Mathematics
Grade 10
15 min
Convert mixed customary units
Convert mixed customary units
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1
Introduction & Learning Objectives
Learning Objectives
Convert side lengths of right triangles given in mixed customary units (e.g., feet and inches) into a single unit (e.g., inches or decimal feet).
Apply the Pythagorean theorem to find a missing side length when dimensions are given in mixed units.
Use trigonometric ratios (SOH CAH TOA) to solve right triangles where measurements are in mixed customary units.
Calculate the perimeter and area of right triangles by first converting all mixed unit measurements to a consistent unit.
Convert a calculated answer from a single unit (e.g., 12.5 feet) back into a mixed customary unit format (e.g., 12 feet 6 inches).
Solve real-world problems involving right triangles and mixed customary units, such as calculating construction dimensions or angles of elevation.
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Key Concepts & Vocabulary
TermDefinitionExample
Mixed Customary UnitsA measurement expressed using two or more related customary units, such as feet and inches or yards and feet.A board that is 7 feet 4 inches long.
Conversion FactorA ratio that expresses the relationship between two different units of measurement, used to convert from one unit to another.The conversion factor to change feet to inches is (12 inches / 1 foot).
Single Unit (Decimal Form)A representation of a measurement using only the larger unit, with the smaller unit expressed as a decimal.7 feet 6 inches can be written as 7.5 feet, because 6 inches is 6/12 or 0.5 of a foot.
Single Unit (Smallest Unit Form)A representation of a measurement using only the smallest unit involved, which avoids decimals in calculations.7 feet 6 inches can be written a...
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Core Formulas
Conversion to a Smaller Unit
Total Small Units = (Number of Large Units × Conversion Factor) + Number of Small Units
Use this to convert a mixed measurement into a single, smaller unit (e.g., feet and inches to just inches). This method is excellent for avoiding decimals during intermediate calculations with the Pythagorean theorem.
Conversion to a Larger Unit (Decimal)
Total Large Units = Number of Large Units + (Number of Small Units / Conversion Factor)
Use this to convert a mixed measurement into a single, larger unit with a decimal (e.g., feet and inches to decimal feet). This is useful for inputting values into a calculator for trigonometric functions.
Key Length Conversion Factors
1 \text{ foot} = 12 \text{ inches} \\ 1 \text{ yard} = 3 \text{ feet} \\ 1 \text{...
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Challenging
A rectangular box has a length of 4 feet, a width of 2 feet 3 inches, and a height of 1 foot 6 inches. What is the length of the space diagonal (the longest interior diagonal connecting opposite corners), in feet and inches, rounded to the nearest inch?
A.4 ft 10 in
B.4 ft 6 in
C.5 ft 1 in
D.5 ft 8 in
Challenging
From a point on the ground, the angle of elevation to the top of a building is 25°. After walking 50 feet 6 inches closer to the building, the angle of elevation is 40°. What is the height of the building, in feet and inches, rounded to the nearest inch?
A.48 ft 3 in
B.55 ft 9 in
C.60 ft 2 in
D.53 ft 0 in
Challenging
A triangle has sides a = 1 yard 2 feet, b = 2 yards 1 foot, and c = 3 yards. Is this a right triangle?
A.Yes, because the side lengths are all whole numbers after conversion.
B.No, because a² + b² ≠ c² after converting to a single unit.
C.Yes, because a + b > c.
D.It is impossible to tell without knowing the angles.
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