Some people love numbers, working with them, playing with them, and thinking about them. Others do not. Many don’t even have a basic understanding of how to use numbers in their work. Here are the basics…
While serving in Iraq, an officer colleague of mine was called upon to estimate the exposure from a radiation source that our soldiers found on a rooftop in Baghdad. He did the calculations and gave them to me to check. This officer was industrious, dedicated, and smart, but he had made a decimal place error and overestimated the exposure by a factor of 1,000. My colleague hadn’t made such calculations for years, and his mistake could have happened to anyone. But had this estimate gone to the commanding general, he would have had to evacuate the area and send many troops back home for medical monitoring.
Working in business and medicine for many years, I have found that many people lack the mathematics skills required for many tasks. Most learned these skills years before but had forgotten them for lack of use. All too often, the same thing happens to me. I have written this article to remind myself, my staff, and my colleagues about math skills long forgotten. I owe a debt to Merriam Webster’s guide to Everyday Math (Brian Burrell, 1998) for this work.
Numbers and Number Systems
Digits – 0,1,2,3,4,5,6,7,8,9 – are symbols in the Indian-Arabic system used to represent numbers. A number is an expression of quantity and order.
There are many different types of numbers and they are divided in systems. Counting numbers ( 1,2,3,4,5,…) are the simplest. Whole numbers include the counting numbers and zero (0,1,2,3,4,5,…). Teachers communicate both counting and whole numbers to young children using living experience; Sally can have zero apples and Ben can have two apples. Fractions are a quotient of two counting numbers, such as ½, 2/3 or even 13/64. They are called rational numbers because they form a ratio of integers (i.e. 7/11). The top number is the numerator and the bottom number the denominator. They are also relatively easy to conceptualize by experience; Sally has ½ of an apple and Ben has ¾ of an apple. In simple fractions, the numerator is less than the denominator (7/8). In improper fractions, the numerator is greater (8/7). Improper fractions reduce to mixed fractions, in which a whole number accompanies a simple fraction (1 1/7).
Counting numbers are not only positive, they can also be negative (-1,-2,-3,-4,-5,…). Integers include positive counting numbers, negative counting numbers, and zero (…,-4,-3,-2,-1,0,1,2,3,4,…) and are frequently plotted on a number line. Fractions form ratios and are therefore considered rational numbers, but some numbers cannot form ratios and are therefore irrational numbers (i.e. π, √2, ℮). They are more difficult to communicate in everyday life.
Decimals are a string of digits that occupy place values. The number 1,234 means one thousand plus two hundreds plus three tens plus four ones, while the number 0.141 means zero ones, one tenth, four hundredths and one thousandth. The number 5,834,689.71205 can be understood as follows:
5 – millions
8 – hundred thousands
3 – ten thousands
4 – thousands
6 – hundreds
8 – tens
9 – ones
7 – tenths
1 – hundredth
2 – thousandths
0 – ten thousandths
5 – hundred thousandths
We can only understand and use numbers if we understand that location matters. We would rather have $1,000.0 than $0.0001.
Percent and Percentages
It is often important to characterize a part out of a whole, for example, the number of girls in a class (50) compared to the total number of people in a class (100). A percent describes a proportion of a whole divided into 100 equal parts. In this case, women comprise 50/100 or 50 percent (50%) of the class. Written as a fraction and decimal, women comprise ½ and 0.5 of the class. The amount to which a percent is applied is known as the base (100 in this case) and the number derived by dividing the percent (50 in this case) by the base (100) is the percentage (50/100 = 0.5 x 100 = 50%). To convert a decimal to a percent, multiply the decimal by 100. Thus 0.11 is 11% and 2.2 is 220%. Sometimes percent and percentage are used interchangeably, but to clarify definitions:
- Percent – A ratio expressed as a number followed by a percent (%) sign. The word percent means “out of one hundred.”
- Percentage – A percent of something.
- Rate – Stated or implied percent
- Base – The quantity that one taking a percentage of
- Basic formula – rate x base = percentage
If you wish to discover what percent one number is of another, you must divide the percentage by the base. If a manager makes $50,000 per year (the base) and he gets a raise of $5,000 (the percentage of the original salary), the percent raise is 5,000/50,000 = 0.1×100 = 10%.
- Solve for percentage: What is 70% of 250? Percentage = rate x base = 0.7 x 250 = 175
- Solve for rate: What percent of 64 is 8? Rate = percentage/base = 8/64 = 0.125 or 12.5%
- Solve for base: 27% of what number is 304? Base = percentage/rate = 304/0.27 = 1,125.9
Percentage and Percentages – Practical Applications
Points and Basis Points – For home mortgages, a point is prepaid interest of 1% of the amount borrowed. On a loan of $100,000, one point is $1,000 and two points is $2,000. When referring to stocks and bonds, a basis point is 1/100 of 1% of the interest rate of earnings (yield).
Majority and Plurality – These are usually used in a political context. When a population is divided into two groups, there are two outcomes: one group can be larger than the other or both groups can be the same size. The larger group (> 50.1% of the total) is the majority. When a population is divided into three or more groups, it is possible that no group will get a majority. In that case and in political usage, the largest group wins, even if they only get 34% of the vote.
Tipping – A standard tip for good service at a restaurant is 15% of the total bill. If a bill is $20.00, the tip would be 20×0.15 = $3.00. The total bill would be $23.00.
Sales Tax – The tax on sales is commonly 5%. For a bill of $50.00, the tax would be $2.50 ($50 x 0.05 = $2.50). The total charge would be $52.50.
Discounts – A discount is an amount, usually a percentage, taken off of a base price. If a sale item is priced at $100 and is marked as 20% off, the new price is $80 ($100 x 0.2 = $20, $100 – $20 = $80). You can use the same formula to calculate that an $80 item reduced by 25% will cost $60. Another way to find the answer is to note that 100%-25% = 75%. Apply this fact to the second example and the equation is as follows: $80 x 0.75 = $60.
Markups – A markup is an amount, usually a percentage, added to a base price. If an item costs $75 and is marked up by 8%, the markup is $75 x 0.08 = $6.00. The new price will be $81. Customers often find markups in big ticket items such as automobiles. A $30,000 car that a dealer has marked up 6% will cost $31,800 ($30,000 X 0.06 + $30,000).
Inflation – Governments track inflation in their economies to guide monetary policy and a host of other decisions. An inflation rate of 2% (the target inflation rate for most developed countries) means that if an item cost $200 in year one, that item would cost $204 in year two ($200 x 0.04 + $200 = $204). We can also discover the change in inflation over a long time. Imagine that in Economy A the Consumer Price Index, a measure of how much a certain set of commonly purchased items costs, was 50 in year one and increased to 93 in year ten. The average inflation over those ten years was 93-50 = 43. Forty-three (43) divided by the number of years gives an annual inflation rate of 4.3%. To calculate the percent increase in that decade, 43/50 = 0.86 x 100 = 86%.
Constant Dollars – Price inflation causes a dollar tomorrow to be worth less than a dollar today, and price deflation causes the opposite. Suppose an item cost $10.00 in year one and the inflation rate was 2%. That item would cost $10.20 ($10 x 0.02 + $10) in year two, and so on as illustrated in the following table:
|Inflation||Price (2% inflation)||Price (2% deflation)|
|Year 1||$ 10.00||$ 10.00|
|Year 2||$ 10.20||$ 9.80|
|Year 3||$ 10.40||$ 9.60|
|Year 4||$ 10.61||$ 9.41|
|Year 5||$ 10.82||$ 9.22|
|Year 6||$ 11.04||$ 9.04|
|Year 7||$ 11.26||$ 8.86|
|Year 8||$ 11.49||$ 8.68|
|Year 9||$ 11.72||$ 8.51|
|Year 10||$ 11.95||$ 8.34|
Deflation is also illustrated above. In an economy with inflation, people spend more money because they expect their dollars to be worth less in the future. To maintain a standard of living in an inflationary economy, wages must increase at the same rate as prices. In an economy with deflation, such as during the Great Depression, people save more money because they expect their dollars to be worth more in the future. Both can be disastrous. Too much inflation causes prices to skyrocket (consider the Weimar Republic in Germany) and too much deflation causes people to hoard their money, which further decreases economic activity. Both can cause economic collapse.
After-Tax Earnings – From before the day that Jesus said “Render unto Caesar what is Caesar’s”, taxes have been an inevitable part of life. Investors in stocks, bonds, annuities and other financial products often get a fixed rate of return as a percentage of their investment. The equation to calculate the rate of return after taxes is I x (1 – t) where I is the annual yield and t is the tax rate. Suppose an investor in a 20% tax bracket purchases a corporate bond with a yield of 4%. The after-tax rate of return will be 0.04 (1 – 0.02) = 0.04 (0.8) = 0.032 x 100 = 3.2%.