Millions of people suffer from an alarmingly common disease: mathophobia (note: I am not a doctor and this may not be a real disease). Infection usually begins around middle school, although some cases begin earlier. The disease makes people feel like they can’t do math, and makes them afraid to even try to calculate a tip, to calculate a growth rate, and other mathematical quandaries.
Fortunately it has been shown that there is now a cure. Taking math concepts one at a time and applying real-life situations produces reasonable estimates that you can use to overcome your fears! Today we’re going to look at three concepts, all about percentages. We’ll see three real-world examples and learn some facts about how percentages work. Next time it’s your turn to figure out the tip, you can guarantee you aren’t parting with your entire money–nor being a rude undertipper.
Percentage Changes
One common issue that comes up in many business contexts–sales, profits, employment, even GDP–is that of the percentage change. What is the growth rate in sales? How quickly did your company shed jobs in the last recession? What was your bounceback from the post-crisis low? These are all questions of percentage changes, and we solve them all in the same way.
To calculate a percent change, you need two numbers: first, you need to know how large the change was (sales increased $3,000 this year) and second, you need to know the starting amount (sales last year were $50,000). Then, we simply divide the first number by the second: 3,000/50,000 or 3,000÷50,000 (÷ and / mean the same thing here!).
However, we aren’t quite done here. Whenever we do anything with a percent, we have to take the fraction that we get–in this case 3,000/50,000 or 3/50– and multiply it by 100. So we now get 100*3/50% = 300/50% = 6%. Easy as pi–er, easy as pie!
Percent of a Number, or How to Tip
One of the most common mathematical challenges most people face is calculating a tip. Because it is common, involves money, and is usually on-the-spot, calculating a tip can be quite stressful! We’ll break it down into just a couple steps so you know what you’re doing.
First, pick a percentage to tip. This isn’t a math question, so you’ll have to pick this number on your own! It varies by country, but a common tip in the US is to tip 20%. We’ll go with that.
Now, suppose that you have a $35 bill and you would like to tip 20%. What to do! Well, calculating a tip involves two steps. First, you take the starting dollar amount (here, $35) and multiply by your percent (here, 20%). So now you are left with 35*20 = 35*2*10 = 70*10 = 700. Second, you have to divide by 100. In our example, 700/100 = $7.
Why did we divide by 100 in this step but multiply by 100 in the first problem? In the first problem, we were turning a number (3,000/50,000) into a percent (6%). In this problem, we were turning a percent (20%) into a number (7). It makes sense that we’d want to multiply in one problem and divide in the other.
Percent Problems
This problem is based on a true story! Suppose you make home-made candles and sell them to a shop. The shop agrees to sell them for $10, and they say they’ll do a 25% markup and give you the rest. They sell 1 candle the first day and cut you a check for $7.50. Did you just get ripped off?
Well, let’s calculate the percents! If the store sells for $10 and gives you $7.50, they’re keeping $2.50 for themselves.The change in amount is $2.50 and the starting amount they received was $10, and so using the first problem above, we can calculate 2.50/10.00 = .25, and multiply by 100 to get 100*.25 = 25%. So they aren’t ripping you off!
Let’s try again with problem two’s method. If they pay you $7.50, how big would a 25% markup be? Following problem two, to calculate 25% of $7.50 we need to find 25*7.5. Well, 25*7.5 = 187.5 (note: use a calculator, that’s what smart phones are for!), and the next step to divide by 100 gives $1.875. So if they are marking up $7.50 by 25%, then they should only charge $7.50+$1.875 = $9.375. But they are charging $10. So they are ripping you off!
Which calculation is correct? Well, as we’ve seen, the percent depends on your starting value: if you start from 10, 25% is 2.5 but if you start from 7.5, 25% is much less! Because the store said they would use a 25% “markup”, they have to be starting from the $7.50 number, so yes: the store is ripping you off!
So now, whether you’re calculating a tip, a markup, or a growth rate, you’ve got the mathematical skills to do it with confidence and get the numbers right!
Bonus question: how much should the shop be paying you?
Bonus answer (select this line of text to see it): $8 per candle!
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I'm more confused now than before I started to read your examples
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