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Convert Units Using
Ratio Reasoning

Learn to transform measurements between units by building and multiplying conversion factors as ratios that equal \(1\).

6th Grade Math

What Is Unit Conversion?

Unit conversion means rewriting a measurement in a different unit without changing the actual amount. Think of it like translating between languages: the meaning stays the same, only the words change.

Same Amount
\(1\) foot and \(12\) inches describe the exact same length.
Different Label
The number changes because the size of each unit is different.

Why Use Ratios?

A ratio compares two quantities. When both sides of a ratio represent the same amount in different units, the ratio equals \(1\). Multiplying by \(1\) never changes a value: it only changes the unit.

12 inches 1 foot = 1 (same length)

The Conversion Factor

A conversion factor is a fraction built from an equivalence between two units. Because the numerator and denominator are equal, the fraction equals \(1\).

\(\dfrac{12\;\text{in}}{1\;\text{ft}}\)
Use when you want to convert feet to inches.
\(\dfrac{1\;\text{ft}}{12\;\text{in}}\)
Use when you want to convert inches to feet.

The Core Rule

\( \text{New Value} = \text{Old Value} \times \text{Conversion Factor} \)

Place the old unit in the denominator of the conversion factor so it cancels. The new unit stays in the numerator.

How Units Cancel

3 ft × 12 in 1 ft = 36 in ft cancels ft inches remain

Larger Units vs. Smaller Units

Larger → Smaller Smaller → Larger
Direction feet → inches inches → feet
Operation Multiply by a number > 1 Divide (or multiply by a fraction < 1)
Number gets Bigger Smaller
Example \(3\;\text{ft} \times 12 = 36\;\text{in}\) \(36\;\text{in} \div 12 = 3\;\text{ft}\)

Step-by-Step Method

1
Identify the unit you have and the unit you want.
2
Write the equivalence between the two units (e.g., \(1\;\text{yd} = 3\;\text{ft}\)).
3
Build the conversion factor: put the wanted unit on top and the old unit on the bottom.
4
Multiply the original quantity by the conversion factor.
5
Check: did the old unit cancel? Does the answer make sense?

Example 1: Yards to Feet

Convert \(5\) yards to feet.

1
Equivalence: \(1\;\text{yd} = 3\;\text{ft}\)
2
Conversion factor: \(\dfrac{3\;\text{ft}}{1\;\text{yd}}\)
3
Multiply: \(5\;\text{yd} \times \dfrac{3\;\text{ft}}{1\;\text{yd}} = 15\;\text{ft}\)

Example 2: Ounces to Pounds

A puppy weighs \(48\) ounces. How many pounds is that?

1
Equivalence: \(1\;\text{lb} = 16\;\text{oz}\)
2
Conversion factor (oz on bottom): \(\dfrac{1\;\text{lb}}{16\;\text{oz}}\)
3
Multiply: \(48\;\text{oz} \times \dfrac{1\;\text{lb}}{16\;\text{oz}} = \dfrac{48}{16}\;\text{lb} = 3\;\text{lb}\)

Example 3: Two-Step Conversion

Convert \(2\) miles to yards. (There is no single "miles to yards" fact, but we know \(1\;\text{mi} = 5{,}280\;\text{ft}\) and \(1\;\text{yd} = 3\;\text{ft}\).)

1
Miles → feet: \(2\;\text{mi} \times \dfrac{5{,}280\;\text{ft}}{1\;\text{mi}} = 10{,}560\;\text{ft}\)
2
Feet → yards: \(10{,}560\;\text{ft} \times \dfrac{1\;\text{yd}}{3\;\text{ft}} = 3{,}520\;\text{yd}\)

Visualizing with a Double Number Line

feet 0 3 6 9 12 yards 0 1 2 3 4

Metric Conversions

The metric system is built on powers of \(10\), making conversions especially clean. The same ratio reasoning applies.

kilo-
\(1\;\text{km} = 1{,}000\;\text{m}\)
centi-
\(1\;\text{m} = 100\;\text{cm}\)
milli-
\(1\;\text{m} = 1{,}000\;\text{mm}\)

Converting Rates

When a measurement has units in both numerator and denominator (like miles per hour), you may need to convert one or both parts.

1
Start: a car goes \(60\) miles per hour. Convert to feet per minute.
2
Convert miles → feet: \(60\;\text{mi} \times \dfrac{5{,}280\;\text{ft}}{1\;\text{mi}} = 316{,}800\;\text{ft}\)
3
Convert hours → minutes: \(\dfrac{316{,}800\;\text{ft}}{1\;\text{hr}} \times \dfrac{1\;\text{hr}}{60\;\text{min}} = 5{,}280\;\text{ft/min}\)

Common Mistakes

✕ Flipping the Factor
Putting the old unit on top instead of the bottom. The old unit never cancels, and you get the wrong unit in your answer.
✕ Forgetting to Multiply
Simply rewriting the number with a new label without actually doing the arithmetic. \(3\;\text{ft}\) is not \(3\;\text{in}\).
✕ Wrong Direction Check
Getting a smaller number when going from a large unit to a small unit. That should make the number bigger.

Reference Table

Type Equivalence Factor (larger → smaller)
Length \(1\;\text{ft} = 12\;\text{in}\) \(\times 12\)
Length \(1\;\text{yd} = 3\;\text{ft}\) \(\times 3\)
Length \(1\;\text{mi} = 5{,}280\;\text{ft}\) \(\times 5{,}280\)
Mass \(1\;\text{lb} = 16\;\text{oz}\) \(\times 16\)
Volume \(1\;\text{gal} = 4\;\text{qt}\) \(\times 4\)
Metric \(1\;\text{km} = 1{,}000\;\text{m}\) \(\times 1{,}000\)

Question 1 of 5

Convert \(7\) yards to feet.

Question 2 of 5

Which conversion factor converts inches to feet?

Question 3 of 5

A bag of apples weighs \(80\) ounces. How many pounds is that? (\(1\;\text{lb} = 16\;\text{oz}\))

Question 4 of 5

Convert \(3.5\) kilometers to meters.

Question 5 of 5

A runner moves at \(6\) miles per hour. What is that speed in feet per hour? (\(1\;\text{mi} = 5{,}280\;\text{ft}\))