The annual growth rate of a city is 2%. If its population in the year 2020 was 1,50,000, then what will its population be in 2023?
1,59,181
Calculating Population Growth Over Time
This problem asks us to find the future population of a city given its current population, annual growth rate, and the number of years into the future.
Population growth, especially over multiple years with a consistent rate, is typically calculated using a compound growth model, similar to how compound interest is calculated. Each year, the growth is applied not just to the initial population but also to the accumulated growth from previous years.
Given Information
- Initial Population (\(P_0\)) in 2020 = 1,50,000
- Annual Growth Rate (\(r\)) = 2% = 0.02
- Time Period (\(t\)) = Number of years from 2020 to 2023
To find the time period \(t\), we subtract the initial year from the target year:
\(t = \text{Target Year} - \text{Initial Year}\)
\(t = 2023 - 2020 = 3 \text{ years}\)
Formula for Compound Population Growth
The formula used to calculate the population (\(P_t\)) after \(t\) years with an initial population \(P_0\) and an annual growth rate \(r\) is:
\(P_t = P_0 (1 + r)^t\)
Where:
- \(P_t\) is the population after \(t\) years.
- \(P_0\) is the initial population.
- \(r\) is the annual growth rate (expressed as a decimal).
- \(t\) is the number of years.
Step-by-Step Calculation
Now, we substitute the given values into the formula:
\(P_{2023} = 1,50,000 (1 + 0.02)^3\)
\(P_{2023} = 1,50,000 (1.02)^3\)
First, we calculate the value of \((1.02)^3\):
\((1.02)^3 = 1.02 \times 1.02 \times 1.02\)
\((1.02)^2 = 1.02 \times 1.02 = 1.0404\)
\((1.02)^3 = 1.0404 \times 1.02 = 1.061208\)
Now, substitute this back into the main formula:
\(P_{2023} = 1,50,000 \times 1.061208\)
\(P_{2023} = 159181.2\)
Since population must be a whole number, we typically round the result. Rounding \(159181.2\) to the nearest whole number gives \(159181\).
Final Result and Option Comparison
The calculated population of the city in 2023 is approximately 1,59,181.
Comparing this result with the given options:
- Option 1: 1,59,181
- Option 2: 1,58,765
- Option 3: 1,58,413
- Option 4: 1,59,000
Our calculated value matches Option 1 exactly.
Revision Table: Key Terms and Formulas
| Term | Description | Value in this problem |
|---|---|---|
| \(P_0\) | Initial Population | 1,50,000 (in 2020) |
| \(r\) | Annual Growth Rate | 2% or 0.02 |
| \(t\) | Time Period (in years) | 3 years (2020 to 2023) |
| \(P_t\) | Population after \(t\) years | Population in 2023 (to be calculated) |
| Formula | \(P_t = P_0 (1 + r)^t\) | |
Additional Information: Simple vs. Compound Growth
It's important to understand why we use the compound growth formula here. Population growth is generally compounded because the growth in one year adds to the base population for the next year's growth calculation.
- Simple Growth: In simple growth, the growth is calculated only on the initial amount. If this were simple growth, the annual increase would be \(2\% \text{ of } 1,50,000 = 0.02 \times 1,50,000 = 3000\) per year. Over 3 years, the total increase would be \(3 \times 3000 = 9000\). The population in 2023 would be \(1,50,000 + 9000 = 1,59,000\). This matches Option 4, but population growth is almost always compounded in such problems.
- Compound Growth: In compound growth, the growth is added to the population each year, and the next year's growth is calculated on this new, larger population. This leads to faster overall growth compared to simple growth over the same period. The formula \(P_t = P_0 (1 + r)^t\) correctly models this compounding effect.
Therefore, using the compound growth formula is appropriate for this population calculation problem.
Important Questions from Percentage
Radha saves 25% of her income. If her expenditure increases by 20% and her income increases by 29%, then her savings increase by;
The income of A is 45% more than the income of B and the income of C is 60% less than the sum of the incomes of A and B. The income of D is 20% more than that of C. If the difference between the incomes of B and D is Rs. 13200, then the income (in Rs.) of C is:
The price of cooking oil increased by 25%. Find by how much percentage a family must reduce its consumption in order to maintain the same budget.
The population of a city increased by 30% in the first year and decreased by 15% in the next year. If the present population is 11,050 then population 2 years ago was:
The income of A is 30% less than the income of B and the income of B is 137.5% more than that of C. If the income of A is Rs. 28500 less than that of B, then the income (in Rs.) of C is:
