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mi km

Rate & Measurement
Problems

Master rates, unit conversions, and measurement word problems to boost your SSAT Quantitative score.

SSAT Middle Level: Quantitative

What Is a Rate?

A rate compares two quantities measured in different units. It tells you how much of one thing corresponds to a certain amount of another.

Speed
\( 55 \) miles per hour
Price
\( \$3.50 \) per pound
Typing
\( 40 \) words per minute

The Unit Rate

A unit rate is a rate where the second quantity is exactly 1. Finding the unit rate makes comparison and calculation much easier.

General Rate
\(150\) miles in \(3\) hours
\(\frac{150\text{ mi}}{3\text{ hr}}\)
 ÷ 3 Unit Rate
\(50\) miles per \(1\) hour
\(50\text{ mi/hr}\)

What Is Unit Conversion?

Unit conversion means changing a measurement from one unit to another without changing the actual quantity. You multiply by a conversion factor: a fraction that equals \( 1 \).

\(3\) feet
 Original ×
\(\frac{12\text{ in}}{1\text{ ft}}\)
 =
\(36\) in
 Converted

Core Formulas You Need

Distance
\( d = r \times t \)
Rate
\( r = \dfrac{d}{t} \)
Time
\( t = \dfrac{d}{r} \)
\(d\)
\(r\)
\(t\)
\(\times\)

Essential Conversion Factors

Type Relationship Conversion Factor
Length \( 1\text{ ft} = 12\text{ in} \) \( \frac{12\text{ in}}{1\text{ ft}} \)
Length \( 1\text{ yd} = 3\text{ ft} \) \( \frac{3\text{ ft}}{1\text{ yd}} \)
Length \( 1\text{ mi} = 5{,}280\text{ ft} \) \( \frac{5{,}280\text{ ft}}{1\text{ mi}} \)
Time \( 1\text{ hr} = 60\text{ min} \) \( \frac{60\text{ min}}{1\text{ hr}} \)
Weight \( 1\text{ lb} = 16\text{ oz} \) \( \frac{16\text{ oz}}{1\text{ lb}} \)
Volume \( 1\text{ gal} = 4\text{ qt} \) \( \frac{4\text{ qt}}{1\text{ gal}} \)

Rate vs. Ratio: Key Distinction

Rate Ratio
Units Different units Same or different units
Example \( 60 \) miles per hour \( 3 \) boys to \( 5 \) girls
"Per" Always includes "per" Uses "to" or ":"
Unit Rate Denominator = \( 1 \) Simplified form (not always \( 1 \))

Step-by-Step Strategy

1
Identify the given rate. Look for "per", "each", or "every" in the problem.
2
Write the rate as a fraction. Put the unit you want to find in the numerator.
3
Check units. If the problem asks for a different unit, set up a conversion factor so unwanted units cancel.
4
Multiply and simplify. Cancel matching units, then compute the arithmetic.
5
Check your answer. Does the unit on your result match what the question asked for?

Example 1: Basic Rate

A car travels 240 miles in 4 hours. What is the car's speed in miles per hour?

\(0\) hr
\(1\) hr
\(2\) hr
\(3\) hr
\(4\) hr
\(0\) mi
\(60\) mi
\(120\) mi
\(180\) mi
\(240\) mi

Example 2: Conversion in a Rate

A faucet fills a bucket at a rate of 2 gallons per minute. How many quarts does it fill in 3 minutes?

1
Find total gallons: \( 2\text{ gal/min} \times 3\text{ min} = 6\text{ gal} \)
2
Convert gallons to quarts: \( 6\text{ gal} \times \dfrac{4\text{ qt}}{1\text{ gal}} = 24\text{ qt} \)
3
Check units: "gal" cancels, leaving 24 quarts ✓

Example 3: Multi-Step Problem

A train travels at 90 miles per hour. How many feet does it travel in 10 seconds?

1
Convert miles to feet: \( 90\text{ mi/hr} \times \dfrac{5{,}280\text{ ft}}{1\text{ mi}} = 475{,}200\text{ ft/hr} \)
2
Convert hours to seconds: \( 475{,}200\text{ ft/hr} \times \dfrac{1\text{ hr}}{3{,}600\text{ s}} = 132\text{ ft/s} \)
3
Multiply by time: \( 132\text{ ft/s} \times 10\text{ s} = \) 1,320 feet

Example 4: Price Rate

Apples cost \$4.80 for 3 pounds. How much would 7 pounds cost?

Unit Rate
\(\frac{\$4.80}{3\text{ lb}}\)
\(=\$1.60/\text{lb}\)
 ×
\(7\) lb
 =
\(\$11.20\)

Common Mistakes

✕ Wrong Unit on Top
Placing the unit you want to cancel in the numerator instead of the denominator. Always put the unwanted unit where it cancels.
✕ Forgetting to Convert
Using miles when the answer asks for feet, or minutes when it asks for hours. Always check the target unit before computing.
✕ Mixing Up d, r, t
Using \( d = \frac{r}{t} \) instead of \( d = r \times t \). Remember: distance equals rate times time, not rate divided by time.

Dimensional Analysis in Action

Chain conversion factors so unwanted units cancel, leaving only the unit you need.

\(5\) yards
 Given ×
\(3\text{ ft}\)
\(1\text{ yd}\)
 ×
\(12\text{ in}\)
\(1\text{ ft}\)
 =
\(180\) in

Summary: Rate & Measurement Toolkit

Concept Formula / Rule When to Use
Unit Rate \( \frac{\text{Total}}{\text{Units}} \) "How much per one?"
Distance \( d = r \times t \) Finding distance traveled
Rate \( r = \frac{d}{t} \) Finding speed
Time \( t = \frac{d}{r} \) Finding duration
Unit Conversion Multiply by \( \frac{\text{new unit}}{\text{old unit}} \) Changing measurement units
Dimensional Analysis Chain conversion factors Multi-step conversions

Question 1 of 5

A cyclist rides 36 miles in 3 hours. What is the cyclist's speed in miles per hour?

Question 2 of 5

How many inches are in 4 feet?

Question 3 of 5

A machine produces 15 widgets per minute. How many widgets does it produce in 2 hours?

Question 4 of 5

Bananas cost \$2.40 for 5 pounds. How much would 8 pounds cost?

Question 5 of 5

A car drives at 60 miles per hour. How many feet per second is this? (Use \( 1\text{ mi} = 5{,}280\text{ ft} \) and \( 1\text{ hr} = 3{,}600\text{ s} \).)