Mastering the SAT: Understanding the Average Rate of Change

The SAT Mathematics section assesses a range of mathematical concepts, and a solid understanding of these concepts is crucial for success. Among these concepts, the average rate of change, particularly in the context of problem-solving and data analysis, plays a significant role. This article delves into the average rate of change, explaining its meaning, calculation, and application through examples relevant to the SAT.

The SAT Math Section: An Overview

While the core mathematical content remains consistent, the SAT Math section has undergone some changes. The number of questions has been reduced to 44, to be completed in 70 minutes, and calculators are permitted for all questions. The test also employs adaptive scoring, where performance in the first module influences the difficulty of the second, with more challenging questions in the second module yielding more points.

The College Board categorizes SAT math questions into four main areas: Algebra, Advanced Math, Problem-Solving and Data Analysis (PSDA), and Geometry and Trigonometry. The PSDA category, which we will focus on, features a variety of topics, including ratios, percentages, rates, measures of center, scatterplots, probability, statistical inference, and margin of error.

Problem-Solving and Data Analysis (PSDA)

The PSDA category contains more topics than any other, and it can be subdivided into arithmetic and statistics. You can expect 5-7 questions from this category.

Ratios

A ratio is a comparison between two or more quantities. It can be expressed as a part-to-part ratio (e.g., the ratio of almonds to walnuts) or a part-to-whole ratio (e.g., the ratio of almonds to the total number of nuts).

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Example: An animal shelter has 14 cats and 49 dogs. There are no other animals in the shelter. The ratio of the number of dogs to the total number of animals in the shelter is which of the following?

Solution: The number of dogs is 49, and the total number of animals is 49 + 14 = 63. The ratio of dogs to the total number of animals is 49/63, which simplifies to 7/9. Therefore, the correct answer is D.

Percentages

The SAT includes percentage-related questions, such as "percent of," "what percent," "percent greater than," and "percent less than." Percent change is a common type of percentage question. The percent change formula is:

Percent change = (New Value - Old Value) / Old Value

Example: Jazlyn baked 496 cookies on Monday and 372 cookies on Friday. What was the percent decrease in the number of cookies she baked from Monday to Friday?

Solution: Using the percent change formula: Percent change = (372 - 496) / 496 = -124 / 496 = -0.25 = -25%. The percent decrease in the number of cookies baked is 25%, so the correct answer is B.

Rates

The basic rate formula is distance = rate x time. Average rate is calculated as:

Average Rate = Total Distance / Total Time

Example: On Tuesday morning, Carson walks 2 miles to school at a rate of 4 mph. After school, he runs home at a rate of 10 mph. What is his average rate on Tuesday?

3 3/7 mph
4 2/3 mph
5 5/7 mph
7 mph

Solution: The total distance is 2 + 2 = 4 miles. The time to get to school in the morning is 2 miles / 4 mph = 1/2 hour. The time to get home in the afternoon is 2 miles / 10 mph = 1/5 hour. The average rate is 4 miles / (1/2 hour + 1/5 hour) = 4 miles / (7/10 hour) = 40/7 mph = 5 5/7 mph. Therefore, the correct answer is C.

Measures of Center

Measures of center, including mean, median, and mode, are used to describe a data set.

Mean = sum / number
The median is the middle value when the data values are listed in order. If there is no unique median, then the median is the average of the 2 middle numbers.
The mode is the data value that occurs most often.

Example: The track coach recorded the following times for finishing a 100-meter race. Each time is rounded to the nearest second.

12 13 16 12 15 15 13 17

After the coach calculated the mean, median, and mode of the 8 data values, he realized that he had failed to include a 9th finishing time of 14 seconds in his data. He then recalculated the mean, median, and mode of the group. Which measure of center had the greatest decrease after the recalculation?

Mean
Median
Mode
None of the measures decreased after the recalculation

Solution: For the original data set: Mean = 113 / 8 = 14.125, Median = (13 + 15) / 2 = 14, Modes are 12, 13, and 15. For the amended data set: Mean = 127 / 9 = 14.111, Median = 14, Modes are 12, 13, and 15. The mean decreased from 14.125 to 14.111. Therefore, the correct answer is A.

Scatterplots

A scatterplot displays the relationship between two numeric variables. It can show a positive relationship (as one variable increases, the other increases), a negative relationship (as one variable increases, the other decreases), or no relationship.

Example: A 401(k) is a type of savings account for retirement. A sample of 20 individuals working at a large company was taken, and each person’s age and the total amount in his or her 401(k) was recorded. The scatterplot above displays the results.

Based on the scatterplot, which of the following statements is true?

There is no relationship between one’s age and the amount of money in one’s 401(k).
The greater the age, the greater the amount in one’s 401(k).
The greater the age, the less the amount in one’s 41(k).
As age increases, the amount in a 401(k) tends to remain constant.

Solution: The scatterplot shows that as age increases, the amount in the 401(k) tends to increase. Therefore, the correct answer is B.

Probability

The SAT often presents probability questions using 2-way tables.

Example: During a busy weekend, a grocery store recorded the grocery bag preference of each customer, noting the gender of each customer and whether he or she preferred paper or plastic. Shoppers who brought their own reusable bags or those who did not express a preference were not included in this survey. If one shopper is randomly selected from those surveyed, what is the probability that the shopper is a female who prefers paper bags?

0.156
0.255
0.398
0.610

Solution: From the table, identify the number of females who prefer paper bags (200) and the total number of respondents (1,286). The probability is 200 / 1,286 = 0.156. Therefore, the correct answer is A.

Statistical Inference

Statistical inference involves generalizing from sample data to an entire population. Important considerations include random sampling, sufficient sample size, and generalizing only to the population from which the sample was taken.

Example: The School Board in a particular county wanted parent input about extending the school day by 15 minutes. The project chairman randomly selected 175 of the 4,000 parents in the county who had children in school and asked them if they were in favor of the proposal or not in favor. All of the parents replied, and 140 of them were in favor of extending the school day by 15 minutes. Which of the following must be true?

The sample size was too small for an accurate estimate to be made about the opinions of all the parents with children in school in the county.
The chairman can report to the School Board that all parents with children in school are in favor of extending the school day by 15 minutes.
The chairman cannot make any conclusive report to the School Board because the parents in the sample were not unanimous in their response to the question.
The chairman can report to the School Board that it is highly likely that the majority of all parents in the county with children in school are favorable to the proposal to extend the school day by 15 minutes.

Solution: The sample size of 175 is more than 30, so it is sufficient. 140/175 are in favor. therefore, the correct answer is D.

The Average Rate of Change Explained

The average rate of change describes how an output quantity changes relative to the change in the input quantity. The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.

[ \dfrac{\Delta y}{\Delta x} = \dfrac{f ( x2 ) − f ( x1 )}{x2 − x1}.

For a linear function with slope ( m ), every increase of 1 in input results in adding ( m ) to the function value.

Understanding Slope

You may recall from Algebra that slope can be described as the ratio [\dfrac{\text {rise}}{\text {run}}.

Using more formal notation, given points (\left(x{0}, y{0}\right)) and (\left(x{1}, y{1}\right)), we use the Greek letter Delta, (\Delta), to write (\Delta y = y{1}-y{0}) and (\Delta x = x{1}-x{0}). In most scientific circles, the symbol (\Delta) means "change in." Hence, we may write[m=\dfrac{\Delta y}{\Delta x}, ]which describes the slope as the rate of change of (y) with respect to (x).

If the slope of a line is positive, then the resulting line is said to be increasing. If it is negative, we say the line is decreasing. A slope of 0 results in a horizontal line, which we say is constant.

Real-World Applications

Many real-world applications are modeled by linear equations.

The slope, 2, means that the height, h, increases by 2 inches when the shoe size, s, increases by 1.
The slope, (\frac{1}{4}), means that the temperature Fahrenheit (F) increases 1 degree when the number of chirps, n, increases by 4.
The slope, 4, means that the weekly cost, C, increases by $4 when the number of pizzas sold, p, increases by 1.

Parallel and Perpendicular Lines

Two lines that have the same slope are called parallel lines. Two lines that have the same slope and different y-intercepts are called parallel lines.

These lines lie in the same plane and intersect in right angles. If we look at the slope of the first line, (m1=\frac{1}{4}), and the slope of the second line, (m2=−4), we can see that they are negative reciprocals of each other.

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