Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Using Ratio Reasoning to Convert Measurement Units

playing your favorite video game and you need to mix 3 potions for every 1 power crystal. The game tells you the ratio. Now imagine real life: you are baking cookies, building a Lego tower, or checking how fast you ran 100 meters. All of these use ratios and units. When we move from one unit to another (like minutes to seconds, meters to centimeters, or dollars per hour to dollars per minute), we are using ratio reasoning to convert measurement units.

Did you know? 🤔 Astronauts, athletes, chefs, and game designers all use unit conversions to make sure things are fair, accurate, and safe.

In this lesson, you will learn how to:

• Understand unit ratios like “1 hour = 60 minutes.”
• Use ratio reasoning to convert between units.
• Use multiplying and dividing to transform units correctly.
• Use tables, double number lines, and equations to organize conversions.
• Apply these skills to real-world problems, from sports to recipes to travel. 🏀🍕✈️

As shown in [Figure 1], many everyday situations, like cooking and sports timing, naturally involve measurements and unit conversions.

1. Ratios and Units: The Basics

A ratio compares two quantities. For example:

• 3 red marbles to 5 blue marbles → ratio is 3:5.
• 1 hour to 60 minutes → ratio is 1:60.

A unit tells us what we are measuring, like centimeters, meters, seconds, minutes, liters, or grams.

When we convert units, we are using a ratio that equals 1, like:

• 1 minute = 60 seconds → \(\frac{60 \textrm{ seconds}}{1 \textrm{ minute}}\)
• 1 meter = 100 centimeters → \(\frac{100 \textrm{ cm}}{1 \textrm{ m}}\)

These fractions each equal 1, because the top and bottom represent the same amount, just in different units.

2. The Big Idea: Multiply by a “Clever One”

To convert units, we multiply by a fraction that equals 1. This fraction is sometimes called a conversion factor.

Example of a conversion factor:

• 1 hour = 60 minutes, so both \(\frac{60 \textrm{ minutes}}{1 \textrm{ hour}}\) and \(\frac{1 \textrm{ hour}}{60 \textrm{ minutes}}\) equal 1.

We choose the one that cancels the unit we want to get rid of.

Key idea: When you multiply by a conversion factor, you aren’t changing the actual amount, only the way it is measured.

3. When Do We Multiply? When Do We Divide?

Think of these guiding questions:

• Am I going from a big unit to a smaller unit (like meters to centimeters)? Then the number gets larger. You multiply by a number greater than 1.
• Am I going from a small unit to a bigger unit (like centimeters to meters)? Then the number gets smaller. You divide by a number greater than 1.

But you can actually just think in terms of the conversion factor and canceling units. The multiply/divide decision happens naturally when you set up your fraction correctly.

4. Using Ratio Tables and Double Number Lines

Sometimes it helps to organize your thinking in a ratio table or a double number line. [Figure 2] shows both a table and a double number line for converting kilometers to meters.

Ratio table example:

• 1 km → 1000 m
• 2 km → 2000 m
• 3 km → 3000 m

Double number line example:

On one line, you mark kilometers. On the other line, you mark meters. You line up 1 km with 1000 m, 2 km with 2000 m, and so on.

These tools help you see patterns and equivalent ratios.

5. Converting Within the Same System (Metric and Customary)

Here are some common unit ratios you should know:

Length – Metric

• 1 kilometer (km) = 1000 meters (m)
• 1 meter (m) = 100 centimeters (cm)
• 1 centimeter (cm) = 10 millimeters (mm)

Length – Customary (U.S.)

• 1 foot (ft) = 12 inches (in)
• 1 yard (yd) = 3 feet (ft)
• 1 mile (mi) = 5280 feet (ft)

Time

• 1 minute (min) = 60 seconds (s)
• 1 hour (h) = 60 minutes (min)
• 1 day = 24 hours

Capacity (Metric)

• 1 liter (L) = 1000 milliliters (mL)

Mass (Metric)

• 1 kilogram (kg) = 1000 grams (g)

6. Step-by-Step Solved Examples

Example 1: Converting a Single Measurement (Meters to Centimeters)

Problem: A character in a game jumps 4.5 meters. How many centimeters is this jump?

Step 1: Write the known conversion.

We know 1 meter = 100 centimeters.

Step 2: Write the conversion factor as a fraction equal to 1.

Use \(\frac{100 \textrm{ cm}}{1 \textrm{ m}}\) because we want to cancel meters and end up with centimeters.

Step 3: Multiply the measurement by the conversion factor.

We have 4.5 m.

Compute:

\(4.5 \textrm{ m} \times \frac{100 \textrm{ cm}}{1 \textrm{ m}} = 450 \textrm{ cm}\)

The unit \(\textrm{m}\) cancels because it is in the numerator and denominator.

Answer: The jump is 450 cm.

Example 2: Time Conversion in a Real-World Context (Minutes to Seconds)

Problem: A YouTube video lasts 8.5 minutes. How many seconds is that?

Step 1: Know the conversion.

1 minute = 60 seconds.

Step 2: Write the conversion factor.

We use \(\frac{60 \textrm{ s}}{1 \textrm{ min}}\) to cancel minutes.

Step 3: Multiply.

\(8.5 \textrm{ min} \times \frac{60 \textrm{ s}}{1 \textrm{ min}} = 510 \textrm{ s}\)

Answer: The video is 510 seconds long.

Example 3: Using a Ratio Table (Miles to Feet)

Problem: Your school track is 0.5 miles long. How many feet is that?

Step 1: Know the conversion.

1 mile = 5280 feet.

Step 2: Build a ratio table.

We want a row for 1 mile and a row for 0.5 miles.

• 1 mile → 5280 feet
• 0.5 mile → ? feet

Step 3: Use the relationship.

0.5 is half of 1, so we take half of 5280.

Half of 5280 is:

\(\frac{5280}{2} = 2640\)

Answer: 0.5 miles = 2640 feet.

We could also use a conversion factor:

\(0.5 \textrm{ mi} \times \frac{5280 \textrm{ ft}}{1 \textrm{ mi}} = 2640 \textrm{ ft}\)

Example 4: Converting Rates (Speed)

Problem: A bicyclist travels 12 meters every second. How many meters does the bicyclist travel in 1 minute at the same speed?

Step 1: Understand the rate.

The rate is 12 meters per second: \(12 \textrm{ m/s}\).

Step 2: Convert the time from seconds to minutes.

1 minute = 60 seconds. The bicyclist travels for 60 seconds.

Step 3: Multiply the rate by the time.

Distance = rate × time.

\(12 \textrm{ m/s} \times 60 \textrm{ s} = 720 \textrm{ m}\)

Notice how seconds cancel, leaving meters.

Answer: The bicyclist travels 720 meters in 1 minute.

Example 5: Multi-Step Conversion (Kilometers per Hour to Meters per Second)

Problem: A car moves at 72 kilometers per hour. What is this speed in meters per second?

This example shows how to manipulate units when multiplying and dividing. It uses two conversions: kilometers to meters and hours to seconds.

Step 1: Write the rate with units.

\(72 \frac{\textrm{km}}{\textrm{h}}\)

Step 2: Convert kilometers to meters.

1 km = 1000 m, so use \(\frac{1000 \textrm{ m}}{1 \textrm{ km}}\).

Multiply:

\(72 \frac{\textrm{km}}{\textrm{h}} \times \frac{1000 \textrm{ m}}{1 \textrm{ km}} = 72000 \frac{\textrm{m}}{\textrm{h}}\)

The km units cancel, leaving meters per hour.

Step 3: Convert hours to seconds.

1 hour = 3600 seconds. We want to go from “per hour” to “per second.”

We divide by 3600:

\(72000 \frac{\textrm{m}}{\textrm{h}} \times \frac{1 \textrm{ h}}{3600 \textrm{ s}} = 20 \frac{\textrm{m}}{\textrm{s}}\)

Hours cancel, leaving meters per second.

Answer: 72 km/h = 20 m/s.

This example shows that you can treat units like numbers when you multiply and divide, canceling them just like factors. ⚡

7. Visualizing Unit Conversion with Double Number Lines

Double number lines give a clear picture of how two units grow together at the same constant rate. In [Figure 3], the double number line shows how minutes and seconds line up.

For example:

• 1 minute aligns with 60 seconds.
• 2 minutes align with 120 seconds.
• 3 minutes align with 180 seconds.

You can use the pattern to find any matching pair. If you know 1 minute ↔ 60 seconds, you can scale up by multiplying both sides of the ratio by the same number.

8. Real-World Applications

Cooking and Baking

Recipes often use units like cups, tablespoons, milliliters, and grams.

• If a recipe is for 4 people but you are cooking for 8, you might double all the ingredients. You also might need to convert 500 mL to liters or 3 teaspoons to tablespoons.

Sports and Fitness

• Running distances: 5 kilometers to meters (for a fitness app), or miles to kilometers.
• Timing: seconds to minutes (lap times), or minutes to hours (total practice time).

Travel and Maps

• Maps might show distance in kilometers, but your car’s dashboard might show miles per hour.
• Plane flights might list speed in km/h, but science problems in class use m/s.

Science Experiments

• Measuring liquids in milliliters and converting to liters.
• Mass in grams and kilograms.

Money and Rates

• Earning 12 dollars per hour and asking, “How much is that per minute?”

To convert 12 dollars per hour to dollars per minute, we use:

1 hour = 60 minutes.

\(12 \frac{\textrm{dollars}}{\textrm{hour}} \times \frac{1 \textrm{ hour}}{60 \textrm{ minutes}} = 0.2 \frac{\textrm{dollars}}{\textrm{minute}}\)

So you earn 0.2 dollars per minute (20 cents per minute).

9. Common Pitfalls and How to Avoid Them

Mistake 1: Using the conversion factor upside down

Example: To convert meters to centimeters, you must multiply by \(\frac{100 \textrm{ cm}}{1 \textrm{ m}}\), not \(\frac{1 \textrm{ cm}}{100 \textrm{ m}}\).

How to avoid: Check that the unit you want to cancel is opposite (top vs bottom) from the original measurement.

Mistake 2: Forgetting which unit is bigger

If you go from a big unit to a small unit (hours to minutes, meters to centimeters), your number should get larger, not smaller.

How to avoid: Ask yourself: “Is this a smaller unit or a larger unit?” Then predict whether the number should go up or down before you calculate.

Mistake 3: Ignoring units in multi-step problems

When converting rates like km/h to m/s, some students only convert the top or only the bottom.

How to avoid: Write the units on every number. Treat them like parts of the fraction. Cancel them step by step.

10. Strategy Checklist for Unit Conversion

1. Identify the given amount and unit. What do you know?
2. Identify the desired unit. What do you want to find?
3. Write the conversion ratio that connects the two units.
4. Choose the correct conversion factor (a fraction equal to 1) so that the old unit cancels.
5. Multiply (or divide) carefully, keeping track of units.
6. Check if the answer makes sense. Should the number be larger or smaller?

11. Key Points to Remember

• Ratios and unit rates describe relationships between quantities, including measurements like time, distance, and mass.

• Converting units is done by multiplying by a conversion factor, which is a fraction equal to 1, such as \(\frac{60 \textrm{ s}}{1 \textrm{ min}}\) or \(\frac{100 \textrm{ cm}}{1 \textrm{ m}}\).

• Units can be treated like algebraic symbols: they can cancel when they appear in both the numerator and denominator.

• Going from larger units to smaller units (hours to minutes, meters to centimeters) makes the number bigger.

• Going from smaller units to larger units (seconds to minutes, centimeters to meters) makes the number smaller.

• Ratio tables and double number lines help visualize and reason about equivalent measurements.

• Multi-step conversions, like changing km/h to m/s, use more than one conversion factor and careful unit cancellation.

• Real-life problems in cooking, sports, travel, science, and money often require you to convert units accurately using ratio reasoning.