Question 15
The graph shows the value of three different annuities over time.
Which of the following statements gives a plausible explanation for the different values after seven terms?
That’s it. Step through the hints below to see why.
Not quite. A shorter interest term, with the same annual rate, means a smaller rate each term, so the annuity grows more slowly per term. That would put Annuities 1 and 2 below Annuity 3, and they finish above it.
Not quite. Look at term 0: all three annuities start at the same value, $500. The graph rules out different initial values.
Not quite. This is backwards. With everything else equal, a higher interest rate makes an annuity grow faster. Annuities 2 and 3 finish below Annuity 1, so they can’t be the ones with the higher rates.
Each option claims one difference. Test it on a model annuity: change that one thing and see whether the annuity ends higher or lower after seven terms.
The model is made up for testing, not read from the graph: it starts at $500 like all three annuities, earns 5% per term and gets a $100 deposit each term.
Pick a change. The graph tells you Annuity 1 finishes highest, then Annuity 2, then Annuity 3, and that all three start at the same value. An explanation is plausible only if it pushes the right annuities up.
A bigger deposit means more money goes in every term, so the annuity ends higher. (A) gives the bigger deposits to Annuities 1 and 2, the two that do finish higher than Annuity 3. Nothing on the graph contradicts it, so (A) is plausible.
A shorter term at the same annual rate halves the rate per term, from 5% to 2.5% here, so after seven terms the annuity ends lower. (B) gives shorter terms to Annuities 1 and 2, the two that finish higher. That’s backwards, so (B) is out.
A lower start does make an annuity end lower. But look at term 0 on the QCAA graph: all three start at $500. The graph itself rules (C) out.
A higher rate makes an annuity end higher. (D) gives the higher rates to Annuities 2 and 3, the two that finish lower. That’s backwards, so (D) is out.
Read two things off the graph before you read the options.
- At term 0 all three annuities have the same value, $500.
- After seven terms Annuity 1 is highest, Annuity 2 is next and Annuity 3 is lowest.
“Plausible” means the statement could explain that order without contradicting the graph. So test each option and throw out the ones that can’t be true.
The formula book’s recurrence relation for an annuity, with $r=1+i$, shows what each option changes:
$$A_{n+1}=rA_n+d$$
- Bigger deposit $d$: more added each term, ends higher.
- Higher rate $i$: more interest each term, ends higher.
- Lower start $A_0$: ends lower.
- Shorter term, same annual rate: smaller $i$ per term, ends lower after the same number of terms.
Now eliminate.
- (C) is false: the annuities all start at $500.
- (D) is backwards: the higher rates would put Annuities 2 and 3 on top.
- (B) is backwards: shorter terms would slow Annuities 1 and 2 down.
(A) is the only statement that fits the order on the graph, so it is the plausible explanation.
All three annuities start at $500 and finish in the order Annuity 1, Annuity 2, Annuity 3. Test each statement against that.
- (C) Lower initial values for Annuities 2 and 3: false, the starting values are equal.
- (D) Higher rates for Annuities 2 and 3: they would then grow faster, but they finish lower.
- (B) Shorter terms for Annuities 1 and 2: a smaller rate per term would slow them down, but they finish higher.
- (A) Higher regular deposits for Annuities 1 and 2: more money in each term, so they finish higher. This fits.
A note on the drawing. In $A_{n+1}=rA_n+d$ the amount added each term grows by the factor $r$, whatever the deposit is. On this graph the increases grow by a factor of about 1.25 each term for Annuity 1, 1.165 for Annuity 2 and 1.12 for Annuity 3, so the curves as drawn really differ in interest rate per term. Bigger deposits alone wouldn’t give those shapes. (A) is still the answer, because the question asks for a plausible explanation and (A) is the only option the graph doesn’t rule out.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2020 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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