Study guide
Word Problems, Statistics, and Data Interpretation
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1 Introduction to the OAR
About the OAR Who Takes the OAR and Why Study Plan and Test-Day Strategy
2 Math Skills Test (MST)
Arithmetic Fundamentals Fractions, Ratios, and Proportions Percentages and Real-World Applications Algebra Essentials Geometry Foundations Word Problems, Statistics, and Data Interpretation
3 Reading Comprehension Test (RCT)
RCT Overview and Core Strategies Main Idea and Purpose Questions Detail, Fact, and Inference Questions Vocabulary in Context and Author Tone Advanced Reading Strategies and Passage Types RCT Practice Passages and Questions
4 Mechanical Comprehension Test (MCT)
Forces and Newton's Laws of Motion Energy, Work, and Power Simple Machines and Mechanical Advantage Gears, Belts, and Rotational Motion Fluids, Pressure, and Hydraulics Electricity, Circuits, and Magnetism Motion, Heat, and Thermodynamics MCT Practice Problems and Review
5 Full Practice Tests and Final Review
Practice Test: Math Skills Practice Test: Reading Comprehension Practice Test: Mechanical Comprehension Final Review and Test-Day Preparation
Word Problems, Statistics, and Data Interpretation
Key Takeaways Explain This Topic AI
Word problems on the OAR require translating English into math — identify the unknown, set up the equation, then solve.
Rate-time-distance (d = rt) and rate-time-work problems are among the most common word problem types.
Mean, median, mode, and range are basic statistics concepts that appear regularly on the OAR.
Probability questions test your ability to calculate simple and compound event likelihood.
Data interpretation questions may involve reading charts, tables, or graphs — extract only the data you need.
Last updated: March 2026
Word problems are where many OAR test-takers lose time. The math is usually not harder — the challenge is translating English into equations efficiently.
Word Problem Translation Guide
- English Phrase Mathematical Operation
- "is," "was," "will be" = (equals)
- "of" × (multiply)
- "per," "each," "every" ÷ (divide)
- "more than," "increased by" + (add)
- "less than," "decreased by" − (subtract)
- "twice," "double" × 2
- "half of" ÷ 2 or × 1/2
- "sum" + (add)
- "product" × (multiply)
- "quotient" ÷ (divide)
- "difference" − (subtract)
- Rate-Time-Distance Problems
- The Core Formula
- Distance = Rate × Time (d = rt)
Rearranged:
- Rate = Distance / Time (r = d/t)
- Time = Distance / Rate (t = d/r)
Example: A patrol boat travels at 24 knots. How far does it travel in 3.5 hours?
- d = 24 × 3.5 = 84 nautical miles
- Relative Speed Problems
- Same direction: Relative speed = difference of speeds Opposite directions: Relative speed = sum of speeds
Example: Ship A travels east at 18 knots and Ship B travels east at 12 knots. Ship B has a 30 nautical mile head start. How long until Ship A catches Ship B?
- Relative speed = 18 - 12 = 6 knots
- Time = 30/6 = 5 hours
- Round Trip Problems
Key insight: Average speed for a round trip is NOT the arithmetic mean of the two speeds.
Example: You drive 60 miles at 30 mph and return at 60 mph. What is your average speed for the whole trip?
- Time going = 60/30 = 2 hours
- Time returning = 60/60 = 1 hour
- Total distance = 120 miles, Total time = 3 hours
- Average speed = 120/3 = 40 mph (not 45 mph!)
- Work Rate Problems
- Combined Work Formula
If Person A can do a job in a hours and Person B in b hours, working together:
- Time together = (a × b) / (a + b)
- Or use rates: 1/a + 1/b = 1/t
Example: Mechanic A can service an engine in 6 hours. Mechanic B can do it in 4 hours. How long together?
- Combined rate: 1/6 + 1/4 = 2/12 + 3/12 = 5/12 per hour
- Time = 12/5 = 2.4 hours (2 hours 24 minutes)
- Mixture Problems
- Price Mixture
Example: A supply officer mixes 10 pounds of Type A coffee at $8/lb with Type B coffee at $12/lb to make a blend costing $9.50/lb. How many pounds of Type B?
Let x = pounds of Type B.
- 10(8) + 12x = 9.50(10 + x)
- 80 + 12x = 95 + 9.5x
- 2.5x = 15
- x = 6 pounds
- Concentration Mixture
Example: How much pure water must be added to 20 liters of 40% salt solution to make a 25% salt solution?
- Salt in the original solution = 0.40 × 20 = 8 liters of salt
- After adding x liters of water: 8/(20 + x) = 0.25
- 8 = 0.25(20 + x)
- 8 = 5 + 0.25x
- 3 = 0.25x
- x = 12 liters
- Age Problems
Example: Maria is 4 times as old as her son. In 8 years, she will be 2.5 times his age. How old is her son now?
Let s = son's current age.
- Maria's age = 4s
- In 8 years: 4s + 8 = 2.5(s + 8)
- 4s + 8 = 2.5s + 20
- 1.5s = 12
- s = 8 years old
- Statistics
- Mean (Average)
- Mean = Sum of all values / Number of values
- Example: Test scores: 78, 85, 92, 88, 72
- Mean = (78 + 85 + 92 + 88 + 72) / 5 = 415 / 5 = 83
- Weighted Average
When values have different weights:
- Example: Midterm (40% weight) = 80, Final (60% weight) = 90
- Weighted average = 0.40(80) + 0.60(90) = 32 + 54 = 86
- Median
The middle value when data is arranged in order. For even number of values, average the two middle values.
- Data: 3, 7, 9, 12, 15 → Median = 9 (middle value)
- Data: 4, 6, 8, 10 → Median = (6 + 8) / 2 = 7
- Mode
The value that appears most frequently. A data set can have no mode, one mode, or multiple modes.
- Range
- Range = Maximum value - Minimum value
- Example: Data: 23, 45, 12, 67, 34
- Range = 67 - 12 = 55
- Probability
- Simple Probability
- P(event) = Favorable outcomes / Total outcomes
Example: A bag contains 4 red, 3 blue, and 5 green marbles. What is the probability of drawing a blue marble?
- P(blue) = 3/12 = 1/4 = 25%
- Complementary Probability
- P(not A) = 1 - P(A)
- Example: Probability of rain = 30%. Probability of no rain = 1 - 0.30 = 70%
- Combined Events
- Type Formula
- Independent events (and) P(A and B) = P(A) × P(B)
- Mutually exclusive events (or) P(A or B) = P(A) + P(B)
- Non-mutually exclusive (or) P(A or B) = P(A) + P(B) - P(A and B)
Example: What is the probability of rolling a 6 on a die AND flipping heads on a coin?