Advanced Multi-Step Conversions

Why Multi-Step Conversions?

You've already learned basic conversions like inches to centimeters. But what if you need to convert miles per hour to meters per second? Or figure out how many cubic centimeters are in a gallon?

That's where multi-step conversions come in. They're like building blocks - you chain together simple conversions to solve complex problems!

💡 Think of it like this: If you can't get somewhere in one step, break the journey into smaller steps. Can't convert miles to meters directly? Go miles → feet → inches → centimeters → meters!

1. The Chain Method (Your New Superpower)

🔗 The Basic Idea

Set up your conversion factors in a chain, and watch the units cancel out like magic!

\[\text{Start} \times \frac{\text{What You Want}}{\cancel{\text{What You Have}}} \times \frac{\text{Next Thing You Want}}{\cancel{\text{What You Have Now}}} = \text{Final Answer}\]

Key Rule: Whatever unit is on the BOTTOM of one fraction must match what's on TOP of the next one (so they cancel).

Example 1: Converting Miles to Centimeters

Problem: Convert 5.2 miles to centimeters.

Step 1: Plan your route
Miles → Feet → Inches → Centimeters

Step 2: Set up the chain

\[5.2 \cancel{\text{ mi}} \times \frac{5280 \cancel{\text{ ft}}}{1 \cancel{\text{ mi}}} \times \frac{12 \cancel{\text{ in}}}{1 \cancel{\text{ ft}}} \times \frac{2.54 \text{ cm}}{1 \cancel{\text{ in}}}\]

Step 3: Multiply across the top, then across the bottom
\[= \frac{5.2 \times 5280 \times 12 \times 2.54}{1 \times 1 \times 1 \times 1} = 836,966.4 \text{ cm}\]

Step 4: Apply sig figs
5.2 has 2 sig figs, so: 840,000 cm or $8.4 \times 10^5$ cm

✅ Pro Tip: Always write units with EVERY number. This helps you catch mistakes before they happen!

2. Squared & Cubed Units (Area & Volume)

📐 The Exponent Rule

When converting area or volume, you must square or cube the ENTIRE conversion factor!

For Area (squared units):

\[\left(\frac{100 \text{ cm}}{1 \text{ m}}\right)^2 = \frac{10,000 \text{ cm}^2}{1 \text{ m}^2}\]

For Volume (cubed units):

\[\left(\frac{100 \text{ cm}}{1 \text{ m}}\right)^3 = \frac{1,000,000 \text{ cm}^3}{1 \text{ m}^3}\]

⚠️ Common Mistake: Don't just square the number!
WRONG: $\frac{100^2 \text{ cm}}{1 \text{ m}^2}$ ❌
RIGHT: $\left(\frac{100 \text{ cm}}{1 \text{ m}}\right)^2 = \frac{10,000 \text{ cm}^2}{1 \text{ m}^2}$ ✅

Example 2: Converting Cubic Inches to Cubic Centimeters

Problem: Convert $4.5 \text{ in}^3$ to $\text{cm}^3$.

Solution: Since 1 in = 2.54 cm, we cube both sides:

\[4.5 \cancel{\text{ in}^3} \times \left(\frac{2.54 \text{ cm}}{1 \cancel{\text{ in}}}\right)^3 = 4.5 \cancel{\text{ in}^3} \times \frac{16.387 \text{ cm}^3}{1 \cancel{\text{ in}^3}}\]

\[= 4.5 \times 16.387 = 73.74 \text{ cm}^3 = \textbf{74 cm}^3\] (2 sig figs)

Example 3: Area of an Oil Slick

Problem: Oil spreads in a thin layer on water called an "oil slick." If the slick covers 200 cm³ of oil in a layer 0.5 nm thick, what area in m² does it cover?

Strategy: Volume = Area × Thickness, so Area = Volume ÷ Thickness

Step 1: Convert everything to meters

\[200 \text{ cm}^3 \times \left(\frac{1 \text{ m}}{100 \text{ cm}}\right)^3 = 200 \text{ cm}^3 \times \frac{1 \text{ m}^3}{1,000,000 \text{ cm}^3} = 0.0002 \text{ m}^3\] \[0.5 \text{ nm} \times \frac{1 \text{ m}}{10^9 \text{ nm}} = 5 \times 10^{-10} \text{ m}\]

Step 2: Calculate area

\[\text{Area} = \frac{0.0002 \text{ m}^3}{5 \times 10^{-10} \text{ m}} = 4 \times 10^5 \text{ m}^2\]

Answer: $4 \times 10^5$ m² (or 400,000 m²) - that's huge!

3. Using Density as a Conversion Factor

⚖️ Density = Mass ÷ Volume

Density can be used to convert between mass and volume!

If density = 2.5 g/mL, then:

\[\frac{2.5 \text{ g}}{1 \text{ mL}} \quad \text{or} \quad \frac{1 \text{ mL}}{2.5 \text{ g}}\]

Choose whichever one cancels the units you want to get rid of!

Example 4: Mass of Milk

Problem: One liter of whole milk has a mass of 1032 g. What is the density in kg/L?

Solution:

\[D = \frac{1032 \text{ g}}{1 \text{ L}} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 1.032 \text{ kg/L}\]

Example 5: Mercury Poisoning Risk

Problem: The accepted toxic dose of mercury is 300 μg/day. If a nurse working in an office is exposed to 2.2 × 10⁻⁶ m³ of mercury vapor per day, is she at risk? (Density of Hg = 13.6 g/mL)

Step 1: Convert volume to mL

\[2.2 \times 10^{-6} \text{ m}^3 \times \left(\frac{100 \text{ cm}}{1 \text{ m}}\right)^3 \times \frac{1 \text{ mL}}{1 \text{ cm}^3}\] \[= 2.2 \times 10^{-6} \times 10^6 \text{ mL} = 2.2 \text{ mL}\]

Step 2: Convert to mass using density

\[2.2 \text{ mL} \times \frac{13.6 \text{ g}}{1 \text{ mL}} = 29.92 \text{ g}\]

Step 3: Convert to μg

\[29.92 \text{ g} \times \frac{10^6 \text{ μg}}{1 \text{ g}} = 2.992 \times 10^7 \text{ μg}\]

YES, she is at SERIOUS risk! She's exposed to ~30 million μg/day, while the safe limit is only 300 μg/day. That's 100,000 times the safe dose!

4. Fun with Custom Units

🐴 Hands, Troy Ounces, and More!

Sometimes scientists (and horse enthusiasts) use weird units. Don't panic - just treat them like any other conversion!

Example 6: Horse Height

Problem: The height of a horse is measured in hands (1 hand = exactly 4 inches). How many meters is a horse that measures 14.2 hands?

Solution: Hands → Inches → Centimeters → Meters

\[14.2 \text{ hands} \times \frac{4 \text{ in}}{1 \text{ hand}} \times \frac{2.54 \text{ cm}}{1 \text{ in}} \times \frac{1 \text{ m}}{100 \text{ cm}}\] \[= 14.2 \times 4 \times 2.54 \times 0.01 = 1.44 \text{ m}\]

Example 7: Gold Coin Value

Problem: The Sacagawea gold-colored dollar coin has a mass of 8.1 g and is 3.5% manganese. What is the mass in ounces (1 lb = 16 oz) and how many ounces of Mn are in this coin?

Part A: Total mass in ounces

\[8.1 \text{ g} \times \frac{1 \text{ lb}}{454 \text{ g}} \times \frac{16 \text{ oz}}{1 \text{ lb}} = 0.285 \text{ oz} \approx 0.29 \text{ oz}\]

Part B: Mass of manganese

\[0.285 \text{ oz total} \times \frac{3.5 \text{ oz Mn}}{100 \text{ oz total}} = 0.00998 \text{ oz Mn} \approx 0.010 \text{ oz Mn}\]

5. Real-World Problem Solving

Example 8: Camels and Straws

Problem: A very strong camel can carry 990 lb. If one straw weighs 1.5 grams, how many straws can the camel carry without breaking his back?

Solution:

\[990 \text{ lb} \times \frac{454 \text{ g}}{1 \text{ lb}} \times \frac{1 \text{ straw}}{1.5 \text{ g}} = 299,640 \text{ straws}\]

Answer: ~300,000 straws (2 sig figs from 990 and 1.5)

So technically, even one more straw (#299,641) could be "the straw that broke the camel's back"! 🐫

Example 9: Gold Nugget Worth

Problem: The largest nugget of gold on record was found in 1872 in New South Wales, Australia, and had a mass of 93.3 kg. Assuming the nugget is pure gold, what is its volume in cubic centimeters? What is it worth by today's standards if gold is $559/oz? (14.58 troy oz = 1 lb, density of Au = 19.3 g/cm³)

Part A: Volume

\[\text{Density} = \frac{\text{mass}}{\text{volume}} \rightarrow \text{volume} = \frac{\text{mass}}{\text{density}}\] \[V = \frac{93.3 \text{ kg} \times 1000 \text{ g/kg}}{19.3 \text{ g/cm}^3} = 4834.7 \text{ cm}^3 \approx 4830 \text{ cm}^3\]

Part B: Value in dollars

\[93.3 \text{ kg} \times \frac{1000 \text{ g}}{1 \text{ kg}} \times \frac{1 \text{ lb}}{454 \text{ g}} \times \frac{14.58 \text{ troy oz}}{1 \text{ lb}} \times \frac{\$559}{1 \text{ troy oz}}\] \[= 93.3 \times 1000 \times \frac{1}{454} \times 14.58 \times 559 = \$1,674,000\]

Answer: The nugget is worth about $1.67 million! 💰

6. Your Success Strategy

🎯 Step-by-Step Game Plan:

Identify what you have and what you need - Read the problem carefully!
Plan your conversion route - What's the path from start to finish?
Set up your chain - Write out all conversion factors
Check that units cancel - Make sure you'll end up with the right units
Calculate - Multiply across top, divide by bottom
Apply sig figs - Round to the least number of sig figs from your starting values
Does it make sense? - Is your answer reasonable?

⚠️ Watch Out For These Mistakes:

Forgetting to square/cube conversion factors for area/volume
Flipping a conversion factor upside down (2.54 cm/in vs in/cm)
Not paying attention to sig figs
Mixing up mass and volume
Forgetting to convert ALL units (like leaving some in cm and some in m)

🧠 Remember:

Units are your friends - they tell you if you're doing it right!
When in doubt, write it out - don't try to do it all in your head
Practice makes perfect - the more you do, the easier it gets
Real-world problems are just puzzles - break them into pieces!