Every other page on this site assumes the arithmetic is the easy part. For a lot of students it is not, and the places it goes wrong are predictable. These four are the ones that cost real marks — and real credibility — in appraisal work. Measuring irregular shapes has its own page, Sketches.
Percent means per hundred. That is the whole definition — the word is doing exactly what it says. 7 percent is 7 per hundred, which is the fraction 7/100, which is 7 divided by 100.
Nothing changes when the number stops being tidy. A rate of 7.7% is 7.7 per hundred, which is 7.7 divided by 100, which is 0.077. These are not two numbers that happen to be related — they are the same number written two ways.
Before any of that, get comfortable with the fact that one quantity has several perfectly correct spellings, and that people switch between them mid-sentence without warning.
All four say the same thing. The first three are numbers and they are what we work in; the fourth is how people talk, and it is where the confusion enters, because “a quarter” and “twenty-five percent” and “point two five” arrive in conversation as though they were different quantities.
| In words | Fraction | Decimal | Percent |
|---|---|---|---|
| a half | 1/2 | 0.50 | 50% |
| a third | 1/3 | 0.3333… | 33.33% |
| a quarter | 1/4 | 0.25 | 25% |
| three quarters | 3/4 | 0.75 | 75% |
| an eighth | 1/8 | 0.125 | 12.5% |
| a tenth | 1/10 | 0.10 | 10% |
| a hundredth | 1/100 | 0.01 | 1% |
A fraction becomes a decimal by doing the division the fraction is asking for. One quarter means one divided by four, and one divided by four is 0.25. That is not a rule to memorize; it is what the notation means. A third comes out 0.3333 and never finishes, which is the first hint that fractions and decimals are not equally useful.
Coming back the other way, reverse both steps: drop the % sign and move the decimal two spots to the left. 7.7% becomes 0.077, which is the form your calculator actually needs when you divide by a rate.
Read the rate below and write it as a percent. Then walk the decimal across one spot at a time and watch what happens to the figure.
| Calculator shows | Written as | Say it as |
|---|---|---|
| 0.0800 | 8.00% | eight percent |
| 0.0775 | 7.75% | seven and three quarters |
| 0.1025 | 10.25% | ten and a quarter |
| 0.0055 | 0.55% | fifty-five hundredths of a percent |
| 1.0800 | 108.00% | a multiplier, not a rate |
That last row is worth a second look. A figure above 1.0 is usually a factor you multiply by, not a rate you divide by. Seeing 1.08 and reading it as an 8% rate is the same mistake wearing a different hat.
In a plain calculation, multiply and divide first, then add and subtract. Work left to right within each group, and do anything in brackets first.
The brackets are not changing the answer — multiplication would happen first regardless. They are making the grouping visible, which is worth doing on your own worksheet until the habit is automatic.
With potential gross income of $240,000 and a vacancy allowance of 6%, that is 240,000 − (240,000 × 0.06) = 240,000 − 14,400 = $225,600. Subtract before multiplying and you would get (240,000 − 240,000) × 0.06 = 0, which at least announces itself. Most order errors are subtler than that.
The same figure comes out of 240,000 × (1 − 0.06) = 240,000 × 0.94 = $225,600, in one step instead of two. Notice the brackets moved, and so did what gets calculated first — that is brackets doing real work rather than decorating.
Potential gross income , vacancy and collection loss , operating expenses of effective gross income. What is net operating income?
| Sale price | Then | Result |
|---|---|---|
| $300,000 | + $20,000 repairs, then × 1.10 market conditions | $352,000 |
| $300,000 | × 1.10 market conditions, then + $20,000 repairs | $350,000 |
Two thousand dollars apart on the same two adjustments. Neither calculation contains an arithmetic error — only one of them follows the required order.
Here is the problem, stated plainly. You have this:
Getting from one to the other is called isolating the unknown. The idea underneath it is simpler than the notation makes it look: the unknown has things attached to it, and every move you make takes one of those things away. When nothing is left attached, you are finished, and whatever sits on the other side of the equals sign is the answer.
What is attached to x? A “+ 5”. So take the 5 away — and take it away from both sides, because the moment you change one side only, the two sides stop being equal and the statement stops being true.
The objective of that move was not “subtract 5”. It was remove the thing attached to x. Subtracting 5 was simply how it was done.
Now b has two things in the way: it is inside a bracket, and the whole bracket is multiplied by 5. You cannot reach b until the 5 is dealt with, so the 5 goes first. That is what working from the outside in means — peel the outermost layer, then the next.
| Move | Objective | Leaves |
|---|---|---|
| ÷ 5, both sides | get rid of the multiplier wrapped around the bracket | 10 = b + 2 |
| − 2, both sides | remove the last thing attached to b | 8 = b |
Two moves, two objectives, and at no point did anyone need to remember a rule about which side things “flip” to. Check it: 5 × (8 + 2) = 5 × 10 = 50.
Solve for x. Two moves, in the order the last example used.
The band of investment builds a capitalization rate from a loan and a down payment. Run left to right, it produces the rate. Run it backwards and you can recover any one of the pieces — and the amount of work depends entirely on which piece you are after.
M is the loan share · Rm is the mortgage constant · Re is the equity dividend rate
Land and building split a property the way debt and equity split its financing, and the band that puts them back together has exactly the same shape. Only the letters change.
B is the building share of value · Rb is the building rate · Rl is the land rate
Mortgage-equity analysis builds an overall rate from the equity yield the investor wants, the financing he uses, how much of the loan he pays off while he holds it, and what he expects the property to be worth when he leaves. Taught on a board it is four rows and a total — where the simple band of investment had two.
Take the rows in the order they were worked and set them down on a single line. Nothing is added and nothing is dropped — the plus and minus signs at the left of each row become the plus and minus signs of the equation.
M loan share · Rm mortgage constant · Ye equity yield · P share of loan paid off · SFF sinking fund factor · Δo change in value
That single line is the thing you can rearrange. The four-row layout is for working a rate out; the equation is for getting a missing piece back. Same statement, two jobs.
An exponent is shorthand for repeated multiplication. 1.053 means 1.05 × 1.05 × 1.05. Nothing more mysterious than that — but the difference between multiplying a rate by the number of periods and raising a factor to the power of the number of periods is money.
| Method | Total change | On $300,000 |
|---|---|---|
| Straight line | ||
| Compounded | ||
| Difference |
Exponents also sit underneath every one of the six functions of a dollar. A future value factor is (1 + i)n; a present value factor is its reciprocal, 1 ÷ (1 + i)n. Once that registers, the six functions stop being six things to memorise and become one idea pointed in different directions.