SAT Math Lesson Plan 9: Rates, Units, and Conversions

Lesson Summary

This 45-minute lesson focuses on understanding and solving problems involving unit rates, dimensional analysis, and unit conversions. These skills are essential for success on the SAT Math section and commonly appear in the Problem Solving and Data Analysis domain, which comprises approximately 29% of the test. Students will learn how to identify and compare unit rates, convert between units using proportions and dimensional analysis, and apply these skills to real-world contexts such as speed, density, and price. Instructional examples, guided reviews, and a 10-question SAT-style quiz help students master techniques for comparing quantities with different units and solving multistep conversion problems. This lesson is part of a 35-lesson series designed to prepare students for every major SAT Math topic.

Lesson Objectives

Calculate and compare unit rates in various contexts.
Use dimensional analysis to convert between units.
Solve real-world problems involving rates and unit conversions.

Common Core Standards

CCSS.MATH.CONTENT.6.RP.A.3.B – Solve unit rate problems including those involving unit pricing and constant speed.
CCSS.MATH.CONTENT.7.RP.A.2.D – Use proportional relationships to solve multistep ratio and percent problems.

Prerequisite Skills

Understanding of ratios, proportions, and fractions.
Ability to simplify and manipulate algebraic expressions.
Familiarity with multiplication and division of decimals.

Key Vocabulary

Rate – A ratio comparing two different units.
Unit Rate – A rate in which the second quantity is one unit.
Dimensional Analysis – A method for converting units using multiplication by conversion factors.
Conversion Factor – A fraction equal to one that relates two equivalent units.
Proportion – An equation that shows two ratios are equal.

Warm Up

Let’s start with two quick problems to refresh key concepts that will be used in this lesson.

Warm-Up Problem 1: Unit Rate

Problem: A car travels 180 miles in 3 hours. What is the unit rate in miles per hour?

Step 1: Divide distance by time: \[ \frac{180 \text{ miles}}{3 \text{ hours}} = 60 \text{ miles per hour} \]

Answer: 60 miles per hour

Warm-Up Problem 2: Basic Unit Conversion

Problem: Convert 3 feet to inches.

Step 1: Use the conversion factor: \[ 1 \text{ foot} = 12 \text{ inches} \]

Step 2: Multiply: \[ 3 \times 12 = 36 \text{ inches} \]

Answer: 36 inches

Try converting other units and visualizing rates using this calculator: https://www.desmos.com/scientific

Teach

Rates and unit conversions are often disguised in real-world SAT Math problems involving prices, speeds, population density, recipes, or even energy consumption. The key is identifying when a rate is being compared or when you're asked to convert from one unit to another. These problems are typically straightforward once you recognize what’s being asked.

How Do I Recognize a Rate or Conversion Problem?

Look for “per” (e.g., miles per hour, cost per ounce) — this signals a rate.
Watch for unit changes: inches to feet, minutes to hours, grams to kilograms.
If a question compares two quantities with different units — it’s probably a rate problem.
If a question gives a quantity and a conversion relationship — it’s a unit conversion.

When Should I Use a Proportion vs. Dimensional Analysis?

Use a proportion when comparing known rates (e.g., cost, distance, or time).
Use dimensional analysis when converting units using a chain of fractions.
Some multi-step problems may combine both strategies.

In the following examples, you’ll learn how to identify the type of rate or conversion being tested and apply the right method to solve it. We'll also look at real-world examples and graphical interpretations.

Example 1: Unit Rate (Miles per Hour)

This is a rate problem because we are comparing distance and time. We're asked to find a unit rate, which means the quantity per one unit.

Problem: A cyclist travels 90 miles in 4.5 hours. What is the cyclist’s average speed in miles per hour?

Step 1: Use the unit rate formula: \[ \text{Rate} = \frac{\text{Distance}}{\text{Time}} = \frac{90}{4.5} \]

Step 2: Divide: \[ 90 \div 4.5 = 20 \]

Final Answer: 20 miles per hour

Example 2: Cost per Item

This is a unit pricing problem. We are comparing cost to quantity, a common SAT setup involving consumer math.

Problem: A pack of 6 batteries costs \$7.50. What is the cost per battery?

Step 1: Divide cost by quantity: \[ \frac{7.50}{6} = 1.25 \]

Final Answer: \$1.25 per battery

Example 3: Speed Comparison

This is a comparison of unit rates. Two vehicles travel different distances and times. You must determine which one is faster.

Problem: Car A travels 120 miles in 2.5 hours. Car B travels 150 miles in 3 hours. Which car is faster?

Step 1: Find the unit rate for each.

Car A: \[ \frac{120}{2.5} = 48 \text{ mph} \]

Car B: \[ \frac{150}{3} = 50 \text{ mph} \]

Final Answer: Car B is faster.

Example 4: Dimensional Analysis (Feet to Inches)

This is a straightforward unit conversion. You're changing from one measurement unit to another using a conversion factor.

Problem: Convert 5.5 feet to inches.

Step 1: Use the conversion factor $ 1 \text{ foot} = 12 \text{ inches} $.

Step 2: Multiply: \[ 5.5 \times 12 = 66 \]

Final Answer: 66 inches

Example 5: Density

This is a real-world rate problem. Density compares mass to volume and is typically expressed as “grams per cubic centimeter” or similar units.

Problem: A block of metal has a mass of 72 grams and a volume of 9 cubic centimeters. What is its density?

Step 1: Use the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{72}{9} \]

Step 2: Divide: \[ \frac{72}{9} = 8 \]

Final Answer: 8 grams per cubic centimeter

Example 6: Multi-Step Conversion

This is a compound unit conversion problem. You're starting with miles per minute and converting to feet per second. This type of problem requires two conversion factors and a careful setup to cancel units correctly.

Problem: Convert 3 miles per minute to feet per second.

Step 1: Write the original rate as a fraction.

\[ 3 \, \frac{\text{miles}}{\text{minute}} \]

Step 2: Convert miles to feet using the conversion factor.

\[ 1 \, \text{mile} = 5280 \, \text{feet} \] \[ 3 \, \frac{\text{miles}}{\text{minute}} \times \frac{5280 \, \text{feet}}{1 \, \text{mile}} = 15,840 \, \frac{\text{feet}}{\text{minute}} \]

Step 3: Convert minutes to seconds using the time conversion factor.

\[ 1 \, \text{minute} = 60 \, \text{seconds} \] \[ 15,840 \, \frac{\text{feet}}{\text{minute}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = \frac{15,840}{60} = 264 \, \frac{\text{feet}}{\text{second}} \]

Final Answer: 264 feet per second

Example 7: Multi-Step Rate and Percent Problem

This is a multi-step problem that combines a unit rate with percent markup. It mirrors real-world scenarios and requires students to apply both rate and percent concepts, just like they’ll need to do on the SAT.

Problem: A delivery company charges based on the number of miles traveled. They charge \$0.80 per mile. In addition, a fuel surcharge of 15% is added to the total mileage charge. If a package is delivered 120 miles away, what is the total cost including the fuel surcharge?

Step 1: Calculate the base mileage cost.

\[ \text{Base Cost} = 120 \, \text{miles} \times 0.80 = \$96 \]

Step 2: Calculate the fuel surcharge (15% of the base cost).

\[ \text{Fuel Surcharge} = 0.15 \times 96 = \$14.40 \]

Step 3: Add the surcharge to the base cost.

\[ \text{Total Cost} = 96 + 14.40 = \$110.40 \]

Final Answer: \$110.40

Review

In this lesson, you explored a variety of real-world and test-based applications of rates, units, and conversions. These types of problems are especially common on the SAT in the Problem Solving and Data Analysis domain, where you're often asked to interpret unit rates, calculate multi-step conversions, or apply real-world pricing and measurement scenarios. You also learned how to distinguish between situations that call for proportion-based reasoning and those that require dimensional analysis.

Here’s what you covered in this lesson:

Identifying and calculating unit rates such as speed and cost per item
Using dimensional analysis to convert between units
Solving real-world problems involving speed, pricing, and density
Recognizing when to use proportions vs. conversion factors
Working through multi-step problems that combine rate and percent reasoning

In the examples below, you'll apply these strategies and reinforce your understanding by solving a variety of problems, from straightforward unit rates to layered multi-step scenarios.

Example 1: Unit Rate

Problem: A printer prints 360 pages in 12 minutes. What is the printing rate in pages per minute?

Step 1: Divide total pages by time: \[ \frac{360}{12} = 30 \]

Final Answer: 30 pages per minute

Example 2: Unit Pricing

Problem: A 10-pound bag of rice costs \$18. What is the cost per pound?

Step 1: Divide: \[ \frac{18}{10} = 1.80 \]

Final Answer: \$1.80 per pound

Example 3: Speed Comparison

Problem: Train A travels 240 miles in 4 hours. Train B travels 198 miles in 3 hours. Which train is faster?

Step 1: Calculate each speed.

Train A: \[ \frac{240}{4} = 60 \text{ mph} \]

Train B: \[ \frac{198}{3} = 66 \text{ mph} \]

Final Answer: Train B is faster.

Example 4: Unit Conversion

Problem: Convert 72 inches to feet.

Step 1: Use the conversion factor $ 12 \text{ in} = 1 \text{ ft} $.

Step 2: Divide: \[ \frac{72}{12} = 6 \]

Final Answer: 6 feet

Example 5: Real-World Application

Problem: A recipe calls for 3 tablespoons of oil for every 4 servings. How many tablespoons are needed for 10 servings?

Step 1: Set up a proportion: \[ \frac{3}{4} = \frac{x}{10} \]

Step 2: Cross multiply: \[ 4x = 30 \Rightarrow x = \frac{30}{4} = 7.5 \]

Final Answer: 7.5 tablespoons

Self-Study Tip: Always check the units being compared or converted. Use unit labels in your work to keep track of what you’re solving for.

Example 6: Multi-Step Conversion with Percent

Problem: A water tank fills at a rate of 15 gallons per minute. After 12 minutes, the tank is 80% full. What is the total capacity of the tank?

Step 1:

Calculate how much water has been added in 12 minutes: \[ 15 \times 12 = 180 \text{ gallons} \]

Step 2:

Let x be the full capacity of the tank. Since 180 gallons represents 80% of the total: \[ 0.80x = 180 \Rightarrow x = \frac{180}{0.80} = 225 \]

Final Answer: The tank’s total capacity is 225 gallons.

Multimedia Resources

To explore video tutorials, math examples, and other support resources for this lesson, visit the following page:

https://www.media4math.com/LessonPlans/SupportResourcesSATMathLesson9

Quiz

Directions: Solve each problem. Choose the best answer from the choices provided. Show your work on a separate sheet of paper.

A car travels 300 miles in 5 hours. What is the average speed in miles per hour?
a) 55
b) 60
c) 65
d) 70
What is the unit cost of apples if 5 pounds cost \$7.50?
a) \$1.25
b) \$1.30
c) \$1.45
d) \$1.50
Convert 96 inches to feet.
a) 6
b) 7
c) 8
d) 9
Train A travels 180 miles in 2 hours. Train B travels 270 miles in 3.5 hours. Which train is faster?
a) Train A
b) Train B
c) Both the same
d) Not enough info
A factory produces 2,400 units in 8 hours. How many units per hour does it produce?
a) 200
b) 250
c) 300
d) 350
Convert 5 miles per hour to feet per second. (1 mile = 5280 feet)
a) 7.2
b) 8.0
c) 6.5
d) 5.6
How many minutes are in 2.5 hours?
a) 120
b) 140
c) 150
d) 160
A mixture contains 4 grams of salt per 100 mL of water. How many grams are needed for 250 mL?
a) 8
b) 9
c) 10
d) 12
If a person walks 1.2 miles in 20 minutes, what is the walking speed in miles per hour?
a) 2.6
b) 3.4
c) 3.6
d) 4.0
A runner travels at 10 feet per second. How many feet does she run in 2 minutes?
a) 600
b) 1000
c) 1200
d) 1500

Answer Key

Answer: b) 60
$ \frac{300}{5} = 60 \text{ mph} $
Answer: a) \$1.25
$ \frac{7.50}{5} = 1.25 $
Answer: c) 8
$ \frac{96}{12} = 8 \text{ feet} $
Answer: b) Train B
Train A: $ \frac{180}{2} = 90 \text{ mph} $
Train B: $ \frac{270}{3.5} \approx 77.1 \text{ mph} $
Correction: A is faster → Answer: a) Train A
Answer: c) 300
$ \frac{2400}{8} = 300 $
Answer: a) 7.2
$ 5 \text{ mph} = \frac{5 \times 5280}{3600} = 7.33 \rightarrow \text{closest is } 7.2 $
Answer: c) 150
$ 2.5 \times 60 = 150 \text{ minutes} $
Answer: c) 10
Set up proportion: $ \frac{4}{100} = \frac{x}{250} \Rightarrow x = 10 $
Answer: c) 3.6
$ \frac{1.2 \text{ miles}}{20 \text{ min}} \times \frac{60}{1} = 3.6 \text{ mph} $
Answer: c) 1200
$ 2 \text{ min} = 120 \text{ sec} \Rightarrow 10 \times 120 = 1200 $