My Math Forum Progression problem

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 April 15th, 2014, 03:49 AM #1 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 Progression problem Two companies, C and D, offer a different salary scale for the same post: Company C:​Starting monthly pay = RM500.00 Thereafter, the monthly salary for a particular month is 10% more than the monthly salary for the preceding month ​ Company D:​Starting monthly pay = RM300.00 Thereafter, the monthly salary for a particular month is 15% more than the monthly salary for the preceding month (a)​Azrin and Aidil start to work for company C and D respectively in the same month. ​After how many months will Aidil’s monthly salary be more than Azrin’s monthly salary? ​Use two methods. (b)​Which salary scale is the better deal? Justify.
April 16th, 2014, 08:02 AM   #2
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Hello, jiasyuen!

Quote:
 Two companies, C and D, offer two different salary scales. Company C:​ Starting monthly pay = \$500. Thereafter, the monthly salary is 10% more than the preceding month. ​ Company D:​ Starting monthly pay = \$300. Thereafter, the monthly salary is 15% more than the preceding month. ​Azrin and Aidil start to work for company C and D resp. in the same month. ​(a) After how many months will Aidil’s monthly salary be more than Azrin’s?

These are geometric sequences.

After $\displaystyle n$ months, Azrin's monthly salary is: $\displaystyle \,500(1.10^n)$ dollars.

After $\displaystyle n$ months, Aidil's monthly salary is: $\displaystyle \,300(1.15^n)$ dollars.

When is Aidil's salary greater than Azrin's?

$\displaystyle \quad 300(1.15^n) \;>\; 500(1.10^n) \quad\Rightarrow\quad \frac{1.15^n}{1.10^n} \;>\;\frac{500}{300}$

$\displaystyle \quad \left(\frac{1.15}{1.10}\right)^n \;>\;\frac{5}{3} \quad\Rightarrow\quad \left(\frac{23}{22}\right)^n \;>\;\frac{5}{3}$

Take logs: $\displaystyle \:\ln\left(\frac{23}{22}\right)^n \;>\; \ln\left(\frac{5}{3}\right) \quad\Rightarrow\quad n\ln\left(\frac{23}{22}\right) \;>\;\ln\left(\frac{5}{3}\right)$

$\displaystyle \quad n \;>\;\frac{\ln\left(\frac{5}{3}\right)}{\ln\left( \frac{23}{22}\right)} \;=\;11.491684...$

Aidil's salary will be greater in 12 months.

Quote:
 (b) ​Which salary scale is the better deal? Justify.

If they plan to work for less than one year,
$\displaystyle \quad$ Company C has the better deal.

Otherwise, Company D's deal is better.

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