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April 12th, 2018, 12:16 AM   #1
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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?
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April 12th, 2018, 01:49 AM   #2
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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.
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April 12th, 2018, 03:36 AM   #3
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Quote:
Originally Posted by romsek View Post
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?
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April 12th, 2018, 10:07 AM   #4
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Quote:
Originally Posted by Student2018 View Post
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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