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Rates & Unit Conversions

Master rate problems and learn to convert between units with confidence: the essential skills for quantitative reasoning.

ISEE Middle Level

What Is a Rate?

A rate compares two quantities that have different units. It tells you how much of one thing happens for every one of another thing.

Speed
\( 60 \) miles per \( 1 \) hour
Price
\( \$3.50 \) per \( 1 \) pound
Typing
\( 45 \) words per \( 1 \) minute

Rate vs. Ratio

Both rates and ratios compare two numbers, but they differ in one important way:

Rate
Compares quantities with different units: miles and hours, dollars and pounds.
Ratio
Compares quantities with the same units (or no units): 3 boys to 5 girls.

The Unit Rate

A unit rate is a rate where the denominator is exactly \( 1 \). To find it, divide both quantities so the bottom number becomes \( 1 \).

Original Rate 150 miles 3 hours ÷ 3 Unit Rate 50 miles 1 hour

The Conversion Factor Rule

A conversion factor is a fraction that equals \( 1 \). Because both the numerator and denominator represent the same amount, multiplying by it changes the units without changing the value.

12 inches 1 foot = 1 60 min 1 hour = 1 1,000 m 1 km = 1

How Unit Cancellation Works

Place units so the one you want to remove appears in the opposite position (top vs. bottom). Then it cancels like a common factor.

5 ft × 12 in 1 ft = 60 inches "ft" cancels top & bottom

Rate Problems vs. Unit Conversions

Rate Problem Unit Conversion
Goal Find distance, cost, time, etc. Change units of measurement
Formula \( \text{amount} = \text{rate} \times \text{quantity} \) \( \text{value} \times \frac{\text{new unit}}{\text{old unit}} \)
Example \( 60 \,\text{mph} \times 3 \,\text{hr} = 180 \,\text{mi} \) \( 5 \,\text{ft} \times \frac{12 \,\text{in}}{1 \,\text{ft}} = 60 \,\text{in} \)
Key Skill Multiply or divide by rate Cancel unwanted units

How to Solve Any Rate or Conversion Problem

1
Identify the units. Write down the units you start with and the units you need.
2
Find or build conversion factors. Write each factor as a fraction equal to \( 1 \).
3
Set up so units cancel. Place the old unit in the opposite position from where it appears.
4
Multiply and simplify. Do the arithmetic, then check that the remaining units match what the question asks for.

Example 1: Finding Total Cost

Problem: Apples cost \( \$2.40 \) per pound. How much do \( 3.5 \) pounds cost?

Rate $2.40/lb × Quantity 3.5 lb = Total $8.40

Example 2: Converting Hours to Minutes

Problem: A movie lasts \( 2.5 \) hours. How many minutes is that?

2.5 hr × 60 min 1 hr = 150 min

Example 3: Two-Step Conversion

Problem: A car travels at \( 45 \) miles per hour. How many miles does it travel in \( 20 \) minutes?

1
Convert minutes to hours: \( 20 \;\text{min} \times \frac{1 \;\text{hr}}{60 \;\text{min}} = \frac{1}{3} \;\text{hr} \)
2
Apply the rate: \( 45 \;\frac{\text{mi}}{\text{hr}} \times \frac{1}{3} \;\text{hr} = 15 \;\text{mi} \)

Example 4: Chain of Conversions

Problem: Convert \( 3 \) yards to inches.

3 yd × 3 ft 1 yd × 12 in 1 ft = 108 in

Must-Know Conversion Facts

Length
\( 1 \;\text{ft} = 12 \;\text{in} \)
\( 1 \;\text{yd} = 3 \;\text{ft} \)
\( 1 \;\text{mi} = 5{,}280 \;\text{ft} \)
Time
\( 1 \;\text{hr} = 60 \;\text{min} \)
\( 1 \;\text{min} = 60 \;\text{sec} \)
\( 1 \;\text{day} = 24 \;\text{hr} \)
Weight / Volume
\( 1 \;\text{lb} = 16 \;\text{oz} \)
\( 1 \;\text{gal} = 4 \;\text{qt} \)
\( 1 \;\text{qt} = 2 \;\text{pt} \)

Using Proportions for Rates

When two rates are equal, you can set up a proportion and cross-multiply to find the unknown.

120 mi 2 hr = x mi 5 hr Cross-multiply 2x = 600 x = 300

Common Mistakes to Avoid

✕ Flipping the Factor
Putting the conversion factor upside down, so units multiply instead of cancelling. Always check which unit is on top.
✕ Mismatched Units
Multiplying rate × quantity when the units don't match (e.g., mph × minutes). Convert units first.
✕ Forgetting "Per"
Treating "\( \$3 \) per pound" as just "\( \$3 \)" without dividing. A rate always involves division.

Summary: Your Toolkit

Strategy When to Use Example
Unit Rate Compare prices, speeds \( \frac{\$7.50}{3 \;\text{lb}} = \$2.50/\text{lb} \)
Conversion Factor Change between units \( 5 \;\text{ft} \times \frac{12 \;\text{in}}{1 \;\text{ft}} \)
Proportion Find unknowns at same rate \( \frac{120}{2} = \frac{x}{5} \)
Chain Conversion Multi-step unit changes yd → ft → in

Question 1 of 5

A printer prints 8 pages per minute. How many pages will it print in 15 minutes?

Question 2 of 5

How many inches are in 4 feet?

Question 3 of 5

A car travels at 60 miles per hour. How far does it go in 45 minutes?

Question 4 of 5

Juice costs \( \$4.80 \) for 6 bottles. What is the unit price per bottle?

Question 5 of 5

Convert 2 yards to inches.