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.

Author: Thomas Skidmore

When it comes to the Digital SAT Math section, the most important number to remember isn’t 3.14, 180, or even 1600 – it’s 95. The section is split into two 35-minute modules consisting of 22 questions each, resulting in an average of roughly 95 seconds per question. A massive time-saver comes in the form of Desmos, the graphing calculator built directly into the Digital SAT’s testing app, Bluebook. This tool can be used on every question in the math section, but its limitations force you to decide whether it is the most efficient use of your already-limited time. Let’s dive deeper into when you should use Desmos, when you should do a problem by hand, and when a problem might require a combination of methods.

The DESMOS “Cheat Code”

The question in this first example often leaves students staring blankly at their screen. Thankfully, DESMOS can make this question quick and easy.

Plugging the inequality directly into DESMOS provides us with the graph that we can now reference while reviewing the answer choices. The unshaded areas of the graph occur in, mostly, the top left corner, which is also known as Quadrant 2. In this quadrant, the y-values are positive (y > 0) and the x-values are negative (x < 0). This gives us a correct answer of D. To solve this by hand, we would need to plug in values for x and y that match each answer choice, eventually finding out that when x < 0 and y > 0, the resulting inequality is untrue. This would not only take much longer, it would also widen the opportunities to make an algebraic mistake.

Let’s look at another example where DESMOS can help us save time and effort.

In this example, we have a system of equations, which is one of the best opportunities to use DESMOS. What happens when we plug the equations in, directly as shown?

You may notice that the second equation gives a small warning sign. This is because we have not defined the variable, t, for DESMOS to recognize. Note that it prompts us to add a slider for t. If we click that button, this will enter into the third line.

There are a couple notable things to see here. First is that it sets t equal to 1, but we can click on and slide the small circle to change the value of t. When we do this, we can see what happens to the graph with each change. Another way to do this is by clicking the play button on the left side. This will automatically slide the slider back and forth for you while you can watch the resulting graph. Doing it this way will allow you to see what effect the variable has on the graph. Here’s the graph when t = 1.

Looking back at the question, we are being asked for the value of t when the system has no solution. When a system of equations has no solutions, it means the lines are parallel and thus never touch. If we move the slider from t=1 to t=8, we notice that they become parallel lines. Our answer is t = 8.

A final way we will look at DESMOS is through regression.Often known as the “Tilde” Method, because we use this symbol, `, usually found in the top left of your keyboard, this method helps us to find multiple variables at a time.

In DESMOS, we will input the question like so:

Notice that we changed x to x1 in every case it showed up, we changed the equal sign to the tilde as mentioned earlier, and we input a second line to tell DESMOS what values to consider for x1. These values do not matter – what does matter is that there is one value for each constant variable. In the question, the variables that represent constants are a and b, so we need two numbers in the list for x1. The final answer here would be 24, because we found a = 4, b = 6, and therefore ab = 4*6 = 24.

When to Solve by Hand

As we saw, DESMOS can be extremely helpful in the right circumstances. Occasionally, the question might be better answered by hand. Let’s look at some examples where you might want to skip out on the calculator and work through some simple algebra on your own.

This is a prime example of when DESMOS cannot help you. Geometry is a topic that appears in many ways on the DSAT and, most of the time, you will want to answer the questions by hand. Here, we are given two parallel lines cut by a transversal and we’re being asked to compare the values of x and y. When given this situation, we can note that all of the acute angles are equal to each other and all of the obtuse angles are equal to each other. Thus, x is an obtuse angle, while y is an acute angle. Because of various geometry rules, these two add up to 180, so if y is less than 65, x must be greater than 115, in order to add up to 180. Our correct answer is B.

Another example of a Geometry question where DESMOS is unable to assist you:

We are unable to tell DESMOS the circumference and how to find the diameter, so we must do this by hand. The equation for circumference is C = ℼd. If the given circumference is 31ℼ, then 31ℼ = ℼd, which gives us d = 31. Our diameter is 31.

Here is another example.

In this example, we may be tempted to use DESMOS because of all of the variables. However, our best bet is to simply distribute and add the exponents using exponent rules. Distributing the first expression to the first term of the second expression gets us this:

This eliminates answer choices A and B. Let’s distribute to the final term:

Our correct answer is D.

Translator vs Solver

The final example we want to take a look at involves setting up the problem first by hand and then using DESMOS to compute the answer. DESMOS struggles to translate problems, but is very good at solving them. If we are able to translate DSAT problems using our own algebra skills, we can then save time by plugging equations into DESMOS.

Here we have a word problem but we are not given the equations. Let’s create them first. In word problems like these, we have one equation to represent the total number of things (in this case, admission) and we have another question to represent the total cost. Our first equation, then, will be the total number of people going on the field trip, 83, which we know has to be equal to the number of students added to the number of chaperones. Therefore, s + c = 83. Our second equation comes from the total cost of $730 being equal to the cost of all student admissions added to the cost of all chaperone admissions. This gives us 8.50s + 12c = 730. We can solve this by hand as well, but here is where we can use the hybrid method to save a little time. Note that we have to change the variables to x and y in order to plug into DESMOS here, so make sure you keep track of which variable represents the students.

Our intersection is at (76, 7), which means our x value is 76 and our y value is 7. We are being asked for the number of students, which we decided was represented by x (this can be seen in the equation on DESMOS where we changed 8.50s to 8.50x). Our final answer is 76.

There are many cases where DESMOS will be your best friend on the DSAT. Here are some popular ones:

1. Solving Equations with One Variable
2. Solving Systems of Equations/Inequalities
3. Finding the Equation of a Line
4. Finding X and Y-Intercepts
5. Finding Mean, Median
6. Equivalent Expressions

Some instances require you to use your algebra or geometry skills without the assistance of a calculator. Certain questions will require you to use a scientific calculator, which can also be through DESMOS or your own handheld calculator. Occasionally, there will be a question that requires some combination of these methods. It is up to you to decide how to spend your time on test day. In the beginning, we discussed the 95-second rule. If you think DESMOS will be quicker than the 95 seconds you are given per question, then you should use it. If you think you can use mental math or some scratch work to solve the problem faster, that should be your method of choice.

If you need help deciding how or when to use DESMOS, or you would like assistance preparing for an upcoming DSAT, please contact us.