Foundations: Problem Solving and Data Analysis

Problem Solving and Data Analysis is the domain where the arithmetic is easy and the reading is not. Five to seven questions on each test ask you to turn a situation into a ratio, a percent, an average or a probability, and almost every wrong answer comes from setting the problem up backwards rather than from a calculation error: the ratio inverted, the percent taken of the wrong number, the median read from an unsorted list, a bar read against the wrong axis.

This lesson gives you one reliable setup for each of those situations. Ratios become proportions with matching units on top and bottom. Every percent question becomes part = percent x whole, with the whole identified first. Measures of center get computed from a sorted list, and from a frequency table without expanding it. Charts get read by finding the axis label before the value. Probability is favorable over total, and both counts come from the same table.

The Medium and Advanced lessons in this domain add percent change chains, scatterplots and lines of best fit, conditional probability, standard deviation and margin of error. Every one of them starts by doing what this lesson does.

A ratio compares two quantities by division. The ratio of a to b can be written a to b, a:b, or a/b, and it is unchanged when both parts are multiplied by the same number: 3:2 is the same ratio as 6:4 and 15:10. A proportion is an equation stating that two ratios are equal, and it is solved by cross-multiplying: if a/b = c/d then ad = bc. The one rule that matters is consistency: whatever quantity is on top in the first ratio must be on top in the second. When a ratio describes a mixture or a group, such as boys to girls 3:2, the parts add to a whole of 5 parts, so boys are 3/5 of the group and girls 2/5; a question that gives the total and asks for one part is solved by dividing the total into that many parts first.

A rate is a ratio of quantities with different units, such as miles per hour or dollars per pound, and a unit rate is a rate whose second quantity is one. Unit rates are found by dividing: 180 miles in 3 hours is 60 miles per hour. Converting units is a chain of multiplications by fractions equal to one, arranged so the unwanted unit cancels: 3 hours x (60 minutes / 1 hour) = 180 minutes. Writing the units and cancelling them is not decoration; it is how you know whether to multiply or divide by the conversion factor.

Percent means per hundred, so 35 percent is 35/100 = 0.35. Every percent problem is one equation, part = percent x whole, with one of the three unknown. To find a percent of a number, multiply: 20 percent of 80 is 0.20 x 80 = 16. To find what percent one number is of another, divide part by whole and multiply by 100: 16 is 16/80 = 0.20 = 20 percent of 80. To find the whole when a part and percent are known, divide: if 16 is 20 percent of a number, the number is 16/0.20 = 80. A percent increase or decrease is computed on the original amount: a price of 80 dollars marked down 25 percent loses 0.25 x 80 = 20 dollars and becomes 60 dollars. Equivalently, multiply once by (1 - 0.25) = 0.75 for a decrease or (1 + 0.25) = 1.25 for an increase; this multiplier form is what the Medium lesson builds on.

A data set of numbers is summarized by measures of center and spread. The mean is the sum divided by the count. The median is the middle value when the data are sorted, or the average of the two middle values when the count is even. The mode is the most frequent value; a set can have more than one mode or none. The range is the largest value minus the smallest. The mean is pulled toward unusually large or small values while the median is not, which is why a question that adds one very large value asks about the mean and expects you to know the median barely moves. When data are given in a frequency table, each value is counted as many times as its frequency: the mean is the sum of value x frequency divided by the total frequency, and the median is found by counting frequencies from the top until half the total is passed.

Data appear on the SAT as tables, bar charts, line graphs, dot plots and histograms. Reading them is a two-step act: identify what is being measured and in what unit from the axis titles or column headers, then read the value at the category or point in question. A two-way table cross-classifies a group by two variables, such as grade level and lunch choice; each cell is a count for one combination, the row and column totals are marginal counts, and the corner is the grand total. Most two-way table questions are read straight from a cell or a total, and the rest are a probability or a percent with the whole taken from the right row or column.

The probability of an event is the number of outcomes in the event divided by the total number of equally likely outcomes, a number from 0 to 1 that may be written as a fraction, decimal or percent. From a table, the probability that a randomly selected member has a property is the count with the property over the count of the group being selected from. The phrase "given that" or "from among those who" shrinks the group, and therefore the denominator, to one row or column. The Medium lesson makes that conditional idea explicit; at the Foundations level the skill is choosing the right two numbers.

Inverting a ratio because the question asks in the opposite order from how the ratio was given. Label both numbers with units.

Using the wrong whole for a percent, especially taking a discount percent of the sale price, or a tax percent of the after-tax total.

Subtracting a percent as if it were dollars: 30 percent off 120 dollars is 36 dollars off, not 30.

Reading the median from an unsorted list, or forgetting to average the two middle values when the count is even.

Expanding a frequency table into a long list and miscounting. Add the frequencies and count down the table.

Reading a bar's height as a percent when the axis is a count, or in units when the axis title says thousands.

Using the grand total as the denominator when the question restricts the group with "from those who" or "given that".