Author: Thomas Skidmore

Welcome to the first installment of our new blog series on test-prep math in the real world! Throughout this series, we’ll explore how the math skills tested on standardized exams are also practical tools we use in everyday life. To kick off this series, we’ll take a closer look at percentages, a common topic on exams like the TACHS, SHSAT, PSAT, and SAT. Percentage questions often appear in real-world situations involving discounts, taxes, tips, interest, grades, sports, and more. Below, we’ll explore a few of these everyday examples and see how the math skills required on test day can help us make sense of the world around us.

Discounts

A laptop is discounted 20%. The sale price is then reduced by 10%. What is the overall percent discount from the original price? Here is an example of a question we use in our SAT Student Workbook. We have a laptop that is first discounted 20%, then reduced by 10%. We are looking to find the overall percent discount from the original price. You might expect that the answer is 30%, but let’s see why that is not true. If we choose $100 as our original price, the first discount would be 20% of $100, which is $20. The sale price after that discount can be found by subtracting this from the original, so $100 – $20 = $80. This is our new sale price after the first discount. Then, the sale price is reduced by 10%. To find 10% of $80, we can simply divide 80 by 10 to get 8. Note that 10% of the original price would be 10% of $100, which is $10, not 8. Because our sale price was reduced by 10%, we are finding the percentage of a different number than our original price. Our next step would be to subtract $8 from $80, resulting in $72. Finally, we subtract the new price from the original price, to get $100 – $72 = $28, which is 28% of the original $100. Note how this compares to the 30% we get from adding 20% and 10%: close, but not perfect! These discounts and sales are all over. All of those BOGO sales that give you 50% off a second item? If we don’t know how to calculate that 50%, we may underestimate (or even overestimate!) how good the deal is for us.

Taxes and Tips

Another common time we use percentages is when we make purchases that involve taxes and tips. Let’s imagine a family of five goes out to a nice Italian dinner. They split a couple appetizers for the table, then each orders an entree. But let’s not forget about dessert! When their bill comes, they see a subtotal of $219.67, so they leave $220 on the table and walk out. Why is this wrong? They forgot the sales tax! In New York, the sales tax is 8.875%, which we will round to 9% for easier calculations. At 9%, the tax for our bill of $219.67 would be approximately $19.77. We find this by taking 9% of 219.67, which we can do by multiplying 0.09 by 219.67. Add these together and we get our new total of $239.44. Note: we can also find this by multiplying 1.09 by 219.67. This would be a one-step way to find the new total, without having to both multiply and add. Is this our final total? Still no! Our server was fantastic and deserves to be tipped well. Traditionally, 18-20% is considered a great tip to give your servers. Because we had such a fantastic experience, we will give our server 20%. To find this, we need to multiply our new total, $239.44 by 0.20, getting a tip of approximately $48. Note that there is a quick way to find this as well. If we round to $240 and take off the last number (the 0), we get 24, which is exactly half of our 20% tip. Double that amount and we get an easy method of finding tips. If we wanted to find our new total, we can use 1.20, similar to how we used 1.09 for the sales tax. However, what if we just want the tip amount because we are paying cash directly to the server? Knowing both methods of percentages will help you make the right decision for your situation.

Sports

Let’s look at a more fun example of where percentages can be found in the real world. We are in the thick of the National Football League season, where percentages are used for so many statistics. One notable NFL statistic is Completion Percentage, which is essentially how accurate a team’s quarterback is when throwing the ball. When the QB throws the ball (a Pass Attempt), it is either caught by a teammate (a Completion) or not caught by a teammate (an Incompletion). For the sake of this exercise, we are going to ignore the possibility of the QB throwing the ball to the other team (an Interception). Completion Percentage is found by dividing the number of Completions by the number of Pass Attempts. Here we can see the stat line from an actual game.

This QB threw the ball 29 times for 29 Pass Attempts. Of those, he completed the pass 23 times. If we divide 23 by 29, we get 0.793, which translates to 79.3%. This is also shown in the table as this QB’s Completion Percentage for the game. These kinds of percentages are found in just about every sport. Whether it’s basketball’s shooting percentage, hockey’s save percentage, or baseball’s batting average, we are always using percentages to help find how well some players are performing.

Voting

The last instance of percents in the real world that we will take a look at is found through voting. Whether we are voting on favorite color, best restaurant, or elections, we can always use percentages to help tell the whole story. Here is an example SAT question. A survey was conducted to predict the results of a school board election. The table shows the survey results. If 2,000 people are expected to vote in the election, by how many votes would Jordan Lee be expected to win, according to the survey?

They give us the survey results of a school board election. Candidate Jordan Lee received 750 survey votes, while Candidate Pat Taylor received 250 survey votes, giving us a total of 1,000 survey votes (this is our sample size – a sample is a small part of a total population). Using these survey results, we can predict how many people will vote for each candidate. If we expect 2,000 people to vote in the election (this is our population size), we can use the percentages found in the survey to estimate how many of these 2,000 people will vote for Jordan Lee. As mentioned, Jordan Lee received 750 of the total 1,000 votes in the survey. To find this percentage, we divide 750 by 1,000, giving us 0.75, or 75%. If the survey is an accurate representation of the total population, then we expect 75% of the 2,000 people to vote for Jordan Lee. To find this percentage, we can multiply 0.75 by 2,000, giving us 1,500 expected votes for Jordan Lee. Note that because of the “easier” numbers, we can actually solve this rather quickly another way. The survey is 1,000 people, which is exactly half of the 2,000 people we expected to vote in the election. Therefore, we can simply double (or multiply by 2) the number of votes each candidate received: 750 x 2 = 1,500. We can find Pat Taylor’s expected votes as well: 250 x 2 = 500, which we can confirm is correct because 1,500 + 500 = 2,000. Therefore, Jordan Lee would be expected to win the election by 1,000 votes (1,500 – 500 = 1,000).

Final Thoughts

In this installment, we discussed how percentages, a staple for standardized testing, are found in everyday situations. If you or someone you know is preparing to take a standardized test, please contact us at Tutoring by Christine to discuss how we could help you improve the math skills tested both in school and in the real world.