Billing Components:
Kilowatt-Hour (kWh) or "Unit": The standard commercial unit of electrical energy. $1 \text{ Unit} = 1 \text{ kWh} = 1000 \text{ watts} \times 3600 \text{ seconds} = 3.6 \times 10^6 \text{ Joules}$.
$$\text{Units Consumed} = \text{Current Meter Reading} - \text{Previous Meter Reading}$$
Telescopic (Tiered) Tariff Slabs: A progressive pricing model designed to promote conservation, where increasing blocks of consumption are billed at progressively higher per-unit rates.
Fixed / Sanctioned Load Charges: A mandatory recurring monthly fee based on the customer's contracted maximum connected electrical load in kilowatts (kW) or kilovolt-amperes (kVA).
Fuel Surcharge Adjustment (FSA): A variable tariff component adjusted periodically by power utilities to reflect fluctuations in the wholesale price of coal and natural gas used for generation.
Electricity Duty: A statutory state government consumption tax levied as a percentage (e.g. 5% to 9%) on energy charges and fixed charges.
Water Consumption Unit: Billed in kilolitres (kL) where $1 \text{ kL} = 1,000 \text{ litres} = 1 \text{ m}^3$. Often includes a sewerage maintenance surcharge (typically 60% of basic water charges).
Mathematical Framework & Tariff Equations
Total monthly utility cost follows an affine piecewise linear function:
$$\text{Total Electricity Bill} = \text{Energy Charges (EC)} + \text{Fixed Charges (FC)} + \text{FSA} + \text{Electricity Duty (ED)} - \text{Rebates}$$
• $\text{Energy Charges } (EC) = \sum_{i=1}^k u_i \cdot r_i$ (where $u_i$ is units in slab $i$, $r_i$ is rate per unit)
• $\text{Fixed Charges } (FC) = \text{Connected Load (kW)} \times \text{Rate per kW}$
• $\text{FSA} = \text{Total Units Consumed} \times \text{FSA Rate per Unit}$
• $\text{Electricity Duty } (ED) = (EC + FC) \times \text{Duty Rate}\%$
Worked Benchmark Examples
Example 1 (Telescopic Electricity Bill Calculation):
A household with a sanctioned load of $3 \text{ kW}$ consumes $340 \text{ units}$ of electricity in a month. The tariff structure is:
• First 100 units @ ₹3.50/unit
• Next 100 units (101-200) @ ₹4.80/unit
• Next 140 units (above 200) @ ₹6.50/unit
Fixed charge is ₹75 per kW per month. Fuel surcharge is ₹0.30 per unit. Electricity duty is 5% on (Energy Charges + Fixed Charges). Calculate the total bill amount.
Step 5: Total Monthly Bill:
$\text{Total Bill} = 1,740 + 225 + 102 + 98.25 = \mathbf{₹2,165.25}$ (Rounded off to ₹2,165).
Example 2 (Municipal Water & Sewerage Bill):
A residential society consumes $45 \text{ kL}$ of water in a billing cycle. Slabs: First $20 \text{ kL}$ @ ₹5/kL; remaining consumption @ ₹12/kL. Sewerage charge is 60% of water charges, plus a flat meter service fee of ₹50. Calculate the total water bill.
Show Step-by-Step Solution
Water Charges:
• First 20 kL: $20 \times 5 = ₹100$
• Next 25 kL (45 - 20): $25 \times 12 = ₹300$
Total Water Consumption Charges = $100 + 300 = \mathbf{₹400}$.
Meter Fee: ₹50. Total Water Bill: $400 + 240 + 50 = \mathbf{₹690}$.
CHECK YOUR PROGRESS 14.3
Graded textbook exercises based directly on CBSE Class 11 Applied Mathematics for Utility Bills. Solve each problem independently before toggling the step-by-step solutions.
Problem 1Meter Reading & Slab Breakdown
An electricity meter displayed a reading of $6,842 \text{ kWh}$ on 1st October and $7,112 \text{ kWh}$ on 1st November. Calculate the energy charges if the tariff is: 0-100 units @ ₹3.00/unit, 101-200 units @ ₹4.50/unit, and above 200 units @ ₹6.00/unit.
Show Step-by-Step Solution
Step 1: Total Units Consumed:
$\text{Units} = 7,112 - 6,842 = \mathbf{270 \text{ units}}$.
Step 2: Telescopic Slab Allocation:
• First 100 units: $100 \times 3.00 = ₹300$
• Next 100 units (101-200): $100 \times 4.50 = ₹450$
• Next 70 units (201-270): $70 \times 6.00 = ₹420$
Total Energy Charges: $300 + 450 + 420 = \mathbf{₹1,170}$.
Problem 2Full Commercial Electricity Bill
A retail shop with a connected load of $5 \text{ kW}$ consumes $520 \text{ units}$ in a month. Commercial tariff:
• Energy charges: Flat ₹7.50 per unit for all consumption.
• Fixed demand charge: ₹120 per kW per month.
• Electricity duty: 6% on total bill before duty.
• A rebate of 1% is offered for online payment before the due date.
Calculate the net amount payable if the shopkeeper pays online on time.
A family consumes $32 \text{ kL}$ of water in a month. Tariff: 0-15 kL @ ₹4/kL, 16-30 kL @ ₹8/kL, above 30 kL @ ₹15/kL. If sewerage surcharge is 60% of water charges and service charge is ₹40, compute the total water bill.
2. Sewerage Surcharge (60%): $0.60 \times 210 = \mathbf{₹126}$. 3. Service Charge: ₹40. Total Water Bill: $210 + 126 + 40 = \mathbf{₹376}$.
Problem 4Piped Natural Gas (PNG) Tariff
A household PNG meter records consumption of $42 \text{ SCM}$ (Standard Cubic Meters) in a two-month cycle. Slabs: First 30 SCM @ ₹48/SCM; beyond 30 SCM @ ₹54/SCM. Fixed network access fee is ₹100. If GST is applicable at 5% on the entire bill, find the total amount payable.
Show Step-by-Step Solution
1. Gas Consumption Charges:
• First 30 SCM: $30 \times 48 = ₹1,440$
• Remaining 12 SCM (42 - 30): $12 \times 54 = ₹648$
Total Gas Charge = $1,440 + 648 = \mathbf{₹2,088}$.
Printable classroom and homework worksheet for Utility Bills. Covers piecewise functions, tiered consumption models, demand charges, and energy economics.
Section A: Conceptual & Objective Drill3 Questions • 1 Mark Each
Q1. Physical Definition of Electrical Unit1 Mark
Define one commercial unit of electricity in terms of power and time, and express it in Joules.
Show Model Solution
One commercial unit is 1 Kilowatt-hour (kWh), which is the electrical energy consumed by a device of 1,000 watts operating continuously for 1 hour.
$1 \text{ Unit} = 1000 \text{ W} \times 3600 \text{ s} = \mathbf{3.6 \times 10^6 \text{ Joules}}$.
Q2. Telescopic vs Non-Telescopic Tariff1 Mark
What is the fundamental mathematical difference between a telescopic tariff and a non-telescopic tariff?
Show Model Solution
In a telescopic tariff, only units falling strictly within each respective slab are charged at that slab's rate. In a non-telescopic tariff, once total consumption crosses a threshold, the ENTIRE consumption from unit 1 is billed at the highest slab rate.
Q3. Connected Load Surcharge1 Mark
If a customer with a 4 kW connected load is billed a fixed charge of ₹90 per kW per month, calculate the annual fixed demand charge.
Section B: Intermediate Applied Calculations2 Questions • 2 Marks Each
Q4. Two-Tier Apartment Complex Billing2 Marks
An apartment complex consumes $1,800 \text{ units}$ through a bulk commercial meter. Tariff: Flat ₹6.80 per unit, fixed charge of ₹1,500, and electricity duty of 7%. Compute the total bill.
Step 2: Electricity Duty & Total [1 Mark]:
Duty ($7\%$) = $0.07 \times 13,740 = \mathbf{₹961.80}$. Total Bill Amount = $13,740 + 961.80 = \mathbf{₹14,701.80}$.
Q5. Fuel Surcharge Sensitivity2 Marks
A manufacturing unit consumes $14,000 \text{ units}$ in June and $16,500 \text{ units}$ in July. In July, the electricity board hikes the Fuel Surcharge Adjustment from ₹0.25 to ₹0.40 per unit. Calculate the net increase in the company's fuel surcharge expense in July compared to June.
Step 2: July Surcharge [1 Mark]:
$\text{FSA}_{\text{July}} = 16,500 \times 0.40 = \mathbf{₹6,600}$. Net Increase: $6,600 - 3,500 = \mathbf{₹3,100}$.
Section C: Comprehensive Board-Level Problem1 Question • 4 Marks
Q6. Comprehensive Municipal Utility Cost Model4 Marks
A small hospital facility has monthly utility records as follows: 1. Electricity: Connected load $15 \text{ kW}$, consumption $2,400 \text{ units}$. Slabs: First 500 units @ ₹4.50, next 1,000 units @ ₹6.00, remaining @ ₹7.50. Fixed charge ₹100/kW/month. FSA ₹0.35/unit. Electricity duty 6% on (Energy + Fixed charges). 2. Water: Consumption $80 \text{ kL}$. First 30 kL @ ₹6/kL, next 50 kL @ ₹14/kL. Sewerage surcharge 60%, meter rent ₹150.
Calculate: (i) Total electricity bill, (ii) Total water bill, and (iii) Aggregate monthly utility expenditure.
Slab Allocation Overflow: When allocating 340 units across slabs of 100, 100, and above, do not multiply 340 by the highest rate! Break it strictly into 100, 100, and 140.
Duty Base: Electricity duty is usually computed on $(EC + FC)$, NOT including the Fuel Surcharge Adjustment (FSA) unless specifically stated.
Section 14.3 Diagnostic Board Mock Examination
CBSE Blueprint Pattern • Time: 45 Mins • Max Marks: 20
10 QuestionsFull Marking Scheme
Section A: Objective & Assertion-Reason (Q1 – Q4)4 Questions • 1 Mark Each
Question 1MCQ • 1 Mark
How many Joules are there in one commercial unit (1 kWh) of electricity?
What is the primary rationale for power boards implementing telescopic tariff structures?
Correct Answer: (A) To charge high-volume consumers higher rates to incentivize conservation Explanation: Telescopic pricing protects lifeline residential consumers while discouraging excessive energy usage.
Question 4Assertion-Reason • 1 Mark
Assertion (A): Even when a consumer has zero energy consumption in a month, an electricity bill with a positive payable balance is generated. Reason (R): Fixed charges based on sanctioned connected load are recurring mandatory capacity fees payable regardless of actual unit consumption.
Correct Answer: (A) Both (A) and (R) are true and (R) explains (A). Explanation: Fixed demand charges are minimum capacity reservation charges.
Section B: Short Answer Type I (Q5 – Q6)2 Questions • 2 Marks Each
Question 52 Marks
Calculate the energy charges for a monthly consumption of $240 \text{ units}$ if: first 100 units are @ ₹3.50/unit and remaining units are @ ₹5.20/unit.
Show Step-by-Step Solution & Marking Scheme
Step 1: First 100 units [1 Mark]: $100 \times 3.50 = ₹350$. Step 2: Remaining 140 units [1 Mark]: $140 \times 5.20 = ₹728$. Total Energy Charges: $350 + 728 = \mathbf{₹1,078}$.
Question 62 Marks
A consumer's bill shows Energy Charges of ₹2,400 and Fixed Charges of ₹250. If Electricity Duty is 5%, compute the duty amount and the total bill.
Show Step-by-Step Solution & Marking Scheme
Base for Duty: $2,400 + 250 = ₹2,650$. Duty (5%): $0.05 \times 2,650 = \mathbf{₹132.50}$. Total Bill: $2,650 + 132.50 = \mathbf{₹2,782.50}$ (₹2,783).
Section C: Short Answer Type II (Q7 – Q8)2 Questions • 3 Marks Each
Question 73 Marks
A household with a connected load of 4 kW consumed 380 units of electricity. Slabs: 0-100 units @ ₹3.50; 101-200 units @ ₹4.50; above 200 units @ ₹6.50. Fixed charge is ₹70/kW/month. FSA is ₹0.30/unit. Electricity duty is 5% on (EC + FC). Calculate the total electricity bill.
A commercial warehouse consumed 48 kL of water. Slabs: 0-20 kL @ ₹6/kL; 21-40 kL @ ₹10/kL; above 40 kL @ ₹18/kL. Sewerage charge is 60% of water charges, and meter service fee is ₹80. Find the total water bill.
Sewerage & Total [1.5 Marks]:
• Sewerage (60%) = $0.60 \times 464 = ₹278.40$.
• Meter fee = ₹80. Total Bill: $464 + 278.40 + 80 = \mathbf{₹822.40}$.
Section D: Case-Based Integrated Problem (Q9 – Q10)2 Sub-parts • 4 Marks Total
Question 9 & 10 Case StudyCase Study • 4 Marks
Case Background: Energy Efficiency Audit in a School Campus
A secondary school campus with a sanctioned load of 20 kW consumed 3,200 units in April under standard halogen lighting. In May, the school retrofitted all classrooms with LED lighting, reducing energy consumption by 25%.
Tariff: Flat ₹7.00/unit; fixed demand charge ₹90/kW/month; FSA ₹0.25/unit; duty 6% on (EC + FC).
Q9 (2 Marks): Calculate the school's total electricity bill for April. Q10 (2 Marks): Calculate the May electricity bill post-LED retrofit and state the net monthly financial savings achieved.
Show Step-by-Step Solution & Marking Scheme
Solution to Q9 (April Bill) [2 Marks]:
• EC ($3,200 \times 7$): ₹22,400.
• FC ($20 \times 90$): ₹1,800.
• FSA ($3,200 \times 0.25$): ₹800.
• Duty (6% of $22,400 + 1,800 = 24,200$): $0.06 \times 24,200 = ₹1,452$. Total April Bill = $22,400 + 1,800 + 800 + 1,452 = \mathbf{₹26,452}$.
Solution to Q10 (May Bill & Savings) [2 Marks]:
• May Consumption (25% reduction): $3,200 \times 0.75 = 2,400 \text{ units}$.
• EC ($2,400 \times 7$): ₹16,800.
• FC ($20 \times 90$): ₹1,800.
• FSA ($2,400 \times 0.25$): ₹600.
• Duty (6% of $16,800 + 1,800 = 18,600$): $0.06 \times 18,600 = ₹1,116$. Total May Bill = $16,800 + 1,800 + 600 + 1,116 = \mathbf{₹20,316}$. Monthly Savings: $26,452 - 20,316 = \mathbf{₹6,136}$.