--- title: "Population 2021 and 2028, incremental method" subject: "Water Supply Engineering" chapter: 3 key: "semester_5/water_supply_engineering:pop-b80" source: https://www.bctengineer.com/q/semester_5/water_supply_engineering/pop-b80 site: BCT Engineering --- # Population 2021 and 2028, incremental method Water Supply Engineering · Chapter 3: Quantity of Water - Asked in: 2080 Bhadra, 8 marks - Video: [Population Forecasting | Incremental Increase Method | Civil Engineering](https://www.youtube.com/watch?v=_CZIlybt2tY) — The Online Tutor - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/water_supply_engineering/pop-b80 - Whole chapter: https://www.bctengineer.com/md/semester_5/water_supply_engineering/chapter-3 ## Question The population of a city obtained from census report is as given below. | Census year | 1911 | 1921 | 1931 | 1941 | 1951 | 1961 | 1971 | 1981 | 1991 | |---|---|---|---|---|---|---|---|---|---| | Population | 20000 | 22000 | 25000 | 27500 | 34100 | 41500 | 47050 | 54500 | 61000 | Estimate the population of the city for the year 2021 and 2028 by the incremental increase method and the changing rate of increase method. ## Short answer Incremental increase method: 2021 = 80,232, 2028 = 85,552. Decreasing rate of growth method: 2021 = 86,800, 2028 = 94,715. ## Answer The census interval is 10 years and the last census is 1991. For 2021, $n = 3.0$ decades; for 2028, $n = 3.7$ decades. "Changing rate of increase" is taken as the decreasing (changing) rate of growth method. ### Population in 2021 (n = 3.0) **Incremental increase method** Increase per decade: 2,000, 3,000, 2,500, 6,600, 7,400, 5,550, 7,450, 6,500; incremental increase: 1,000, -500, 4,100, 800, -1,850, 1,900, -950. $\bar x = 5{,}125.00$, $\bar y = 642.86$ $$ P_{2021} = P_n + n\bar x + \frac{n(n+1)}{2}\bar y = 61{,}000 + 3(5{,}125.00) + \frac{3(4)}{2}(642.86) = 80{,}232 $$ **Decreasing rate of growth method** | Decade | Increase | Growth rate (%) | Decrease in rate (%) | |---|---|---|---| | 1911-1921 | 2,000 | 10.00 | - | | 1921-1931 | 3,000 | 13.64 | -3.64 | | 1931-1941 | 2,500 | 10.00 | 3.64 | | 1941-1951 | 6,600 | 24.00 | -14.00 | | 1951-1961 | 7,400 | 21.70 | 2.30 | | 1961-1971 | 5,550 | 13.37 | 8.33 | | 1971-1981 | 7,450 | 15.83 | -2.46 | | 1981-1991 | 6,500 | 11.93 | 3.91 | Average decrease in rate $= -0.275\%$ per decade. | Year | Growth rate (%) | Population | |---|---|---| | 2001 | 12.20 | 68,443 | | 2011 | 12.48 | 76,983 | | 2021 | 12.75 | 86,800 | $P_{2021} = 86{,}800$ ### Population in 2028 (n = 3.7) **Incremental increase method** Increase per decade: 2,000, 3,000, 2,500, 6,600, 7,400, 5,550, 7,450, 6,500; incremental increase: 1,000, -500, 4,100, 800, -1,850, 1,900, -950. $\bar x = 5{,}125.00$, $\bar y = 642.86$ $$ P_{2028} = P_n + n\bar x + \frac{n(n+1)}{2}\bar y = 61{,}000 + 3.7(5{,}125.00) + \frac{3.7(4.7)}{2}(642.86) = 85{,}552 $$ **Decreasing rate of growth method** | Decade | Increase | Growth rate (%) | Decrease in rate (%) | |---|---|---|---| | 1911-1921 | 2,000 | 10.00 | - | | 1921-1931 | 3,000 | 13.64 | -3.64 | | 1931-1941 | 2,500 | 10.00 | 3.64 | | 1941-1951 | 6,600 | 24.00 | -14.00 | | 1951-1961 | 7,400 | 21.70 | 2.30 | | 1961-1971 | 5,550 | 13.37 | 8.33 | | 1971-1981 | 7,450 | 15.83 | -2.46 | | 1981-1991 | 6,500 | 11.93 | 3.91 | Average decrease in rate $= -0.275\%$ per decade. | Year | Growth rate (%) | Population | |---|---|---| | 2001 | 12.20 | 68,443 | | 2011 | 12.48 | 76,983 | | 2021 | 12.75 | 86,800 | | 2031 | 13.03 | 98,108 | Population at 2028 is obtained by linear interpolation between the decade values: $P_{2028} = 94{,}715$. #### Summary | Method | 2021 | 2028 | |---|---|---| | Incremental increase | 80,232 | 85,552 | | Changing rate of increase (decreasing growth rate) | 86,800 | 94,715 | **Answer: Incremental increase: 2021 = 80,232, 2028 = 85,552. Changing rate of increase: 2021 = 86,800, 2028 = 94,715.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which formula is the incremental increase method? A. P = Pn(1+r)^n B. P = Pn + n x + n(n+1)/2 y C. P = Pn + n x D. P = Pn(1 + r_avg - n d) Answer: B. x is the average increase per decade and y the average incremental increase. (Key point: Incremental method formula) 2. What is n for 2028 with last census 1991? A. 3.0 B. 2.8 C. 4.7 D. 3.7 Answer: D. (2028-1991)/10 = 3.7 decades. (Key point: Value of n) 3. What is done in the decreasing rate of growth method? A. Adds a constant increase each decade B. Applies the average decrease in growth rate to successive decades C. Uses the geometric mean of populations D. Plots a logistic curve Answer: B. Each decade's growth rate is the previous rate adjusted by the average decrease. (Key point: Decreasing rate) 4. How is the 2028 value obtained by the decreasing rate method? A. From 2021 using 7 years of the same rate only B. By linear interpolation between decade values C. By repeating the 2031 rate D. By the incremental formula Answer: B. 2028 is interpolated between 2021 (86,800) and 2031 (98,108). 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