1. Logical tests

2. Logical tests
Consider an income tax system in which a fixed allowance of £4000 is awarded against gross income, i.e. no tax is paid on first £4000 earned.
First £16000 is taxed at 20%
Over £16000 the rate is 25%

3. Continued… If you earn £30000
Taxable income (TI) = £ £4000 = £26000
First £16000 of this is taxed at 20%; 20/100*£16000 = £3200
The remaining £10000 is taxed at 25%; 25/100*£10000 = £2500
Therefore total tax bill is £3200+£2500=£5700
Net income is £30000-£5700=£24300

4. In Excel… We can use the IF statement Let G be gross income
A be the tax free allowance
Taxable income, TI=G-A
Check to see where any tax should be paid
=IF(G-A<0,0, G-A)

5. … Now to calculate the amount of tax
If TI is less than the upper limit, M, then the tax is simply = lower rate * TI
If TI is greater than M, then the tax is lower rate*TI + upper rate * (TI-M)
Translating into an IF statement, =IF(G-A<=M, lower*(G-A), lower*M+upper*(G-A-M))

6. Naming cells
To assign a name to a cell, highlight the cell and from the menu choose INSERT>NAME>DEFINE
This can be very useful in keeping track of entries in formulae, especially as it means you do not have to use $ to fixed the cell you are referring to.

7. …
Note that the cells A, tl, th, M, G, TI, TD and NI are all named cells
Note lower rate = tl, higher rate =tu
Try this on your own without naming the cells, what do you think will happen?

8. Vertical lookups
How do we deal with situations where there may be more than two or three embedded IF statements, when it may get rather messy?
Imagine a firm is selling kitchen units offers a range of discount rates depending on the size of the order.

9. Example For orders less than 10 units, there is no discount
Orders >=10 but <25 a 4% discount is offered
Orders >=25 but <45 a 6% discount is offered
Orders >=45 but <75 a 7% discount is offered
Orders >=75but <100 a 7.5% discount is offered
Orders >=100 a 9% discount is offered

10. …
We use the VLOOKUP statement

11. Define the lookup table
The order size is stored in cell B3
The different discount rates are stored in cells D3:E8
We want to return the values from column 2 of the lookup table
The statement is therefore = VLOOKUP(B3, D3:E8, 2)