We've already learnt about the compound interest formula but we have been using it mostly to find the total amount, $A$A. However we can also used this formula to find the principal amount, $P$P, the interest rate, $r$r, or the time duration, $n$n.
Remember the compound interest formula is:
$A=P\left(1+r\right)^n$A=P(1+r)n
If we want to find an unknown other than $A$A, we substitute in the values we know, then just rearrange the equation to change the subject of the formula, then solve the equation as usual.
Worked example
At what annual compound interest rate, $r$r, must Joanne invest $\$220$$220 if she wishes to triple her money in $17$17 years? Give your answer as a percentage correct to two decimal places.
Think: How much is triple the principal?
Do:
| $220\times3$220×3 | $=$= | $\$660$$660 | |
| $A$A | $=$= | $P\left(1+r\right)^n$P(1+r)n | |
| $660$660 | $=$= | $220\times\left(1+r\right)^{17}$220×(1+r)17 | (divide both sides by $220$220) |
| $3$3 | $=$= | $\left(1+r\right)^{17}$(1+r)17 | |
| $\sqrt[17]{3}$17√3 | $=$= | $1+r$1+r | (subtract $1$1 from both sides) |
| $r$r | $=$= | $0.0667$0.0667 ... | |
| $r$r | $=$= | $6.68%$6.68% | (to $2$2 d.p.) |
Practice question
Find the amount, $P$P, that would need to be invested at $6%$6% p.a. compounded monthly to accumulate $\$5600$$5600 in $9$9 years. Give your answer to the nearest dollar.