Unit
Conversions
Master the art of converting between units within a measurement system: length, weight, volume, and time.
SSAT Upper Level: Quantitative
What Is a Unit Conversion?
A unit conversion changes the unit of measurement without changing the actual quantity. For example, \(12\) inches and \(1\) foot describe the same length: only the label changes.
Why Conversions Matter
Standardization
Problems may give data in feet but ask for the answer in inches. You need a consistent unit to compare or compute.
SSAT Context
SSAT quantitative sections test whether you know standard conversion factors and can apply them quickly under time pressure.
The Four Measurement Types
Length
inch, foot, yard, mile
Weight
ounce, pound, ton
Volume
cup, pint, quart, gallon
Time
second, minute, hour, day
The Conversion Factor Rule
Essential Conversion Factors
| Type | Relationship |
|---|---|
| Length | \(1\text{ ft} = 12\text{ in}\) |
| Length | \(1\text{ yd} = 3\text{ ft}\) |
| Length | \(1\text{ mi} = 5{,}280\text{ ft}\) |
| Weight | \(1\text{ lb} = 16\text{ oz}\) |
| Weight | \(1\text{ ton} = 2{,}000\text{ lb}\) |
| Volume | \(1\text{ gal} = 4\text{ qt}\) |
| Volume | \(1\text{ qt} = 2\text{ pt}\) |
| Volume | \(1\text{ pt} = 2\text{ cups}\) |
| Time | \(1\text{ hr} = 60\text{ min} = 3{,}600\text{ s}\) |
| Time | \(1\text{ day} = 24\text{ hr}\) |
Going Up vs. Going Down
Smaller → Larger Unit
Going "up" the ladder.
Divide by the conversion factor.
Example: \(36\text{ in} \div 12 = 3\text{ ft}\)
Divide by the conversion factor.
Example: \(36\text{ in} \div 12 = 3\text{ ft}\)
Larger → Smaller Unit
Going "down" the ladder.
Multiply by the conversion factor.
Example: \(3\text{ ft} \times 12 = 36\text{ in}\)
Multiply by the conversion factor.
Example: \(3\text{ ft} \times 12 = 36\text{ in}\)
Dimensional Analysis in Action
Step-by-Step Method
1
Identify the starting unit and the target unit.
2
Write the conversion factor as a fraction with the target unit on top and the starting unit on the bottom.
3
Multiply the given quantity by the fraction. Cancel matching units.
4
Compute the arithmetic and label your answer with the new unit.
5
Sanity check: Does the answer make sense? A bigger unit should yield a smaller number.
Example 1: Inches to Feet
Problem: Convert \(54\) inches to feet.
1
Start: \(54\) inches. Target: feet.
2
Conversion factor: \(\dfrac{1\text{ ft}}{12\text{ in}}\).
3
Multiply: \(54 \;\cancel{\text{in}} \times \dfrac{1\text{ ft}}{12\;\cancel{\text{in}}}\).
4
Compute: \(\dfrac{54}{12} = 4.5\text{ ft}\).
Example 2: Pounds to Ounces
Problem: A recipe calls for \(3.25\) pounds of flour. How many ounces is that?
1
Start: \(3.25\) lb. Target: oz.
2
Factor: \(\dfrac{16\text{ oz}}{1\text{ lb}}\).
3
Multiply: \(3.25\;\cancel{\text{lb}} \times \dfrac{16\text{ oz}}{1\;\cancel{\text{lb}}} = 52\text{ oz}\).
Example 3: Multi-Step Chain
Problem: How many cups are in \(2\) gallons?
1
No direct gallon-to-cup factor, so we chain: gallon → quart → pint → cup.
2
\(2\;\cancel{\text{gal}} \times \dfrac{4\;\cancel{\text{qt}}}{1\;\cancel{\text{gal}}} \times \dfrac{2\;\cancel{\text{pt}}}{1\;\cancel{\text{qt}}} \times \dfrac{2\text{ cups}}{1\;\cancel{\text{pt}}}\)
3
Compute: \(2 \times 4 \times 2 \times 2 = 32\text{ cups}\).
Example 4: Time Conversions
Problem: A train travels for \(2.5\) hours. How many seconds is that?
1
Chain: hours → minutes → seconds.
2
\(2.5\;\cancel{\text{hr}} \times \dfrac{60\;\cancel{\text{min}}}{1\;\cancel{\text{hr}}} \times \dfrac{60\text{ s}}{1\;\cancel{\text{min}}}\)
3
Compute: \(2.5 \times 60 \times 60 = 9{,}000\text{ seconds}\).
Common Mistakes
✕ Flipped Factor
Putting the wrong unit on top. If you want feet from inches, the factor must be \(\frac{1\text{ ft}}{12\text{ in}}\), not \(\frac{12\text{ in}}{1\text{ ft}}\).
✕ Forgetting a Step
Skipping an intermediate unit in a chain. Going from gallons directly to cups without converting through quarts and pints.
✕ Arithmetic Errors
Multiplying when you should divide, or vice versa. Always double-check: did the number go in the expected direction?
Quick-Reference Summary
| Direction | Operation | Number Changes |
|---|---|---|
| Large → Small | Multiply | Gets bigger |
| Small → Large | Divide | Gets smaller |
| Multi-step | Chain multiply | Cancel units step by step |
Question 1 of 5
How many inches are in \(7\) feet?
Since \(1\text{ ft} = 12\text{ in}\), we compute \(7 \times 12 = 84\text{ inches}\).
Question 2 of 5
A package weighs \(48\) ounces. How many pounds is that?
Since \(1\text{ lb} = 16\text{ oz}\), we divide: \(48 \div 16 = 3\text{ pounds}\).
Question 3 of 5
How many quarts are in \(3\) gallons?
Since \(1\text{ gal} = 4\text{ qt}\), multiply: \(3 \times 4 = 12\text{ quarts}\).
Question 4 of 5
A runner completes a race in \(7{,}200\) seconds. How many hours is that?
Since \(1\text{ hr} = 3{,}600\text{ s}\), divide: \(7{,}200 \div 3{,}600 = 2\text{ hours}\).
Question 5 of 5
How many cups are in \(5\) quarts?
Chain: \(1\text{ qt} = 2\text{ pt}\) and \(1\text{ pt} = 2\text{ cups}\), so \(1\text{ qt} = 4\text{ cups}\). Therefore \(5 \times 4 = 20\text{ cups}\).