Exam prep · SAT · Unit guide

SAT: Math: Problem-Solving and Data Analysis

What ratios, percentages, probability, and data displays actually mean

On the exam: Ratios, percentages, probability, reading tables and scatterplots. About 15%.

Think a ratio and a percent are the same thing? On the SAT, that mix-up costs real points in Problem-Solving and Data Analysis, about fifteen percent of the Math section. We'll cover ratios, percentages, probability, tables, and scatterplots, then work through a typical question step by step.

How Much of the Test This Covers

Problem-solving and data analysis makes up about fifteen percent of this section of the test, around nine or ten questions out of fifty-eight. Instead of pure algebra, these questions test whether you can read a table, compute a percent change, or figure out a probability from real-world data. That makes this topic less about formulas and more about careful reading and arithmetic.

15% share of the Math test devoted to problem-solving and data analysis

Key points.

• About 15% of this test's Math questions
• Covers ratios, percentages, probability
• Covers tables and scatterplots
• Tests reading and reasoning, not just formulas

Ratios, Rates, and Proportions

A ratio compares two quantities directly, like three cups of flour to two cups of sugar, written three to two. A rate compares a quantity to one unit, like fifty-five miles per hour. A proportion sets two ratios equal so you can solve for an unknown: if three is to two as nine is to what, you cross multiply, three times what equals two times nine, so what equals six.

Key points.

• Ratio compares two quantities, like 3 to 2
• Rate compares quantity to one unit, like mph
• Proportion sets two ratios equal
• Cross-multiply to solve for unknown

Percent Change and Percent Of

Let's write the percent change formula: new minus old, divided by old, times one hundred. Say a shirt's price goes from forty dollars to fifty dollars. Fifty minus forty is ten, divided by forty is point two five, times one hundred gives twenty-five percent increase. For 'percent of', just multiply: twenty percent of one hundred fifty is point two zero times one hundred fifty, which is thirty.

% change = (new−old)/old × 100
old=$40, new=$50
(50−40)/40×100 = 25%
part = percent × whole
20% of 150 = 0.20×150 = 30

Key points.

• Percent change = (new − old) ÷ old × 100
• Shirt example: $40 → $50 is a 25% increase
• Percent of = percent × whole
• 20% of 150 = 30

Probability Basics

On the SAT, probability almost always comes from this one idea: favorable outcomes over total outcomes. Roll a six-sided die, and the probability of landing on a four is one out of six. Often the SAT gives real data, like a bag with twenty marbles where five are red; then probability of picking red is five over twenty, one fourth, or twenty-five percent.

Probability equals the number of favorable outcomes divided by the total number of possible outcomes. SAT Math definition

Key points.

• Probability = favorable outcomes ÷ total outcomes
• Ranges from 0 (impossible) to 1 (certain)
• Can come from theory or real data tables
• Written as fraction, decimal, or percent

Two-Way Tables and Conditional Probability

Two-way tables sort data by two categories at once, like grade level and whether a student plays a sport. The edges of the table give marginal totals, the overall count for each row or column. Conditional probability means zooming into just one row or column: if fifty freshmen total, and thirty of them play sports, the probability a freshman plays a sport is thirty over fifty, or sixty percent.

Key points.

• Rows and columns track two categories at once
• Row/column totals are called marginal totals
• Conditional probability restricts to one row or column
• Divide the cell value by that total

Working Through a Typical Question, Step by Step

Here's a typical table question. A survey of two hundred students shows how many play a sport, by grade. Freshmen playing sport: thirty. Sophomores playing sport: forty. So seventy total play a sport. The question asks: probability a sport-player is a freshman? Take freshman-and-sport, thirty, divide by total sport-players, seventy, giving three sevenths, about forty-three percent.

1. Read the table: sport vs grade, find totals
2. Question asks: probability a sport-player is a freshman
3. Total who play a sport = 70
4. Freshmen who play a sport = 30
5. Divide: 30 ÷ 70 = 3/7 ≈ 0.43

Scatterplots and Lines of Best Fit

A scatterplot plots pairs of data, like hours studied against test score, one dot per student. Dots trending upward show a positive relationship; downward shows negative. A line of best fit summarizes that trend: its slope gives the rate of change, say five extra points per hour studied, and its y-intercept predicts the score at zero hours. Points far from the line are outliers.

Key points.

• Each dot represents one data pair
• Upward trend = positive relationship, downward = negative
• Slope of best-fit line = rate of change
• Points far from the line are outliers

Ratio (Part-to-Part) vs Proportion (Part-to-Whole)

A classroom with three boys and two girls gives a ratio of three to two, boys to girls, a part-to-part comparison. But if you're asked what fraction of the class is boys, that's three out of five, or sixty percent, a part-to-whole comparison. Mixing these up is one of the most common errors: always check whether the question compares two groups to each other or one group to the total.

Remember this

The one thing to remember: always check whether a question wants a ratio, a percent, or a probability, since the SAT loves testing that exact confusion. Try a few practice problems now and see how these ideas show up.

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