Page 4 - Math: Problem Solving and Data Analysis Study Guide for the SAT Exam

More Data Concepts

Determine Simple and Compound Interest

Interest is the amount paid or earned for the use of money, such as in money borrowed or invested. Here’s an example: Leila borrows \(\$10,000\) from a bank that charges at interest rate of \(10\%\) per year.

Here are the terms that we need to understand:

Principal—the amount borrowed or invested. In the example, the principal is \(\$10 \text{,} 000\).

Interest rate—the ratio of the amount paid for the use of money to the principal; it is customary to present this as percentage per year. In the example, the interest rate (also often called the annual interest rate) is \(10\%\).

Time—the time or duration that the principal amount is borrowed or invested.

Interest—the amount paid for borrowing money or earned for investing money. Interest can be computed as simple interest or compound interest.

Simple interest—the amount paid for the use of money based on an annual interest rate. The formula for simple interest is:

\[\text{Interest} = \text{principal} \cdot \text{rate} \cdot \text{time}\]

where rate is the annual interest rate expressed in decimal form, and time is expressed in some measurement of time (normally years).

In the example for a loan duration of one year, the interest charged on the loan is:

\[\text{Interest} = 10\text{,}000 \cdot 0.10 \cdot 1 = $1\text{,}000.00\]

Take note that, if a loan’s duration is \(2\) years, the interest charged would be \(\$2\text{,}000\).

Questions involving compound interest basically entail these steps:

1) Solve for the interest in the first period.

2) Add the computed interest in the first period to the principal.

3) Use this total to solve for the interest in the second period.

4) Add the computed value to the previous total.

5) Repeat the procedure as many times as the number of years desired (such as \(5\) times for the interest in \(5\) years, compounded annually).

6) Add all interest for the different periods to get the total interest of the amount borrowed compounded annually.

Using the same example as above, the interest for the first year will be the same at \(\$1\text{,}000\).

For the second year, it will be:

\[\text{Interest} = (10\text{,}000 + 1\text{,}000) \cdot 0.10 \cdot (1) = 1,100\]

Adding all interests for \(2\) years, the total interest charged would be \(\$2\text{,}100\). Take note that this is a bigger amount than the simple interest on the same principle for the same period of time.

However, when the period of time (loan duration or term or investment term) involves bigger numbers, it is more practical to use this formula:

\[A = P \cdot (1 + r)^t\]

where \(A\) is the future value to be paid, \(P\) is the principal, \(r\) is the annual interest rate in decimal form, and \(t\) is time or period in years.

When a question states ”\(10\%\) monthly compounding”, it does not mean \(10\%\) per month. Rather, it means, a monthly interest of \(0.83\%\) (\(10\%\) divided by \(12\)), compounded monthly.

The formula varies a bit:

\[A = P \cdot (1 + r/n)^{nt}\]

where \(n\) is the number of times the interest compounds each year.

In the given example of monthly compounding interest and using the same principle (\(\$10\text{,}000\)), the interest for \(2\) years is \(\$2\text{,}107.00\).

\[A = 10\text{,}000 \cdot (1 + 0.10/12)^{12 \cdot 2} = 12\text{,}107\] \[\text{Interest} = 12\text{,}107 – 10\text{,}000 = 2\text{,}107\]

Apply Data Concepts to Real-Life Experiences

Data analysis is more applicable to real life than many people think. When an ad for a pet lice shampoo says that “This shampoo is recommended by \(9\) out of \(10\) vets”, what does it mean? Does it mean that \(90\%\) of all vets recommend that particular shampoo for their pets? Or is that a valid claim at all? How was that statement arrived at?

A public opinion survey says that \(65\%\) of people support a tax increase. What was the age group surveyed? How representative was it of the whole population?

In a study that says that the leading cause of low productivity in a workplace is absenteeism, see first how the data was collected. It is highly likely that this was done using an observational study. This method of data collection is very useful for understanding how one variable affects another. Because the method lacks the feature of randomness, however, it will be inappropriate to generalize the results to the larger population.

In real life, you get a better grasp of data and how they are being applied by taking them with a grain of salt and asking how those data were produced, collected, presented, and analyzed.

Determine Probability of Simple and Compound Events

Probability is the likelihood of an event occurring. The value is given as a fraction or a decimal number, and it is always less than \(1\). The closer a probability is to \(1\), the higher its probability of occurring. It is computed with this formula:

\[\text{Probability} = \frac{\text{number of desired outcomes}}{\text{total number of possible outcomes}}\]

To determine the probability of randomly picking a green shirt from a hamper that contains \(2\) green shirts, \(3\) red shirts, \(5\) yellow shirts, and \(5\) blue shirts, we apply the formula and calculate probability:

\[P (\text{green shirt}) = \frac{2}{15}\]

There is a \(2/15\) or \(0.133\) chance of picking up a green shirt.

To determine the probability of two or more events, we multiply their individual probabilities. The probability of randomly picking a green shirt and then a red shirt from the same example is calculated as follows:

\[\text{Probability} = \frac{2}{15} \cdot \frac{3}{14} = \frac{6}{210} = \frac{1}{35}\]

Note that when calculating the total number of shirts in the hamper for determining the probability of choosing a red shirt, the total number is \(14\) because a green shirt was already selected and removed from the hamper.

Extra Tip: Know When to Use the Calculator

Even though all of the questions of this type will be located in the “calculator allowed” portion of the SAT Math section, using it for every single calculation may be a waste of time.

While you’re preparing for the SAT, it is best to do as many practice questions without the calculator as with it. This means you have to:

  • know the multiplication tables** at least from \(1\) through \(10\). Knowing them beyond \(10\) can be even more helpful.

  • know the most basic squares and square roots.

  • know the decimal equivalents of common fractions. It would be great if you instantly recognize \(0.25\) as \(\frac{1}{4}\) and \(0.5\) as \(\frac{1}{2}\). They make multiplying and dividing much simpler.

With ample practice on questions of the data analysis type, you would recognize on sight questions that are better off solved without a calculator. Here are some examples:

1) Remember unit conversion? It’s also referred to as dimensional analysis or factor-label analysis. The canceling technique in that method is a lot of fun, not only in the cancellation of units but also in canceling out some numbers.

Say, you need to convert \(500\text{,}000\) inches to kilometers:

\[500\text{,}000\;\text{in} \cdot \frac{2.54\;\text{cm}}{1\;\text{in}} \cdot \frac {1\;\text{m}}{100\;\text{cm}} \cdot \frac{1\;\text{km}}{1\text{,}000\;\text{m}} = \__ \; km\]

Cancel all alike units and numbers that occur both as the numerator and the denominator. This includes the zeros in the value for inches, cm, and m. After canceling out terms and numbers, the whole thing is actually reduced to \(5 \cdot 2.54 \;km\) without using the calculator. If you need to turn to the calculator at this point, it would take you less time, and the chance of erroneously picking the wrong key is minimized.

What about this one?

\[A = 0.20 \pi (5)^2\]

If you have developed the knack of looking at \(0.20\) and seeing it the same as \(\frac {1}{5}\) and \(5^2\) as \(5 \cdot 5\), then the equation becomes:

\[A = \frac{1}{5} \pi 5 \cdot 5\]

The \(5\) in \(\frac {1}{5}\) and one of the \(5\)s in \(5 \cdot 5\) cancel out, leaving you with:

\[A = 5 \pi\]

You may need to use the calculator to multiply pi and \(5\), but again, maybe not. Check the answer choices first. If they’re given in terms of pi, then you’ve arrived at the answer. If not, then maybe it’s time to get the calculator. But hey, check the answer choices again. Are the values spaced so far apart? Maybe, you can guestimate at this point. Pi is around \(3.1416\) or barely \(3\). Multiply \(5\) by \(3\) and the answer is \(15\). If there’s an answer around \(15\) and the other choices are, say, \(17, 20,\) and \(14,\) then feel safe to save a few seconds and tick off the answer that’s around \(15+\).

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