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Calculate New Users for Music Website with 40% Annual Growth (2017-2020)
Mathematics
Grade 9
Question Content
At the end of 2015, a music website had 100 000 users. The number of users grew at a rate of 40% per year until the end of 2020. How many more users joined the website between the end of 2017 and the end of 2020?
Correct Answer
194480
Detailed Solution Steps
1
Step 1: Determine the growth factor. A 40% annual growth rate means each year's user count is 140% (or 1.4 as a decimal) of the previous year's count.
2
Step 2: Calculate the number of users at the end of 2017. From 2015 to 2017 is 2 years of growth. Use the compound growth formula: Final Amount = Initial Amount × (Growth Factor)^Number of Years. So users at end of 2017 = 100000 × (1.4)^2 = 100000 × 1.96 = 196000.
3
Step 3: Calculate the number of users at the end of 2020. From 2015 to 2020 is 5 years of growth. Users at end of 2020 = 100000 × (1.4)^5 = 100000 × 5.37824 = 537824.
4
Step 4: Find the number of new users between the end of 2017 and 2020 by subtracting the 2017 user count from the 2020 user count: 537824 - 196000 = 194480.
Knowledge Points Involved
1
Compound Growth Formula
The formula for compound growth is \\(A = P(1 + r)^t\\), where \\(A\\) is the final amount, \\(P\\) is the initial principal amount, \\(r\\) is the annual growth rate (as a decimal), and \\(t\\) is the number of time periods. It is used when a quantity increases by a fixed percentage rate over consecutive equal time intervals, such as population growth, investment returns, or user base expansion for businesses.
2
Percentage to Decimal Conversion
To use a percentage growth rate in calculations, convert it to a decimal by dividing the percentage by 100. For a 40% growth rate, this becomes 0.4. Adding 1 to this decimal gives the growth factor (1.4), which represents the total proportion of the original amount plus the growth.
3
Exponentiation for Periodic Growth
Exponentiation is used to calculate growth over multiple periods. Raising the growth factor to the power of the number of years applies the growth rate repeatedly for each year. For example, \\((1.4)^2\\) applies the 40% growth twice, equivalent to multiplying by 1.4 two times.
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