7.1 Rates & Speed/Distance/Time

7.1 Rates & Speed/Distance/Time

Quick Answer: A rate compares two quantities with different units, like kilometers per hour or dollars per pound. Speed is the rate distance ÷ time. To find any one of speed, distance, or time, cover that variable in the formula triangle and the remaining two tell you the operation to perform.

What Is a Rate?

A rate is a ratio that compares two quantities measured in different units. When you see the word "per," the phrase "for every," or a fraction bar between unlike units, you are working with a rate. Common NEX examples include dollars per kilogram, miles per gallon, and milliliters per hour.

A unit rate simplifies a rate so the denominator is 1. If 12 ounces of infant formula cost $3.00, the unit rate is $3.00 ÷ 12 oz = $0.25 per ounce. Unit rates let you compare options side by side: the better buy is the one with the smaller unit price.

Always reduce a rate to its simplest fractional form first. Writing the units into the fraction keeps you honest — when the units cancel, the answer lands in the right unit.

The Speed Triangle

Speed, distance, and time form a tight relationship:

A memory triangle helps. Write Distance on top, Speed and Time side by side below it. Cover the quantity you need; the remaining two show the operation. Cover Distance → Speed and Time sit side by side, so multiply. Cover Speed → Distance sits over Time, so divide. Cover Time → Distance sits over Speed, so divide.

Worked example 1. An ambulance travels 168 miles in 3 hours at a constant speed. Find the speed.

Cover Speed in the triangle: Distance ÷ Time. 168 mi ÷ 3 h = 56 mi/h. The speed is 56 mph.

Worked example 2. An IV pump delivers 120 mL/h. How much volume is infused in 4.5 hours?

Cover Distance (here, volume): Speed × Time. 120 mL/h × 4.5 h = 540 mL. Check the units: h cancels, leaving mL.

Worked example 3. A patient walks 2.4 km at 0.8 km/h. How long does the walk take?

Cover Time: Distance ÷ Speed. 2.4 km ÷ 0.8 km/h = 3 h.

Converting Units Inside a Rate

Rates on the NEX frequently arrive in mixed units. Convert before dividing or multiplying so the rate matches what the question asks.

Worked example 4. A runner covers 400 meters in 50 seconds. Give the speed in km/h.

First convert distance: 400 m = 0.4 km. Then speed = 0.4 km ÷ 50 s = 0.008 km/s. To switch seconds to hours, multiply by 3600 (the number of seconds in one hour): 0.008 × 3600 = 28.8 km/h. Alternatively, convert time first: 50 s = 50/3600 h, then speed = 0.4 km ÷ (50/3600) h = 28.8 km/h. Same result.

The useful shortcut: 1 m/s = 3.6 km/h. So 400 m ÷ 50 s = 8 m/s, and 8 × 3.6 = 28.8 km/h.

Nursing Drip-Rate Basics

IV flow rates are rates — nothing new mathematically. The core relationship is:

Rate = Volume ÷ Time

If an order is 1,000 mL over 8 hours, the pump rate is 1,000 ÷ 8 = 125 mL/h. If a 500 mL bag must run over 10 hours, the rate is 50 mL/h.

For gravity drips (no pump), the rate is expressed in drops per minute (gtt/min):

gtt/min = (Volume × drop factor) ÷ Time in minutes

The drop factor (gtt/mL) is printed on the tubing package — common values are 10, 15, and 60. The NEX does not require you to memorize drop factors; the question gives you one.

Worked example 5. Infuse 1,200 mL over 6 hours using tubing with a drop factor of 15 gtt/mL. Find the drip rate.

Time in minutes = 6 × 60 = 360 min. Volume × drop factor = 1,200 × 15 = 18,000 gtt. Drip rate = 18,000 ÷ 360 = 50 gtt/min.

Work Problems (Combined Rates)

When more than one worker or pump works together, add their rates, not their times.

Worked example 6. One pump fills a tank in 6 hours; a second pump fills the same tank in 3 hours. How long do they take together?

Rates: 1/6 tank per hour and 1/3 tank per hour. Combined rate = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 tank per hour. Time to fill one tank = 1 ÷ (1/2) = 2 hours.

If three identical pumps each fill a tank in 6 hours, together their rate is 3 × (1/6) = 1/2 tank per hour, so the fill time is again 2 hours.

Common Traps

• Unit mismatch in numerator/denominator. Distance in miles but time in minutes — convert one before dividing.
• Dividing the wrong way. Speed is distance ÷ time, not time ÷ distance. The larger quantity (usually distance) goes on top.
• Adding times instead of rates in work problems. Add rates (tanks per hour), never the hours themselves.
• Forgetting to convert minutes to hours when the rate is per hour but the time is given in minutes.
• Drop-factor confusion. gtt/mL is the number of drops per 1 mL — multiply by volume, then divide by minutes.