Word Problems, Statistics, and Data Interpretation

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

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

Example: What is the probability of rolling a 6 on a die AND flipping heads on a coin?

P = 1/6 × 1/2 = 1/12 ≈ 8.3%