Example 4 of 7 · Finance: a yearly cost

AI HL · Mathematics

A gym membership costs CHF 80 per month, correct to the nearest CHF 10. Find the maximum percentage error in the cost for a year.

Smallest true value

12 × 75 900
\[\begin{aligned}T_2&=12\times75=900\\[6pt]\class{mjx-pe-s6}{\%\text{ error}}&\class{mjx-pe-s6}{{}=\left|\frac{960-900}{900}\right|\times100}\\[6pt]&\class{mjx-pe-s6}{{}=\class{mjx-pe-key}{\boldsymbol{6.67\%}}}\end{aligned}\]

Approximate value

12 × 80 960
\[\begin{aligned}A&=12\times80=960\\[6pt]\class{mjx-pe-s5}{\%\text{ error}}&\class{mjx-pe-s5}{{}=\left|\frac{A-T}{T}\right|\times100}\end{aligned}\]

Largest true value

12 × 85 1020
\[\begin{aligned}T_1&=12\times85=1020\\[6pt]\class{mjx-pe-s7}{\%\text{ error}}&\class{mjx-pe-s7}{{}=\left|\frac{960-1020}{1020}\right|\times100}\\[6pt]&\class{mjx-pe-s7}{{}=\class{mjx-pe-key}{\boldsymbol{5.88\%}}}\end{aligned}\]

Maximum percentage error = 6.67%. The smallest true value gives the bigger error, because we divide by it.