--- title: "Population 2021 and 2026 by three methods" subject: "Water Supply Engineering" chapter: 3 key: "semester_5/water_supply_engineering:pop-c68" source: https://www.bctengineer.com/q/semester_5/water_supply_engineering/pop-c68 site: BCT Engineering --- # Population 2021 and 2026 by three methods Water Supply Engineering · Chapter 3: Quantity of Water - Asked in: 2068 Chaitra, 8 marks - Video: [Population Forecasting | Geometrical Increase Method | Civil Engineering](https://www.youtube.com/watch?v=ERu9DmpIBBg) — 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-c68 - Whole chapter: https://www.bctengineer.com/md/semester_5/water_supply_engineering/chapter-3 ## Question Determine the population of the town in the year 2021 and 2026 by (i) Arithmetical increase method (ii) Geometrical increase method and (iii) Decreased rate of growth method. | Year A.D | 1961 | 1971 | 1981 | 1991 | 2001 | 2011 | |---|---|---|---|---|---|---| | Population | 18000 | 27000 | 38000 | 51000 | 66000 | 83000 | ## Short answer For 2021: arithmetic 96,000, geometric 112,678, decreased rate 99,348. For 2026: arithmetic 102,500, geometric 131,286, decreased rate 106,122. ## Answer Census interval 10 years; last census 2011. For 2021, $n = 1$ decade; for 2026, $n = 1.5$ decades. ### Population in 2021 (n = 1) **Arithmetical increase method** Average increase per decade: $\bar x = \dfrac{83{,}000-18{,}000}{5} = 13{,}000.0$ $$ P_{2021} = P_n + n\bar x = 83{,}000 + 1\times 13{,}000.0 = 96{,}000 $$ **Geometrical increase method** Average growth rate per decade (geometric mean): $r = \left(\dfrac{83{,}000}{18{,}000}\right)^{1/5}-1 = 35.757\%$ $$ P_{2021} = P_n(1+r)^n = 83{,}000\times(1+0.35757)^{1} = 112{,}678 $$ **Decreasing rate of growth method** | Decade | Increase | Growth rate (%) | Decrease in rate (%) | |---|---|---|---| | 1961-1971 | 9,000 | 50.00 | - | | 1971-1981 | 11,000 | 40.74 | 9.26 | | 1981-1991 | 13,000 | 34.21 | 6.53 | | 1991-2001 | 15,000 | 29.41 | 4.80 | | 2001-2011 | 17,000 | 25.76 | 3.65 | Average decrease in rate $= 6.061\%$ per decade. | Year | Growth rate (%) | Population | |---|---|---| | 2021 | 19.70 | 99,348 | $P_{2021} = 99{,}348$ ### Population in 2026 (n = 1.5) **Arithmetical increase method** Average increase per decade: $\bar x = \dfrac{83{,}000-18{,}000}{5} = 13{,}000.0$ $$ P_{2026} = P_n + n\bar x = 83{,}000 + 1.5\times 13{,}000.0 = 102{,}500 $$ **Geometrical increase method** Average growth rate per decade (geometric mean): $r = \left(\dfrac{83{,}000}{18{,}000}\right)^{1/5}-1 = 35.757\%$ $$ P_{2026} = P_n(1+r)^n = 83{,}000\times(1+0.35757)^{1.5} = 131{,}286 $$ **Decreasing rate of growth method** | Decade | Increase | Growth rate (%) | Decrease in rate (%) | |---|---|---|---| | 1961-1971 | 9,000 | 50.00 | - | | 1971-1981 | 11,000 | 40.74 | 9.26 | | 1981-1991 | 13,000 | 34.21 | 6.53 | | 1991-2001 | 15,000 | 29.41 | 4.80 | | 2001-2011 | 17,000 | 25.76 | 3.65 | Average decrease in rate $= 6.061\%$ per decade. | Year | Growth rate (%) | Population | |---|---|---| | 2021 | 19.70 | 99,348 | | 2031 | 13.64 | 112,896 | Population at 2026 is obtained by linear interpolation between the decade values: $P_{2026} = 106{,}122$. #### Summary | Method | 2021 | 2026 | |---|---|---| | (i) Arithmetical increase | 96,000 | 102,500 | | (ii) Geometrical increase | 112,678 | 131,286 | | (iii) Decreased rate of growth | 99,348 | 106,122 | **Answer: 2021: arithmetical 96,000, geometrical 112,678, decreased rate 99,348. 2026: arithmetical 102,500, geometrical 131,286, decreased rate 106,122.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which is the geometric rate per decade? A. (83,000/18,000)^(1/5) - 1 = 35.757% B. (83,000/18,000)^(1/6) - 1 C. (83,000-18,000)/18,000 D. (83,000/18,000)^5 - 1 Answer: A. Five decade intervals, so the fifth root of the ratio. (Key point: Geometric growth rate) 2. Which formula gives the 2026 geometric result? A. 83,000 x (1.35757)^15 B. 83,000 x (1.35757)^1.5 C. 83,000 x (1.35757)^1 D. 83,000 + 1.5 x 0.35757 Answer: B. P = Pn(1+r)^n with n = 1.5 decades. (Key point: Geometric formula for 2026) 3. What is the 2021 growth rate in the decreasing rate method? A. 25.76% + 6.061% B. 6.061% C. 25.76% - 6.061% = 19.70% D. 50% - 6.061% Answer: C. The latest decade rate is reduced by the average decrease. (Key point: Decreasing rate) 4. Why does the geometrical method give the highest value (131,286 for 2026)? A. It assumes the growth rate stays constant while actual rates are falling B. It uses the largest census data C. It adds the incremental increase twice D. It includes the design period twice Answer: A. Real rates fall each decade, so constant-rate growth overestimates. 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