© 2011-19 Stu James and Management by the Numbers, Inc Author: Stu James PERCENTAGES Understanding and calculating percentages is an essential skill in business – market share, growth rates, interest rates, and many ratios are expressed as a percentage. © 2011-19 Stu James and Management by the Numbers, Inc

INTRODUCTION “Baseball is 90 percent mental & the other half is physical” Yogi Berra, Baseball Player and Coach Without context, numbers mean very little. Percentages are one important way to provide context to a value in a single calculation. Let’s illustrate this using 4 numbers: 100,000 16 800 2 Million Are these numbers significant?

INTRODUCTION 100,000 of what? The first question we might ask is: 100,000 of what? So, let’s add that and see if it helps. 100,000 Votes 16 Wins 800 Computers $2 Million of Revenues It does help some because the item in question provides certain associations, but is it sufficient? Consider the following regarding 100,000 votes. Jane received 100,000 votes for mayor of the city. 100,000 people voted for the independent presidential candidate for the country.

INTRODUCTION We can easily see how the context of size of voting population could have a large impact on how we view the number 100,000. Let’s add two important values to those statements to further enhance the meaning. Jane received 100,000 votes for mayor of the city out of 200,000 total votes cast. 100,000 people voted for the independent presidential candidate for the country out of 10 million total votes cast.

INTRODUCTION By dividing the number of votes the two candidates received by the total votes cast, this gives us an idea of each candidate’s popularity relative to the voting population. Jane won 100,000 / 200,000 votes = .50 of votes cast. The independent presidential candidate won 100,000 / 10,000,000 votes = .01 This “ratio” (votes received relative to voting population) tells us that Jane is a fairly popular mayoral candidate (or had very weak or little competition), whereas the independent candidate for president was not particularly popular.

CALCULATING PERCENTAGES To convert this ratio to percentages we multiply the ratio by 100 and add a percent sign (%) to the result. So ¼ = .25 becomes 25%. When working with percentages, the whole may be considered 100, and percentages part of that 100. Percent is actually from the Latin “per centum” meaning by the hundred. This leads us to our definition. Percent (%) = 100 * (Part / Whole) Or perhaps easier than multiplying by 100, is to realize that a decimal is equivalent to the percent (e.g. 20 / 100 =.20 = 20%) Going back to our voting examples, let’s calculate what percent of total votes each candidate won: Jane won 100 * (100,000 / 200,000) = 50% of votes cast. The independent presidential candidate won 100 *(100,000 / 10,000,000) = 1% of votes cast.

PERCENTAGES: SAMPLE PROBLEMS Question 1: There are 20 apples in total. Joe has 5 apples. Joe’s apples represent what percent of all the apples? After you’ve answered the question, click the mouse or press enter to check your work Answer: First calculate the ratio of Joe’s Apples to Total Apples Joe’s Apples / Total = 5 / 20 = .25 Then multiply by 100 and add percent sign .25 * 100 = 25% Therefore, Joe’s Apples represents 25% of the total apples

CALCULATING PERCENT GROWTH Another common use of percentages is for calculating growth rates. That is, what is the percentage change (growth rate %) compared to a previous time period? This is really a variation on the previous definition, but since growth doesn’t really qualify as a portion, let’s use slightly different terminology. Percent (%) Growth = 100 * (Change / Base) Or = 100 * (New – Base) / Base INSIGHT Notice that the underlying equation for calculating the percent is really the same for both definitions, only the context is different.

PERCENTAGES: SAMPLE PROBLEMS Question 2: In 2015, sales of the Carp-O-Matic were 800K units. In 2016, sales reached 1 million units. What was the percent growth from 2015 to 2016? After you’ve answered the question, click the mouse or press enter to check your work Answer: Growth % = 100 * (Change in sales from 2015 to 2016) / 2015 sales = 100 * (1,000,000 – 800,000) / 800,000 = 100 * (200,000 / 800,000) = 100 * .25 = 25% Growth

CALCULATING PERCENT GROWTH We can also use the percent growth rate to project future values by changing the formula around. Projected = Base + Base * Growth % or = Base * (1 + Growth %) INSIGHT Managers might make use of this calculation to answer questions such as, “if our company grows at the same rate as last year, what will our sales be?”

PERCENTAGES: SAMPLE PROBLEMS Question 3: If Carp-O-Matic maintains the same growth rate (%) for the following year, what will sales be in 2017? After you’ve answered the question, click the mouse or press enter to check your work Answer: Projected = Base + Base * Growth % = 2016 Sales + 2016 Sales * Growth % = 1,000,000 + 1,000,000 * 25% = 1,000,000 + 1,000,000 * .25 = 1,000,000 + 250,000 = 1,250,000 Units for 2017 Now let’s return to our votes example to see how growth rates might change the significance of these values.

PERCENTAGES: SAMPLE PROBLEMS Question 4: In the previous mayoral race, Jane received 150,000 votes for mayor of the city against the same challenger (now 100,000 votes). In the previous presidential race, 20,000 people voted for the same independent presidential candidate (now 100,000 votes). Calculate the growth rate of each candidate based on the previous election results. After you’ve answered the question, click the mouse or press enter to check your work Answer: Jane’s growth (%) = 100 * (100,000 – 150,000) / 150,000 = 100 * -50,000 / 150,000 = - 33.3% growth Pres. Candidate % = 100 * (100,000 – 20,000) / 20,000 = 100 * 80,000 / 20,000 = 400% growth Based on this information, it appears that Jane’s popularity is on the decline while the independent candidate is growing rapidly. Again, another change in perspective, which may be important to consider.

PERCENTAGES: SAMPLE PROBLEMS Time to revisit our three remaining values from the beginning of the tutorial, but now let’s use percentages to help us appreciate the context. To remind us of the examples: 16 Wins (for an NFL team, a 16 game season vs. for a NBA team, an 82 game season). 800 Computers for a university with 1,000 students vs. a university with 16,000 students. $2 Million of a product’s revenues in the current year if last year’s product revenues were $200,000 and total revenues were $400 million. Question 5: Calculate the appropriate percentages and then explain in your own words how the percentages changed the meaning of the numeric value. The answers are provided on the following pages. Please proceed to check your work.

Answer: 16 Wins (for an NFL team – 16 game season vs. for a NBA team – 82 game season). NFL Team (%) = 100 * (16 / 16) = 100 * 1 = 100% winning percentage NBA Team (%) = 100 * (16 / 82) = 100 * .195 = 19.5% winning percentage Based on this information, we might conclude that the NFL team was likely the best in their league as they won every game (100%) and that the NBA team was probably one of the poorest in their league (they only won 19.5% of their games), even though each team had 16 wins. The answer to the next context (computers) is provided on the following page. Please proceed to check your work.

Answer: 800 Computers for a university with 1,000 students vs. a university with 16,000 students. 1,000 Student U (%) = 100 * (800 / 1000) = 100 * .80 = 80% of the student body 16,000 Student U (%) = 100 * (800 / 16000) = 100 * .05 = 5% of the student body We can conclude that the 800 computers is more significant relative to the student population in the first university. However, we don’t really know what this represents. Does it mean that 80% and 5% of the students have computers? Perhaps it represents how many computers had to be purchased upon arrival on campus. Or maybe it refers to a brand of computer. We really don’t know. Without more information, we really can’t conclude too much. Proceed with caution! The answer to the next context (revenues) is provided on the following page. Please proceed to check your work.

Answer: $2 Million of a product’s revenues in the current year if last year’s product revenues were $200,000 and total revenues were $400 million. Growth (%) = 100 * (2,000,000 – 200,000) / 200,000) = 100 * 9.0 = 900% Growth % of Total Revenues = 100 * (2,000,000 / 400,000,000) = 100 * .005 = .5% of overall revenues This one is particularly interesting. We can see the product’s revenues grew considerably over the previous year’s (900% or 10x last year). On the other hand, that product’s overall contribution to revenues is only .5% (that is, the product represents .5% of the company’s total revenues or sales). So certainly, due to the year over year growth, this could be an exciting new product, but right now, not very significant.

PERCENTAGES: SAMPLE PROBLEMS Question 6 (Follow-up on Revenue Context): If the product grew by the same amount next year (900% growth on $2 million) and total revenues for the company remained constant at $400 million, what would be the product’s revenues next year and what % of total revenues would that represent? After you’ve answered the question, click the mouse or press enter to check your work Answer: Next Year’s Revenues = 2,000,000 * (1 + 900%) = 2,000,000 * (1 + 9.00) = 2,000,000 * 10 = 20,000,000 of Product Revenues % of Total Revenues = 100 * (20,000,000 / 400,000,000) = 100 * .05 = 5% of Overall Company Revenues By projecting next year’s revenues, we can see how quickly this product could become an important part of the company’s overall revenues. The question is, then, whether that growth rate will continue into the future!

PERCENTAGES - FURTHER REFERENCE MBTN Modules on Growth Rates, Market Share Metrics I, and Financial Metrics I all contain many additional examples and problem sets that extensively use percentage calculations.