SAT Math Problem Solving and Data Analysis
Ultra exam-oriented. Master proportional reasoning and data inference. Expect multi-step setups and interpretation under time pressure.
Problem Solving & Data Analysis — SAT Topic Page
Ultra exam-oriented. Master proportional reasoning and data inference. Expect multi-step setups and interpretation under time pressure.
Problem Solving & Data Analysis (Mastery Level)
| Domain | Elite Formula / Cue | Advanced Insight (Hover for Balloon) |
|---|---|---|
| Growth & Percent | \( \text{Final} = \text{Initial} \times (1 \pm r)^t \) | Compound Multipliers SAT Trick: Never add percents. Two 10% increases = \( 1.1 \times 1.1 = 1.21 \) (a 21% increase). For “no change,” multipliers must be reciprocals. |
| Complex Rates | \( \frac{1}{T_{total}} = \frac{1}{t_1} + \frac{1}{t_2} + \dots \) | The Harmonic Mean MIT Insight: For “Work” or “Average Speed” over same distance, use rates (1/time). Avg Speed = \( \frac{2v_1v_2}{v_1+v_2} \). |
| Data Spread | \( \sigma \approx \frac{\text{Range}}{4} \) (Rule of Thumb) | Standard Deviation SAT won’t make you calculate \( \sigma \). Just know: More “spread out” from the mean = Higher \( \sigma \). Adding a constant doesn’t change \( \sigma \). |
| Conditional Prob. | \( P(A|B) = \frac{n(A \cap B)}{n(B)} \) | Restricted Domain “Given that…” means the denominator is NO LONGER the grand total. It is the total of the specific row or column mentioned. |
| Sample Inference | \( \text{Margin of Error} \propto \frac{1}{\sqrt{n}} \) | Confidence Intervals To decrease error, increase sample size \(n\). Results only generalize to the population the sample was randomly drawn from. |
| Regression | \( \hat{y} = mx + b \) | Residual Analysis Residual = Observed – Predicted. If the residual plot has a pattern (like a curve), a linear model is not appropriate. |
Advanced Worked Examples (15) — (Data & Problem Solving)
Example 1 (Hard)
A stock’s price decreases by 20% in January, then increases by 30% in February. By what total percentage did the price change from the start of January to the end of February?
Show detailed solution
1. **Avoid the Trap:** Do not just add the percentages (-20 + 30 = 10% is wrong).
2. **Method 1 (Multipliers):** A 20% decrease is a multiplier of \(0.80\). A 30% increase is a multiplier of \(1.30\).
3. **Compound:** \(0.80 \times 1.30 = 1.04\).
4. **Convert back:** \(1.04\) represents a **4% increase** from the original price.
Answer: 4% increase
Example 2 (Ivy)
In a group of 400 people, the ratio of adults to children is 3:5. If 20% of the adults and 10% of the children wear glasses, how many people in total wear glasses?
Show detailed solution
1. **Find total parts:** \(3 + 5 = 8\) parts.
2. **Value per part:** \(400 / 8 = 50\).
3. **Count Adults & Children:** Adults = \(3 \times 50 = 150\); Children = \(5 \times 50 = 250\).
4. **Calculate Glasses:** Adults with glasses = \(0.20 \times 150 = 30\). Children with glasses = \(0.10 \times 250 = 25\).
5. **Sum:** \(30 + 25 = 55\).
Answer: 55
Example 3 (MIT)
How many liters of a 10% saline solution must be added to 5 liters of a 25% saline solution to create a 15% saline solution?
Show detailed solution
1. **Define Variable:** Let \(x\) be the liters of 10% solution added.
2. **Salt Balance Equation:** \((0.10)x + (0.25)5 = (0.15)(x + 5)\).
3. **Expand:** \(0.1x + 1.25 = 0.15x + 0.75\).
4. **Isolate x:** \(1.25 – 0.75 = 0.15x – 0.1x \Rightarrow 0.50 = 0.05x\).
5. **Solve:** \(x = 0.50 / 0.05 = 10\).
Answer: 10 Liters
Example 4 (Hard)
A printer produces 50 pages every 2 minutes. How many hours will it take to print 4,500 pages?
Show detailed solution
1. **Find Rate:** \(50 \text{ pages} / 2 \text{ minutes} = 25 \text{ pages per minute}\).
2. **Find Total Minutes:** \(4,500 \text{ pages} / 25 \text{ ppm} = 180 \text{ minutes}\).
3. **Convert to Hours:** \(180 \text{ minutes} / 60 = 3 \text{ hours}\).
Answer: 3 hours
Example 5 (Ivy)
A car travels at 60 miles per hour. What is its speed in feet per second? (1 mile = 5,280 feet)
Show detailed solution
1. **Set up conversion chain:** \( \frac{60 \text{ miles}}{1 \text{ hour}} \times \frac{5,280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3,600 \text{ seconds}} \).
2. **Multiply across:** \( 60 \times 5,280 = 316,800 \text{ feet per hour} \).
3. **Divide by seconds:** \( 316,800 / 3,600 = 88 \text{ feet per second} \).
Answer: 88 ft/s
Example 6 (Hard)
The mean of 5 numbers is 20. If a 6th number is added, the new mean becomes 22. What is the 6th number?
Show detailed solution
1. **Original Total:** \(5 \times 20 = 100\).
2. **New Total:** \(6 \times 22 = 132\).
3. **Difference:** \(132 – 100 = 32\).
Answer: 32
Example 7 (Ivy)
A survey with a margin of error of 3% reports that 45% of voters support a certain candidate. If the total population is 10,000, what is the range of the predicted number of supporters?
Show detailed solution
1. **Find Percentage Range:** \(45\% \pm 3\% = [42\%, 48\%]\).
2. **Convert to raw numbers:** \(0.42 \times 10,000 = 4,200\) and \(0.48 \times 10,000 = 4,800\).
3. **Result:** Between 4,200 and 4,800 supporters.
Answer: 4,200 to 4,800
Example 8 (MIT)
A researcher finds a correlation coefficient of \(r = -0.92\) between “time spent gaming” and “hours of sleep.” Which of the following is true?
Show detailed solution
1. **Interpret Magnitude:** \(0.92\) is close to 1, indicating a **strong** linear relationship.
2. **Interpret Sign:** The negative sign indicates an **inverse** relationship (as gaming increases, sleep decreases).
3. **Causation Warning:** Even with a high \(r\), we **cannot** conclude that gaming *causes* less sleep without further experimentation.
Answer: Strong negative linear association.
Example 9 (Hard)
In a box of 50 pens, 20 are blue, 15 are red, and the rest are black. What is the probability of picking a pen that is NOT red?
Show detailed solution
1. **Find NOT red count:** \(50 \text{ total} – 15 \text{ red} = 35\).
2. **Probability:** \(35 / 50 = 0.7\) or 70%.
Answer: 7/10 or 0.7
Example 10 (MIT)
In a two-way table, 80 students were asked if they like Math or English. 30 like Math, 40 like English, and 10 like both. Given that a student likes Math, what is the probability they also like English?
Show detailed solution
1. **Identify the Condition:** The group is limited to those who “like Math.”
2. **Denominator:** Total Math lovers = 30.
3. **Numerator:** Those in the Math group who also like English (Both) = 10.
4. **Calculate:** \(10 / 30 = 1/3\).
Answer: 1/3
Example 11 (Ivy)
A bacterial colony doubles every 3 hours. If there are 100 bacteria initially, how many will there be after 12 hours?
Show detailed solution
1. **Find number of cycles:** \(12 \text{ hours} / 3 \text{ hours per cycle} = 4 \text{ cycles}\).
2. **Apply Growth:** \(100 \times 2^4 = 100 \times 16 = 1,600\).
Answer: 1,600
Example 12 (MIT)
The line of best fit for a dataset is \(y = -2.5x + 40\). If the actual value at \(x=4\) is 32, what is the residual at this point?
Show detailed solution
1. **Calculate Predicted Value (\(\hat{y}\)):** \(-2.5(4) + 40 = -10 + 40 = 30\).
2. **Identify Actual Value (\(y\)):** 32.
3. **Residual Formula:** \( \text{Actual} – \text{Predicted} = 32 – 30 = 2\).
Answer: 2
Example 13 (Hard)
A map scale is 1 inch = 50 miles. If two cities are 3.5 inches apart on the map, what is the actual distance between them?
Show detailed solution
1. **Set up proportion:** \( \frac{1 \text{ in}}{50 \text{ mi}} = \frac{3.5 \text{ in}}{x \text{ mi}} \).
2. **Solve for x:** \( x = 3.5 \times 50 = 175 \text{ miles} \).
Answer: 175 miles
Example 14 (Ivy)
In a set of 11 consecutive integers, the median is 25. What is the sum of the smallest and largest integers in the set?
Show detailed solution
1. **Median Position:** In 11 numbers, the 6th number is the median (25).
2. **Find Smallest:** 5 steps back: \(25 – 5 = 20\).
3. **Find Largest:** 5 steps forward: \(25 + 5 = 30\).
4. **Sum:** \(20 + 30 = 50\). (Note: In any symmetric set, sum = \(2 \times \text{median}\)).
Answer: 50
Example 15 (MIT)
A test has a mean score of 70 and a standard deviation of 10. If every student’s score is increased by 5 points, what are the new mean and standard deviation?
Show detailed solution
1. **New Mean:** Adding a constant increases the mean by that constant: \(70 + 5 = 75\).
2. **New Standard Deviation:** Adding a constant to every value **does not change** the spread (distance between values). The S.D. remains 10.
Answer: Mean = 75, S.D. = 10
Elite Practice: Problem Solving & Data Analysis (30 Questions)
- 1. A computer’s price is increased by 20%, then decreased by 20%, and finally increased by 10%. What is the net percentage change from the original price?
Show Detailed Solution
1. Use multipliers: \(1.20 \times 0.80 \times 1.10\).
2. Calculate step-by-step: \(0.96 \times 1.10 = 1.056\).
3. Subtract 1 to find the decimal change: \(1.056 – 1 = 0.056\).
4. Result: 5.6% increase. - 2. The ratio of red marbles to blue marbles in a jar is 4:7. If there are 132 marbles in total, how many blue marbles must be added to make the ratio 1:3?
Show Detailed Solution
1. Original parts: \(4+7=11\). Each part = \(132/11 = 12\).
2. Current counts: Red = 48, Blue = 84.
3. New Ratio (1:3): Let \(x\) be added blue marbles. \( \frac{48}{84+x} = \frac{1}{3} \).
4. Cross-multiply: \(144 = 84 + x \Rightarrow x = 60\). - 3. A factory increases its production rate by 25%. If it now produces 2,000 units in 8 hours, what was the original rate in units per hour?
Show Detailed Solution
1. Current rate: \(2,000 / 8 = 250\) units/hr.
2. Since this is 25% higher: \(1.25 \times \text{Original} = 250\).
3. Original = \(250 / 1.25 = 200\) units/hr. - 4. A map scale is 1:250,000. If two cities are 12 cm apart on the map, what is the actual distance in kilometers?
Show Detailed Solution
1. Map distance: \(12 \times 250,000 = 3,000,000\) cm.
2. Convert to meters: \(3,000,000 / 100 = 30,000\) m.
3. Convert to km: \(30,000 / 1,000 = 30\) km. - 5. If 3 workers can build 2 fences in 5 hours, how many hours will it take 5 workers to build 4 fences?
Show Detailed Solution
1. Find “Worker-Hours per fence”: \((3 \text{ workers} \times 5 \text{ hours}) / 2 \text{ fences} = 7.5\) worker-hours/fence.
2. Target: 4 fences \(\times 7.5 = 30\) worker-hours total needed.
3. Time for 5 workers: \(30 / 5 = 6\) hours.
- 6. In a data set of 20 values, the mean is 50 and the standard deviation is 0. If one value is replaced by 70, what is the new mean?
Show Detailed Solution
1. If \(SD=0\), all 20 values are exactly 50.
2. Total sum = \(20 \times 50 = 1,000\).
3. Remove one 50 and add 70: \(1,000 – 50 + 70 = 1,020\).
4. New mean = \(1,020 / 20 = 51\). - 7. A poll has a margin of error of \(\pm 4\%\). If the poll reports 48% “Yes,” which statement must be true about the population?
Show Detailed Solution
1. The range is \(44\%\) to \(52\%\).
2. Correct Logic: It is “plausible” that the majority (over 50%) supports “Yes,” but we cannot say for certain. Results only apply to the population from which the sample was randomly drawn. - 8. In a two-way table for 200 students, 60% are girls. 40% of the girls play sports, while 50% of the boys play sports. What is the probability that a randomly chosen sports-player is a girl?
Show Detailed Solution
1. Girls = 120; Boys = 80.
2. Girls in sports = \(120 \times 0.40 = 48\).
3. Boys in sports = \(80 \times 0.50 = 40\).
4. Total sports players = 88.
5. \(P(Girl | Sports) = 48 / 88 = 6 / 11 \approx 0.545\). - 9. A dataset has 101 values. If the smallest value is decreased and the largest value is increased, what happens to the median and the range?
Show Detailed Solution
1. Median: The middle (51st) value remains unchanged. No effect.
2. Range: The distance between Max and Min increases. Range increases. - 10. A residual plot for a linear model shows a clear U-shaped pattern. What does this indicate?
Show Detailed Solution
A pattern in the residuals (non-randomness) indicates that a linear model is not appropriate for the data and a non-linear model (like quadratic) would be better.
- 11. The mean of \(x, y, z\) is 12. The mean of \(x, y, z, w\) is 15. Find \(w\).
Show Detailed Solution
1. Sum(3) = \(3 \times 12 = 36\).
2. Sum(4) = \(4 \times 15 = 60\).
3. \(w = 60 – 36 = 24\). - 12. A probability of 0.35 is assigned to event A, and 0.45 to event B. If A and B are independent, what is \(P(A \text{ or } B)\)?
Show Detailed Solution
1. Independent: \(P(A \cap B) = 0.35 \times 0.45 = 0.1575\).
2. Addition Rule: \(0.35 + 0.45 – 0.1575 = 0.6425\). - 13. Boxplot A has a longer “whisker” on the right than Boxplot B. What does this suggest?
Show Detailed Solution
It suggests that the data in A is positively skewed (skewed right) or has more extreme outliers on the high end compared to B.
- 14. Sample A (n=100) and Sample B (n=1000) are taken from the same population. Which will likely have a smaller margin of error?
Show Detailed Solution
Sample B. Margin of error is inversely proportional to the square root of the sample size (\(1/\sqrt{n}\)). Larger sample = smaller error.
- 15. If the median of 5 consecutive integers is \(k\), what is the mean of these integers?
Show Detailed Solution
In any perfectly symmetric distribution (like consecutive integers), the mean is equal to the median. So, Mean = \(k\).
- 16. A car covers 100 miles at 40 mph and the next 100 miles at 60 mph. What is the average speed for the whole 200 miles?
Show Detailed Solution
1. Time 1 = \(100/40 = 2.5\) hrs.
2. Time 2 = \(100/60 = 1.66\) hrs.
3. Total Time = 4.166… hrs.
4. Avg Speed = \(200 / 4.166… = 48\) mph. (Harmonic Mean). - 17. A container has 4L of 10% juice. A second has 6L of 30% juice. If mixed, what is the final percentage?
Show Detailed Solution
1. Juice in A = 0.4L; Juice in B = 1.8L.
2. Total Juice = 2.2L. Total Vol = 10L.
3. Percent = \(2.2 / 10 = 22\%\). - 18. If \( x \) is the mean of a set, what is the value of the sum of deviations \( \sum (x_i – x) \)?
Show Detailed Solution
By definition of the mean, the sum of deviations from the mean is always 0.
- 19. A pump fills a tank in 3 hours, but a leak can empty it in 5 hours. How long to fill the tank if both are active?
Show Detailed Solution
1. Rates: Fill = \(1/3\), Leak = \(1/5\).
2. Net Rate = \(1/3 – 1/5 = 2/15\) tanks/hr.
3. Time = \(1 / (2/15) = 7.5\) hours. - 20. A sequence follows \( a_{n} = a_{n-1} \times 1.05 \). If \( a_1 = 100 \), find the smallest \(n\) such that \( a_n > 200 \).
Show Detailed Solution
1. This is \(100(1.05)^{n-1} > 200 \Rightarrow 1.05^{n-1} > 2\).
2. Using “Rule of 72”: \(72 / 5 \approx 14.4\) periods.
3. Let’s use log: \((n-1) \ln(1.05) > \ln 2 \Rightarrow (n-1) > 14.2 \Rightarrow n = 16\). - 21. A set of numbers has a range of 20 and a mean of 50. If every number is multiplied by 3 and then 10 is added, what are the new range and mean?
Show Detailed Solution
1. New Mean: \(50 \times 3 + 10 = 160\).
2. New Range: \(20 \times 3 = 60\) (Addition doesn’t affect spread). - 22. Given \( P(A|B) = 0.4 \) and \( P(B) = 0.5 \), find \( P(A \cap B) \).
Show Detailed Solution
Conditional Formula: \( P(A \cap B) = P(A|B) \times P(B) = 0.4 \times 0.5 = 0.2 \).
- 23. In a class of 30, the mean height is 160 cm. If the tallest person (190 cm) and shortest person (130 cm) leave, what is the new mean?
Show Detailed Solution
1. Old sum = \(30 \times 160 = 4,800\).
2. New sum = \(4,800 – 190 – 130 = 4,480\).
3. New mean = \(4,480 / 28 = 160\). (Mean is unchanged because the removed values were symmetric around it). - 24. A data point with a z-score of -2.5 is how many standard deviations from the mean?
Show Detailed Solution
Z-score represents the number of SDs. So, **2.5 standard deviations below** the mean.
- 25. In a normal distribution, what approximate percentage of data lies within 2 standard deviations of the mean?
Show Detailed Solution
Following the 68-95-99.7 rule, approximately 95%.
- 26. If the ratio of area of two circles is 4:9, what is the ratio of their circumferences?
Show Detailed Solution
1. Area ratio = \(r^2\) ratio = 4:9.
2. Radius ratio = \(\sqrt{4}:\sqrt{9} = 2:3\).
3. Circumference ratio = radius ratio = 2:3. - 27. A population triples every 10 years. What is the annual growth rate?
Show Detailed Solution
1. \((1+r)^{10} = 3\).
2. \(1+r = 3^{1/10} \approx 1.116\).
3. Rate \(\approx 11.6\%\). - 28. If 5 coins are flipped, what is the probability of getting exactly 3 heads?
Show Detailed Solution
1. Total outcomes = \(2^5 = 32\).
2. Success (5 Choose 3) = \(10\).
3. Prob = \(10/32 = 5/16\). - 29. A scatterplot with \(r=0\) suggests what?
Show Detailed Solution
It suggests there is no linear relationship between the two variables. It does NOT mean there is no relationship at all (could be a circle or curve).
- 30. A group of 10 people have an average age of 25. If the oldest person is 40, what is the maximum possible average age of the remaining 9 if no one is younger than 18?
Show Detailed Solution
1. Total sum = 250.
2. Sum of remaining 9 = \(250 – 40 = 210\).
3. Average = \(210 / 9 \approx 23.33\).
Next Steps
You have completed the elite practice for Data Analysis. To reach 800, focus on the “why” behind the statistics. Ready to move to Advanced Math (Functions & Quadratics)?