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 April 12th, 2018, 12:16 AM #1 Member   Joined: Apr 2018 From: On Earth Posts: 34 Thanks: 0 Salary and percentages question Tax is collected annually in the following manner: On the first £5000 earned, no tax is paid On the next £5000 earned, 25% tax is paid On the next £10000 earned, 40% tax is paid On any further income, 60% tax is paid. A sailor pays 33% of his salary in tax. What is his salary? Can anyone solve this?
 April 12th, 2018, 01:49 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,501 Thanks: 1372 Let the salary be $s$ $tax = \begin{cases} 0 &0\leq s \leq 5000 \\ (0.25)(s-5000) &5000 < s \leq 10000 \\ 1250+(0.4)(s-10000) &10000 < s \leq 20000 \\ 5250+ (0.6)(s-20000) &20000 < s \end{cases}$ let's try each case. We see 0 is a solution, but probably not the one they are looking for. if $5000< s \leq 10000$ then $(0.25)(s-5000) = (0.33)(s)$ $-1250 = 0.8s$ which gives us a negative salary so $s > 10000$ If $10000 < s \leq 20000$ $1250 + (0.4)(s-10000) = 0.33s$ $1250 - 4000 = -0.07s$ $s=39285.7$ but this violates $s < 20000$ so $s > 20000$ $5250 + (0.6)(s-20000) = 0.33s$ $5250 - 12000 = -0.27s$ $s = 25000$ and this is your answer. Thanks from Student2018
April 12th, 2018, 03:36 AM   #3
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 Originally Posted by romsek Let the salary be $s$ $tax = \begin{cases} 0 &0\leq s \leq 5000 \\ (0.25)(s-5000) &5000 < s \leq 10000 \\ 1250+(0.4)(s-10000) &10000 < s \leq 20000 \\ 5250+ (0.6)(s-20000) &20000 < s \end{cases}$ let's try each case. We see 0 is a solution, but probably not the one they are looking for. if $5000< s \leq 10000$ then $(0.25)(s-5000) = (0.33)(s)$ $-1250 = 0.8s$ which gives us a negative salary so $s > 10000$ If $10000 < s \leq 20000$ $1250 + (0.4)(s-10000) = 0.33s$ $1250 - 4000 = -0.07s$ $s=39285.7$ but this violates $s < 20000$ so $s > 20000$ $5250 + (0.6)(s-20000) = 0.33s$ $5250 - 12000 = -0.27s$ $s = 25000$ and this is your answer.
Thanks a lot, but might there be a slightly simpler way to solve it, or is this the only method?

April 12th, 2018, 10:07 AM   #4
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 Originally Posted by Student2018 Thanks a lot, but might there be a slightly simpler way to solve it, or is this the only method?
you can start with $s > 20000$ and that will get you the answer on the first try.

I purposefully showed you that the other ranges for $s$ didn't work.

In general though you have to try each range until you find the right one.

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