Assuming that the interest is paid as compound interest.
Final value = Intial Value × (Multiplier)ⁿ
Multiplier is 100% + the percentage increase/decrease
The letter 'n' is for the number of years
We have:
Initial Value = 1000
Final Value = 18600
Multiplier = 100% + 6.75% = 1 + 0.0675 = 1.0675
We are looking to find:
The number of years 'n'
Substituting the values we have into the formula
18600 = 1000 × (1.0675)ⁿ
18600 ÷ 1000 = 1.0675ⁿ
18.6 = 1.0675ⁿ ⇒ From here you can use either trial and error method or you can use logarithm law
Using trial and method
Let n = 1; 1.0675¹ = 1.0675 ⇒ Not the result we want
Let n = 5; 1.0675⁵ = 1.39 ⇒ Not the result we want
Let n = 10; 1.0675¹⁰ = 1.92 ⇒ Not the result we want
Let n = 25; 1.0675²⁵ = 5.12 ⇒ Still too small
Let n = 55; 1.0675⁵⁵ = 36.33 ⇒ Values are too big
Let n = 40; 1.0675⁴⁰ = 13.64 ⇒ Too small (we narrowed down the value of n to be between 40 and 55)
Let n = 48; 1.0675⁴⁸ = 23.00 ⇒ Too big
Let n = 45; 1.0675⁴⁵ = 19.00 ⇒ Let's try n = 44 to see which value are closer to 18.60
Let n = 44; 1.0675⁴⁴ = 17.70 ⇒ This a little far from 18.6
So we have our answer, n = 45
Conclusion: Tina needs to save for 45 years to get $1000 to $18600 with compound interest 6.75% paid yearly.
