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SL 1.4—Financial apps – compound interest, annual depreciation - IB Questionbank | RevisionDojo

Practice SL 1.4—Financial apps – compound interest, annual depreciation with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

Two savings plans for the same horizon $n$ years with annual rate $r>0$ :

Plan A: invest all $P today.
Plan B: invest $\frac{P}{n}$ at each year-end for $n$ years.

Write the future values $V_A,V_B$ and simplify $V_B$ .

[3]

Prove that $V_A>V_B$ for $n\ge2$ using $(1+r)^n>1+nr$ .

[3]

Show that the percentage shortfall $\dfrac{V_A-V_B}{V_A}$ equals $1-\dfrac{(1+r)^n-1}{nr(1+r)^n}$ and decreases as $r$ decreases or $n$ increases (qualitative).

[3]

Question 2

SLPaper 1

An account credits a fixed annual interest rate $r>0$ compounded once per year.

A lump sum $P$ is invested today. Write the value after $n$ years.

[2]

A target $k>1$ is set so that the balance must reach $kP$ . Find the least integer $n$ required, using logarithms.

[3]

Two different rates $r_1,r_2>0$ are offered for the same horizon $n$ . Prove using log/exponent laws that $P(1+r_1)^n>P(1+r_2)^n$ iff $r_1>r_2$ .

[3]

Question 3

SLPaper 1

A company records annual revenue as $R_n=R_0(1+g)^n$ with growth rate $g$ (could be negative).

If $g>0$ , find $n$ so that $R_n=2R_0$ .

[2]

If $g<0$ (decline), show that the half-life $H$ (time until $R_H=\tfrac12 R_0$ ) equals $H=\dfrac{\log(\tfrac12)}{\log(1+g)}$ . Explain why $H>0$ .

[3]

Prove the law $\log_a(xy)=\log_a x+\log_a y$ from the exponential identity $a^{u+v}=a^u a^v$ using part 1’s reasoning structure.

[3]

Question 4

SL & HLPaper 2

After $7$ years of investing in an account that pays a fixed annual compound interest rate of $4.5\%$ , the total value of the investment has grown to $9500.

Calculate the initial amount invested, correct to the nearest dollar.

[3]

Question 5

SL & HLPaper 2

Sania invests $6000 at a rate of $3.1\%$ per year compound interest.

Work out the interest earned on the investment at the end of $2$ years.

[3]

Question 6

SL & HLPaper 2

An investor has two options:

Option 1: Invest $P at an annual compound interest rate of $r_1\%$ . After $12$ years, the investment grows to $2P.

Option 2: Invest $P at an annual compound interest rate of $r_2\%$ , compounded semi-annually.

Determine the value of $r_2$ , correct to $2$ decimal places, such that the total value of the investment under Option 2 after $10$ years is equal to the total value of the investment under Option 1 after $10$ years.

[6]

Question 7

SL & HLPaper 2

David invests $1200 at a rate of $2.5\%$ per year compound interest.

Calculate the amount David has after $8$ years.

[2]

Question 8

SLPaper 1

A gadget’s value follows $V_n=V_0(1-d)^n$ with $0<d<1$ . A warranty requires $V_n\ge \beta V_0$ for $n$ years with $\beta\in(0,1)$ .

Find the largest integer $n$ satisfying the warranty.

[3]

If the firm upgrades to a two-stage schedule: first $k$ years at rate $d_1$ , then at $d_2$ ( $0<d_1,d_2<1$ ). Show
$V_n=\begin{cases} V_0(1-d_1)^n,& n\le k,\\ V_0(1-d_1)^k(1-d_2)^{\,n-k},& n>k. \end{cases}$

[3]

Prove $\log_a\!\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$ and use it to linearize the inequality for the warranty in the two-stage case.

[3]

Question 9

SL & HLPaper 2

A company purchases a machine for $25000. The machine depreciates at a rate of $8\%$ per year.

Show that, at the end of 5 years, the machine will be worth approximately $16477, correct to the nearest dollar.

[2]

Another company also purchases a machine for $25000. At the end of 3 years, this machine is worth $18000. Find the yearly depreciation rate for this machine.

[3]

Question 10

SLPaper 2

Maria is planning for her future and wants to invest in an account that offers a 4% interest rate per year, compounded annually.

Calculate the amount Maria needs to invest today to reach a future value of $20,000 in 10 years. Give your answer to the nearest dollar.

[4]

After investing, Maria finds another fund that offers a 5% interest rate, compounded annually. Calculate how much less she would need to invest today with this higher interest rate to reach the same goal of $20,000 in 10 years.

[3]

SL 1.4—Financial apps – compound interest, annual depreciation Questionbank