Rates, Speed & Unit Conversions | Quantitative Reasoning for UCAT PDF Download
| Table of contents |
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| 1. Rates |
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| 2. Speed |
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| 3. Unit Conversions |
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| Advanced Tips for UCAT Preparation |
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This document is designed for UCAT students preparing for the Quantitative Reasoning section, focusing on Rates, Speed, and Unit Conversions. These topics test your ability to solve numerical problems involving time-based relationships and conversions, often in medical or dental contexts like infusion rates, emergency travel times, or dosage calculations. Each topic is explored in depth, covering definitions, methods, variations, and multiple worked examples to ensure thorough understanding.
1. Rates
Rates measure one quantity per unit of another, such as mL per hour for drug administration or samples processed per minute. In the UCAT, rate problems often involve medical scenarios, requiring you to calculate, compare, or apply rates under time pressure.
1.1 Key Concepts
- Definition: A rate is a ratio of two quantities with different units, often involving time (e.g., 60 mL/hour or 30 patients/day).
- Calculating Rates: Divide the total quantity by the time taken (e.g., 300 mL over 5 hours = 300 ÷ 5 = 60 mL/hour).
- Comparing Rates: Determine which rate is higher or lower by converting to the same units or comparing directly (e.g., is 50 mL/hour faster than 0.8 mL/minute?).
- Time-Based Rates: Solve problems where rates involve time, such as processing rates (e.g., a machine processes 120 samples in 4 hours = 120 ÷ 4 = 30 samples/hour).
- Combining Rates: Calculate total rates for multiple entities working together (e.g., two machines processing samples at different rates).
1.2 Methods for Solving Rate Problems
- Direct Division: Divide the total quantity by the time to find the rate.
- Unit Conversion: Convert rates to consistent units for comparison or calculation.
- Proportional Scaling: Use the rate to find quantities over different time periods (e.g., if 60 mL/hour, then 2 hours = 60 × 2 = 120 mL).
1.3 Formulas
Quantity = Rate × Time
Time = Quantity ÷ Rate
1.4 Examples
Example 1: Calculating a Rate
A nurse administers 500 mL of saline over 4 hours. What is the infusion rate in mL per hour?
Solution:
- Identify quantities: Quantity = 500 mL, Time = 4 hours.
- Calculate rate: Rate = 500 ÷ 4 = 125 mL/hour.
Answer: The infusion rate is 125 mL/hour.
Note: Ensure units are consistent. Here, mL and hours are directly compatible.
Example 2: Comparing Rates
Doctor A treats 48 patients in 6 hours. Doctor B treats 35 patients in 5 hours. Which doctor treats patients at a faster rate?
Solution:
- Calculate Doctor A’s rate: 48 ÷ 6 = 8 patients/hour.
- Calculate Doctor B’s rate: 35 ÷ 5 = 7 patients/hour.
- Compare: 8 patients/hour > 7 patients/hour, so Doctor A is faster.
Answer: Doctor A treats patients at a faster rate.
Note: Comparing rates requires the same units (patients/hour). Convert if necessary.
Example 3: Time-Based Rate Application
A machine processes 240 samples in 8 hours. How many samples can it process in 3 hours?
Solution:
- Calculate the rate: 240 ÷ 8 = 30 samples/hour.
- Apply the rate: For 3 hours, samples = 30 × 3 = 90.
Answer: The machine processes 90 samples in 3 hours.
Note: Use the rate to scale quantities for different time periods.
Example 4: Combining Rates
Two machines process samples. Machine A processes 20 samples/hour, and Machine B processes 30 samples/hour. How many samples do they process together in 5 hours?
Solution:
- Combine rates: Total rate = 20 + 30 = 50 samples/hour.
- Calculate for 5 hours: Samples = 50 × 5 = 250.
Answer: Together, they process 250 samples in 5 hours.
Note: Add rates directly when entities work simultaneously, assuming no overlap or interference.
2. Speed
Speed measures distance traveled per unit of time, commonly in km/h or m/s. In the UCAT, speed problems often involve medical scenarios like ambulance travel times or equipment movement, requiring calculations of speed, time, or distance.
2.1 Key Concepts
- Speed Calculation: Speed = Distance ÷ Time (e.g., 60 km in 1 hour = 60 km/h).
- Time and Distance Problems: Solve for time (Time = Distance ÷ Speed) or distance (Distance = Speed × Time).
- Relative Speed: The combined speed of objects moving toward (add speeds) or away from (subtract speeds) each other.
- Average Speed: Total distance ÷ Total time, accounting for different speeds over segments.
- Unit Consistency: Ensure distance and time units match (e.g., convert minutes to hours for km/h).
2.2 Methods for Solving Speed Problems
- Formula Application: Use Speed = Distance ÷ Time and its rearrangements.
- Unit Conversion: Convert units (e.g., minutes to hours) before calculating.
- Segment Analysis: Break trips into segments for average speed or relative speed problems.
2.3 Formulas
Distance = Speed × Time
Time = Distance ÷ Speed
Average Speed = Total Distance ÷ Total Time
Relative Speed (toward) = Speed1 + Speed2
Relative Speed (away) = |Speed1 - Speed2|
2.4 Examples
Example 5: Calculating Speed
An ambulance travels 120 km in 90 minutes. What is its speed in km/h?
Solution:
- Convert time: 90 minutes = 90 ÷ 60 = 1.5 hours.
- Calculate speed: Speed = 120 ÷ 1.5 = 80 km/h.
Answer: The speed is 80 km/h.
Note: Convert time to hours for km/h calculations to ensure unit consistency.
Example 6: Time Calculation
A doctor travels 200 km at 50 km/h to reach a clinic. How long does the journey take?
Solution:
- Use the formula: Time = Distance ÷ Speed = 200 ÷ 50 = 4 hours.
Answer: The journey takes 4 hours.
Note: Ensure units match (km and km/h are compatible here).
Example 7: Relative Speed
Two ambulances, 100 km apart, travel toward each other. Ambulance A travels at 60 km/h, and Ambulance B at 40 km/h. How long until they meet?
Solution:
- Calculate relative speed: They move toward each other, so relative speed = 60 + 40 = 100 km/h.
- Calculate time: Distance = 100 km. Time = 100 ÷ 100 = 1 hour.
Answer: They meet in 1 hour.
Note: Add speeds for objects moving toward each other. Subtract for objects moving apart.
Example 8: Average Speed
A nurse travels 60 km at 80 km/h and returns the same distance at 40 km/h. What is the average speed for the round trip?
Solution:
- Calculate total distance: 60 + 60 = 120 km.
- Calculate time for each segment: Outbound: 60 ÷ 80 = 0.75 hours. Return: 60 ÷ 40 = 1.5 hours.
- Calculate total time: 0.75 + 1.5 = 2.25 hours.
- Calculate average speed: Average speed = 120 ÷ 2.25 ≈ 53.33 km/h.
Answer: The average speed is approximately 53.33 km/h.
Note: Average speed is not the average of the speeds but total distance ÷ total time.
3. Unit Conversions
Unit conversions involve changing quantities between different units, such as length, mass, volume, time, or rates. In the UCAT, conversions are common in medical contexts, like converting dosages (mg to g) or speeds (km/h to m/s).
3.1 Key Concepts
- Length Conversions: Convert between kilometers, meters, centimeters, or miles (e.g., 1 km = 1000 m, 1 mile ≈ 1.609 km).
- Mass Conversions: Convert between kilograms, grams, milligrams, or pounds (e.g., 1 kg = 1000 g, 1 g = 1000 mg, 1 kg ≈ 2.2046 pounds).
- Volume Conversions: Convert between liters, milliliters, or cubic units (e.g., 1 L = 1000 mL).
- Time Conversions: Convert between hours, minutes, seconds, or days (e.g., 1 hour = 60 minutes, 1 day = 24 hours).
- Rate Conversions: Convert rates between units (e.g., km/h to m/s: multiply by 1000/3600 = 5/18).
- Complex Conversions: Combine multiple conversions (e.g., mg/kg/hour to g/kg/day).
- Precision: Use approximate conversions (e.g., 1 mile ≈ 1.6 km) when exact values are not provided.
3.2 Methods for Solving Conversion Problems
- Multiplication/Division: Multiply or divide by the conversion factor (e.g., to convert km to m, multiply by 1000).
- Unit Cancellation: Set up conversions so units cancel out (e.g., 60 km/h × (1000 m/1 km) × (1 h/3600 s) = 16.67 m/s).
- Step-by-Step Conversion: Break complex conversions into manageable steps.
3.3 Common Conversion Factors
Mass: 1 kg = 1000 g, 1 g = 1000 mg, 1 kg ≈ 2.2046 pounds
Volume: 1 L = 1000 mL
Time: 1 hour = 60 minutes, 1 minute = 60 seconds, 1 day = 24 hours
Speed: 1 km/h = (1000/3600) m/s = 5/18 m/s
Rate: 1 unit/hour = 24 units/day
3.4 Examples
Example 9: Length Conversion
A patient’s height is 175 cm. What is it in meters?
Solution:
- Apply conversion: 1 m = 100 cm, so 175 cm = 175 ÷ 100 = 1.75 m.
Answer: The height is 1.75 m.
Note: Divide by 100 to convert cm to m, or multiply by 0.01.
Example 10: Mass Conversion
A medication dosage is 0.4 g. How many milligrams is this?
Solution:
- Apply conversion: 1 g = 1000 mg, so 0.4 g = 0.4 × 1000 = 400 mg.
Answer: The dosage is 400 mg.
Note: Multiply by 1000 to convert g to mg, a common conversion in dosage calculations.
Example 11: Rate Conversion
An ambulance travels at 72 km/h. What is its speed in m/s?
Solution:
- Apply conversion: 1 km/h = (1000 m)/(3600 s) = 5/18 m/s.
- Calculate: 72 × 5/18 = (72 × 5) ÷ 18 = 360 ÷ 18 = 20 m/s.
Answer: The speed is 20 m/s.
Note: Memorize the km/h to m/s conversion factor (5/18) for quick calculations.
Example 12: Complex Conversion
A dosage is prescribed at 2 mg/kg/hour. Convert this to g/kg/day.
Solution:
- Convert mg to g: 1 g = 1000 mg, so 2 mg = 2 ÷ 1000 = 0.002 g.
- Convert hour to day: 1 day = 24 hours, so 1 hour = 1/24 day. Rate per day = 0.002 g × 24 = 0.048 g.
- Combine: The rate is 0.048 g/kg/day.
Answer: The dosage is 0.048 g/kg/day.
Note: Break complex conversions into steps: convert mass, then time, and combine.
Advanced Tips for UCAT Preparation
- Master Time Management: UCAT Quantitative Reasoning gives ~40 seconds per question. Practice solving rate, speed, and conversion problems in 30–40 seconds.
- Use Mental Math: Memorize common conversions (e.g., 1 km = 1000 m, 1 g = 1000 mg, 5/18 for km/h to m/s) to save time.
- Practice Medical Scenarios: Focus on UCAT questions involving infusion rates, ambulance travel, or dosage conversions.
- Check Units: Ensure units are consistent before calculating (e.g., convert minutes to hours for km/h).
- Verify Calculations: For conversions, double-check by reversing the process (e.g., 400 mg = 0.4 g, then 0.4 × 1000 = 400 mg).
- Use Official Resources: Practice with UCAT’s official question bank and mock tests to align with the test format.
- Simplify Calculations: For average speed, calculate total distance and time separately. For conversions, use approximate values (e.g., 1 mile ≈ 1.6 km) if exactness isn’t required.
Tip: Create a mental checklist: For rates, calculate quantity ÷ time. For speed, ensure distance and time units match. For conversions, memorize key factors and verify units.
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