Compound interest and percentage growth

10 Layla invests money in the bank and receives compound interest. The amount \(\pounds L\) that she has after \(t\) years is given by the equation \(\mathrm { L } = 2800 \times 1.023 ^ { \mathrm { t } }\).

1. 1. State the amount she invests.
   2. State the annual rate of interest. Amit invests \(\pounds 3000\) and receives \(2 \%\) compound interest per year. The amount \(\pounds A\) that he has after \(t\) years is given by the equation \(\mathrm { A } = \mathrm { ab } ^ { \mathrm { t } }\).
2. Determine the values of the constants \(a\) and \(b\).
3. Layla and Amit invest their money in the bank at the same time. Determine the value of \(t\) for which Layla and Amit have equal amounts in the bank. Give your answer correct to \(\mathbf { 1 }\) decimal place.

3. Every \(\pounds 1\) of money invested in a savings scheme continuously gains interest at a rate of \(4 \%\) per year. Hence, after \(x\) years, the total value of an initial \(\pounds 1\) investment is \(\pounds y\), where $$y = 1.04 ^ { x }$$

1. Sketch the graph of \(y = 1.04 ^ { x } , x \geq 0\).
2. Calculate, to the nearest \(\pounds\), the total value of an initial \(\pounds 800\) investment after 10 years.
3. Use logarithms to find the number of years it takes to double the total value of any initial investment.

4 The value, \(\pounds V\), of an initial investment, \(\pounds P\), at the end of \(n\) years is given by the formula $$V = P \left( 1 + \frac { r } { 100 } \right) ^ { n }$$ where \(r \%\) per year is the fixed interest rate.
Mr Brown invests \(\pounds 1000\) in Barcelona Bank at a fixed interest rate of \(3 \%\) per year.

1. 1. Find the value of Mr Brown's investment at the end of 5 years. Give your value to the nearest \(\pounds 10\).
   2. The value of Mr Brown's investment will first exceed \(\pounds 2000\) after \(N\) complete years. Find the value of \(N\).
2. Mrs White invests \(\pounds 1500\) in Bilbao Bank at a fixed interest rate of \(1.5 \%\) per year. Mr Brown and Mrs White invest their money at the same time. The value of Mr Brown's investment will first exceed the value of Mrs White's investment after \(T\) complete years. Find the value of \(T\).

15. Robert wants to deposit \(\pounds P\) into a savings account. He has a choice of two accounts.

• Account \(A\) offers an annual compound interest rate of \(1 \%\).
• Account \(B\) offers an interest rate of \(5 \%\) for the first year and an annual compound interest rate of \(0.6 \%\) for each subsequent year.

After \(n\) years, account \(A\) is more profitable than account \(B\). Find the smallest value of \(n\).
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