There are few real math whizzes in the world. As the old saying goes, how often are you going to trot out even high school algebra in most of what you do? But people use regular arithmetic all the time, whether at work, in school, out shopping, checking your pace on the treadmill, or even cooking.
Some basics are what you need most of the time. Make a mistake and people may think you're just not all that smart. Intelligence may go far beyond ease with numbers, but life isn't fair. The first step to getting past them is to stop beating yourself. Math is often taught badly in school.
If you concentrate on getting the basics, you'll be able to follow business discussions more easily and be more confident in front of others.
How to figure a percentage
Some people have a hard time managing percentages. Remember that all you're doing is saying what part of one amount (A) you'd need to get another (B). For example, 50 is more than 10, so you wouldn't need all of a 50 gallon tank to fill a 10 gallon one. But how much would you need?
Grab a calculator and divide B by A to get a fraction. In this case, B is the final amount and A is the amount you're comparing it to. For the 50 gallons versus 10 gallons, you're winding up with 10 gallons and comparing it to 50. Take a calculator and divide 10 by 50 to get 0.2.
A percentage is just a way of writing that relationship instead of writing the fraction 0.2. What you do is multiple the fraction by 100, so you get 0.2 * 100, or 20 percent (we'll use % instead of "percent" from here on). What confuses many people is that step of multiplying by 100. It doesn't do anything to the relationship. It's just another way to write it -- a convention that someone came up with many years ago. Instead of saying that 10 is 20% of 50, you could just as well say that 10 is 0.2 times 50. In a way, it's like comparing a sentence written in cursive to the same sentence in block letters. They look different but both say the same thing. Confusing? Yes, and you can blame whomever came up with the percentage convention.
Percentages greater than 100%
We just talked about how one value can be represented as a part of another, but there was an assumption in the explanation. The final number, B, was smaller than A, the amount you were comparing it to. But what happens when B is larger? You do the same thing, with dividing B by A and then multiplying by 100 to get the percentage.
If B in this case is 160 gallons and A is still 50 gallons, you do the same thing as before, dividing 160 by 50. The answer is 3.2. The part left of the decimal point shows how many times you'd have to use all of A. The part on the right is the portion that's less than a full A you'd additionally need. So, 3.2 represents 3 full As. The .2 is 0.2 part of A. Multiply the 3.2 by 100 to see that B is 320% of A. It's the same as 300% (3 full As multiplied by 100, according to that odd convention) and another 20% of an A.
It surprised many people, but math ultimately comes down to language. It's a shorthand for expressing certain types of ideas. Unfortunately, things can get confusing when you switch back and forth between math and English.
A case in point is when you hear that something grew by some percentage. The confusion comes because there are, as usual, three numbers you can look at: the initial amount, the amount it increased by, and the final amount. As an example, say that the sales of purple thingamabobs grew from 1,000 units last year to 5,000 units this year.
There is the starting point sales of 1,000 units. To express this year's sales of 5,000 units as a percentage of last year's sales, you divide 5,000 by 1,000, which equals 5, and multiple 100 to get 500%. So this year's sales are 500% of last years.
However, the amount of growth, which is 4,000 units, is different when expressed as a percentage. You divide 4,000 by 1,000 for 4, because you had to add 4 times last year's unit sales to get the unit sales for this year, and then multiply 4 by 100 for 400%. While this year's sales are 500% of last year's, the unit sales grew by only 400%. Keep in mind the starting point, the ending point, and the amount you had to add (or maybe subtract if things shrunk). When you ask about growth as a percentage, you're always interested in the amount added to get to the final point.
Percentage versus percentage point
Back to percentages again. You see how one amount can be expressed as a percentage of another. Sometimes you'll hear the term percentage points, like "interest rates are up 0.3 percentage points." This is when you're discussing change as a series of steps. Instead of saying that today's interest rate of 3% is some percentage above the 1% it was last month, you talk about the new rate as being up two percentage points.
You'll typically hear the term used when there is a fixed scale -- interest rates, pressure, or temperatures, for example. People often use percentage points in such cases because the scale offers a reference, so you no longer need the beginning value to act as a reference.
Which fraction is bigger?
You probably know that 1/2 is bigger than 1/3 (look at a measuring cup). But, in general, it's handy to know whether one fraction is bigger than another. That can help you know whether one drill bit is larger or smaller than the size you need or if a company is trying to pretend that you're getting a bigger discount than you used to.
There's an easy trick: cross-multiplication. Call the fractions a/b and c/d. Put each fraction on one side of a question mark and then move the d crosswise to multiply the a and the b crosswise to multiple the c. If a*d is larger than b*c, then a/b is larger than c/d. Also, if b*c is larger, than c/d is larger.
Compound interest versus simple
In simple interest, you invest money and whoever holds it for you pays you a fixed percentage every year on that initial principal. So, if you've put in $100 and get 5 percent annual simple interest, every year you get an additional $5.
Compound interest is better if you're the one being paid. Not only do you get interest on the principle, but on the interest payments you've received over time. After the first year, the $100 brings you $5 in interest. At the end of the second year, you get 5 percent on $105, which is another $5.25. At the end of the third year you get interest on the total to date, which is now $110.25. And so it goes, which is why compound interest can pay off so well.
After 10 years with 5 percent simple interest, your $100 has brought in an additional $50 ($5 a year), so you get $150. With compound interest calculated once a year (using a calculator from Investor.gov), you'd have $162.89.
Compound annual growth rate versus average growth rate
If you start with one amount of money -- call it $X -- and end after some number of years with another amount, $Y, you know there's been growth. The question is how much.
There are two ways to calculate it. One is using average growth. You subtract the initial number from the final one to get how much it increased. Now you figure the amount of increase as a percentage of the first amount. If you start with $110 and end up with $153, the amount of increase is $43. That $43 is 39 percent of the initial $110 -- 100*(39/110). Divide the percentage of increase by the number of years and you have the average growth. If the $110 grew to $153 over 7 years, you divide the 39 percent by 7 to get about 5.6 percent a year for each of the 7 years.
If you're talking about an investment, typically it's compound interest, not simple. That's what CAGR does. Instead of calculating an average simple interest percentage, you figure the percentage that, when treated as compound interest over the time period, gets you the final figure. It's going to be lower than the average simple interest because you get the advantage of compounding. Using a CAGR calculator, plug in $110 as the initial value, $153 as the final, and set 7 as the number of periods to get a CAGR of 4.83 percent.