2.4 Multi-Step Conversions

Now that we learned the factor label method for conversions, we can now try more complex conversions. Sometimes we need to convert several items in one problem.  We can do it the same, just follow the units.  

Example: Convert 55 mph to m/s

We want to change miles → meters and hours → seconds.

That means we need two conversion factors:

• 1 mile = 1,609 meters
• 1 hour = 3,600 seconds

Though it’s two changes, we use the same factor-label method—just with two fractions!

 Step-by-Step Using Factor-Label

Start with what you know:

$\text{55}\frac{\mathrm{miles}}{\mathrm{hour}}$

Step 1: Convert miles → meters

Use the fraction that cancels "miles":

$\text{55}\frac{\cancel{\mathrm{miles}}}{\mathrm{hour}}\text{×}\frac{1,609\mathrm{meters}}{1 \cancel{\mathrm{mile}}}$

→ "Miles" cancels out. Now we have meters/hour.

Step 2: Convert hours → seconds

We have “hours” in the denominator, so we need a fraction with “hours” on top to cancel it:

$\text{×}\frac{1 \mathrm{hour}}{3,600 \mathrm{seconds}}$

→ Now “hours” cancels too!

Put it all together:

$\text{55}\frac{\cancel{\mathrm{miles}}}{\cancel{\mathrm{hour}}}\text{×}\frac{1,609 \mathrm{meters}}{1 \cancel{\mathrm{mile}}}\text{×}\frac{1 \cancel{\mathrm{hour}}}{3,600 \mathrm{seconds}}$

Here's another way to look at it:

Now multiply the numbers:

$\frac{55\times 1,609}{3,600}=\frac{88,495}{3,600}\approx 2.46$

 Final answer: 55 mph ≈ 24.6 meters/second

 Key Tips for Multi-Step Conversions

1. Treat compound units (like mph) as two separate units:
   → “miles per hour” = miles ÷ hours, so convert numerator and denominator separately.
2. Always arrange conversion fractions so unwanted units cancel:
   • Want to cancel miles? Put miles in the denominator of your conversion factor.
   • Want to cancel hours? Put hours in the numerator.
3. You can chain as many steps as needed:
   Example: gallons → liters → milliliters → cm³ → m³… just keep adding fractions!

 Real-World Why It Matters

• Scientists and engineers always use metric units (like m/s), but speed limits are in mph in the U.S.
• Knowing how to convert helps you understand car safety data, physics problems, or even video game physics!
• 24.6 m/s is about how fast a major league fastball travels—so 55 mph is roughly fast-pitch softball speed!

 Quick Check: Does the answer make sense?

• 1 m/s ≈ 2.24 mph
• So 55 mph ÷ 2.24 ≈ 24.5 m/s → matches our result! 

When your estimate lines up, you know you’re on the right track.

Final Thought

Multi-step conversions might look intimidating at first—but with the factor-label method, you just add one fraction at a time, let the units cancel, and follow the math.

No memorizing “multiply or divide”—just let the units guide you!