Using Equation to solve Tax Questions

[Jasmine123]

Hello!
I'm a tax student struggling with a reverse engineered tax question, based on algebraic principles.
Please could someone answer by question by breaking the answer down into a series of steps solving each part at a time as the answer I have from the ICAEW skips too many steps for me to follow.
I can solve these reverse engineering questions when it does not include thresholds to which the tax does not apply but I cannot figure out how to factor in the personal allowance for income tax and primary thresholds for NIC. For simpler questions which only include a rate of tax to the whole amount, I would do Net x (100/100-rate of tax) so for example:
net receipt needs to be 20k, income tax is suffered at 40% and NIC at 2%. What is the gross payment?
Answer: 40+2 = 42%. tax suffered
100-62 = 58%
Gross = 20k x 100/58 = £34,483

Using this principal that I understand, can anyone help me re-arrange the following question which then goes on to include the thresholds which are tax free?

The question is:

If Sarah wishes to receive after-tax (net) income of £20,000, what is the pre-tax (gross) amount the company will have to pay her?
PAYE is 20% of (gross-12,500) (first 12,500 earned is tax-free)
NIC is 12% of (gross-8,636) (first 8636 is NIC tax free)

So far I've got to:
G = 20k + (20% x (G-12,500)) + (12% x (G-8636))

Then it all goes downhill from there

Thank you!

 

[Dr.Peterson]

I know nothing about PAYE or NIC, but I'll assume your description is right, and so is your equation (though I would not have written it initially as an equation for G, but as G - (20% x (G-12,500)) + (12% x (G-8636)) = 20,000 ).

What is it that "goes downhill"? To solve the equation, you need to replace percentages with decimals (e.g. 20% with 0.20), then distribute and collect like terms. Please show your work, so we can see what you are doing wrong.

 

[Jasmine123]

Hi, thank you for your response, I feel like I got a bit further with it this time from your suggestion but you will see from the below descent into madness that I have completely failed to remember any GCSE maths.

I can solve the questions posed to me in basic algebra tutorials, yet cant solve this!

I attach the answer from the book and my chaotic workings. I know what I'm trying to get to, I just keep getting lost on the way.

 

[Dr.Peterson]

I suggested that you distribute ("multiply out", or "expand") and then collect like terms. You tried to collect like terms (by adding in an illegal way) before there were any like terms! "Like terms" are terms like, say, 0.2a and 0.12a, which have the same variable and different coefficients. Those terms didn't exist yet when you combined things by adding numbers that can't be added.

Multiply out first, so that there are no parentheses. Then there will be terms you can combine.

Give it another try.

 

[Jasmine123]

A bit closer this time after remembering some of the rules. Lost it at step 3 though

 

[Dr.Peterson]

One little mistake: when you have something like A - (B + C), you have to distribute the "-" over the parenthesis (brackets), like A - B - C. That is, you're multiplying each term inside by -1. In your case, it's A - (B - C) = A - B + C. Do you see why?

But you're definitely getting there. Thanks for patiently extracting memories as I try to jog them.

 

[Jasmine123]

YES! IVE GOT IT!
Thank you for patiently helping me to remember what I had lost. There were a few rules there I had forgotten how to apply logically to maths problems and it was more helpful to work it out logically that be instructed on the steps again.
Thank you so much! All of my tutors kept saying it's a very hard past exam question, just explain the calculation and come up with a rough figure, students got marks for knowing the basic principles! but I knew it was fairly basic algebra once upon a time, so why not get those skills back if my brain was once capable of it.
Getting these logic skills back will greatly help me in my role as an auditor as well as getting through exams so thank you very much for your help

 

[Dr.Peterson]

This is what we wish happened all the time! It validates the policy of just helping, not doing, when we have someone willing to struggle and win.

 

[Jasmine123]

It's how more teaching should be done! I'm sure half of the stuff I learnt at school is gone because I learned it by rote for the exams without ever fully understanding it. Being able to convert any accounting logical question with a known into a solvable equation is priceless for my work, but of course seemed like just playing around with numbers and letter at school!

Thank you for your time!