MEASUREMENT UNITS

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UNIT 7

• MEASUREMENT UNITS

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MEASUREMENT UNITS

• Linear Measurement expresses distance between two
  points or measurement of lengths
• Examples of English units of linear measure
• 1 foot (ft) 12 inches (in)
• 1 yard (yd) 3 feet (ft)
• 1 yard (yd) 36 inches (in)
• 1 rod (rd) 16.5 feet (ft)
• 1 mile (mi) 5,280 feet (ft)
• 1 mile (mi) 1,760 yards (yd)
• 1 mile (mi) 320 rods (rd)

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CONVERTING UNITS OF MEASURE

• Unity fraction method is used to convert between
  different units of measure
• Method multiplies given unit of measure by a
  fraction equal to one
• Unity fraction is set up in such a way that
  original unit cancels out and unit being
  converted to remains. Examples on following
  slides will demonstrate use of unity fraction
  when converting units of measure

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CONVERSION EXAMPLE 1

• Express 43 inches as feet
• Use 12 inches 1 foot equivalence to set up
  unity fraction
• Since we started with 43 inches, the inches must
  cancel out
• We will make our unity fraction

3.6 ft Ans Note Rounded to 2
significant digits
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CONVERSION EXAMPLE 2

• Express 4.56 miles as rods
• Use 1 mile 320 rods equivalence to set up unity
  fraction
• Since we started with miles, they must cancel out
• We will make our unity fraction

1460 Ans Note Rounded to 3
significant digits
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ADDITION OF COMPOUND NUMBERS

• To add compound numbers, arrange like units in
  same columns, then add each column. When
  necessary, simplify sum
• Add

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ADDITION OF COMPOUND NUMBERS

• Arrange like units in
  same columns
• Add each column
• Simplify

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SUBTRACTION

• To subtract compound numbers, arrange like units
  in same columns, then subtract each column
  starting from right. Regroup as necessary
• Subtract 3 ft 14 in 1 ft 21 in
• Arrange like units in
  same columns
• 21 inches cannot be
  subtracted from 14 inches
  so you need to
  borrow
• Now finish subtraction

1 ft 5 in Ans
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MULTIPLICATION

• To multiply compound numbers, multiply each unit
  by the multiplier. When necessary, simplify
  product
• Multiply 2 mi 540
  yd
• ? 5

10 mi 2700 yd

• Multiply each unit by 5

• Simplify product

2700 yd 1 mi 940 yd

• Add 10 mi 1 mi 940 yd 11 mi 940 yd Ans

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DIVISION

• To divide compound numbers, divide each unit by
  divisor starting at left. If a unit is not
  exactly divisible, express remainder as next
  smaller unit and add to given number of smaller
  units
• Continue process until all units are divided
• Divide

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DIVISION

• Write as division problem and start dividing at
  left 15 ? 2 7

7 yd
2 ft

• 2 ? 7 14 write 14 under the 15 and subtract

14 yd 1 yd
3 ft 4 ft

• Convert 1 yd into 3 feet and add it to the 1 foot
  in the dividend

4 ft

• Complete division

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METRIC UNITS OF LINEAR MEASURE

• Based on powers of 10
• Most frequently used units of length
• kilometer
• meter
• centimeter
• millimeter

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METRIC UNITS OF LINEAR MEASURE

• Following list shows several units of linear
  measure and their lengths in meters

1 kilometer (km) 1,000 meters (m) 1
hectometer (hm) 100 meters (m) 1
dekameter (dam) 10 meters (m) 1 meter
(m) 1 meter (m) 1
decimeter (dm) 0.1 meter (m) 1
centimeter (cm) 0.01 meter (m) 1
millimeter (mm) 0.001 meter (m)
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CONVERTING METRIC LENGTHS

• Since the metric system is based on powers of
  ten, the easiest method of converting between
  metric units is by moving the decimal point
• To express a given unit as a larger unit, move
  decimal point to left the number of places
  between the two units
• To express a given unit as a smaller unit, move
  decimal point to right the number of places
  between the two units

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CONVERTING METRIC LENGTHS

• Express 75 decimeters as meters

• Move decimal 1 place to left since a meter is
  next larger unit to a decimeter

75 dm 7.5 m Ans

• Express 1.95 centimeters as millimeters

• Move decimal 1 place to right since a millimeter
  is next smaller unit to a centimeter

1.95 cm 19.5 mm Ans
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SURFACE AREA (ENGLISH)

• Two-dimensional computed as product of two
  linear measures and expressed in square units
• Several equivalent units of area measure in the
  English measurement system are

1 square foot (sq ft) 144 square inches (sq
in) 1 square yard (sq yd) 9 square feet (sq
ft) 1 square rod (sq rd) 30.25 square yards
(sq yd) 1 acre (A) 160
square rods (sq rd) 1 acre (A)
43,560 square feet (sq ft) 1 square mile (sq
mi) 640 acres (A)
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SURFACE AREA CONVERSIONS

• Unity fractions are used for surface area
  conversions in same manner as for linear
  conversions
• Express 15 square yards as square feet

• Use 1 square yard 9 square feet equivalence to
  set up unity fraction

• Square yards must cancel out
• We will make our unity fraction

140 sq ft Ans
Note Rounded to 2 significant digits
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SURFACE AREA (METRIC)

• As with linear measure, the easiest method of
  converting between metric units of surface area
  is by moving the decimal point
• To express given unit as next larger unit, move
  decimal point two places to left. Note Move two
  places instead of one to account for the fact
  that surface area is two-dimensional
• To express given unit as next smaller unit, move
  decimal point two places to right

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METRIC AREA CONVERSIONS

• Table below reviews some common metric units,
  listing them from larger to smaller in order from
  left to right
• Express 84.5 square centimeters as square
  decimeters

• Square dekameter is next smaller unit to a square
  hectometer
• Move decimal point 2 places to right
• 5.6 hm2 560 dam2 Ans

• Express 5.6 square hectometers as square
  dekameters

• Square decimeter is next larger unit to a square
  centimeter
• Move decimal point 2 places to left
• 84.5 cm2 .845 dm2 Ans

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ENGLISH VOLUME UNITS

• Three-dimensional computed as product of three
  linear measures and expressed in cubic units
• Equivalent units of volume measure in English
  measurement

1 cubic foot (cu ft) 1,728 cubic inches (cu
in) 1 cubic yard (cu yd) 27 cubic feet (cu ft)
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ENGLISH VOLUME UNITS

• Use unity fractions to express 12,500 cubic
  inches as cubic yards
• No direct equivalence between cubic inches and
  cubic yards
• We will have to use more than one unity fraction

.268 cu yd Ans
Note Rounded to 3 significant digits.
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METRIC VOLUME UNITS

• As with linear and area measure, easiest method
  of converting between metric units of volume is
  by moving decimal point and following the rules
  discussed earlier. Now, however, decimal will be
  moved three places in order to account for all
  three dimensions
• Express 38,500 cubic millimeters as cubic
  decimeters

• Since cubic decimeter is 2 units larger than
  cubic millimeter, decimal point is moved 2 3 or
  6 places to left

38,500 mm3 .0385 dm3 Ans
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CAPACITY AND WEIGHT

• Unity fractions and tables below are used in
  examples on following slide to convert between
  various units of capacity and weight

English/Metric Units of Capacity 16 ounces (oz)
1 pint (pt) 2 pints (pt) 1 quart
(qt) 4 quarts (qt) 1 gallon (gal) 1
gallon (gal) 231 cubic inches 7.5 gallons
(gal) 1 cubic foot (cu ft) 1000 milliliters
(mL) 1 liter (L) 1 milliliter (mL) 1 cubic
centimeter 1000 liters (L) 1 cubic meter
(m3)
English/Metric Units of Weight 16 ounces (oz)
1 pound (lb) 2000 pounds (lb) 1 short
ton 2240 pounds (lb) 1 long ton 1000 milligram
(mg) 1 gram (g) 1000 grams (g) 1 kilogram
(kg) 1000 kilograms 1 metric ton
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CAPACITY/WEIGHT CONVERSIONS

• Express 1.5 quarts as cubic inches
• No direct conversion between quarts and cubic
  inches, so more than one unity fraction will be
  needed

87 cu in Ans
Note Rounded to 2 significant digits

• Express 200 cm3 as liters

0.2 L Ans
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COMPOUND UNITS

• Products or quotients of two or more basic units
  such as mi/hr or lb/sq in
• Unity fractions used to convert compound units by
  placing units on opposite sides of fraction so
  that they cancel out
• Express 60 miles/hour as feet/second

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PRACTICE PROBLEMS

• Convert numbers 13 as indicated. Round each
  answer to the same number of significant digits
  as in the original quantity
• 1. 34.52 feet as yards
• 2. 2.36 miles as feet
• 3. 4.5 yards as inches

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PRACTICE PROBLEMS

• Perform the indicated operations in problems
  411. Simplify all answers
• 4. 1 mi 250 rd 3 mi 225 rd
• 5. 7 ft 2 in 4 ft 9 in
• 6. 4 rd 14 ft 2 rd 12.5 ft 3 rd 7 ft
• 7. 5 yd 1 ¾ ft 2 yd 2 ½ ft
• 8. 4 yd 2 ft 4
• 9. 5 mi 110 rd ? 3
• 10. 6 mi 210 rd 7.5 yd 3
• 11. 6 ft 6 in ? 4

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PRACTICE PROBLEMS
12. Express 63 meters as centimeters 13. Express
4.75 dekameters as kilometers 14. Express 0.0076
hectometers as millimeters 15. Express 49.3
decimeters as dekameters 16. Express 163.2 square
inches as square feet 17. Express 543 square rods
as square yards 18. Express 75 square miles as
acres 19. Express 0.084 square decimeters as
square centimeters 20. Express 1.09 square
dekameters as square hectometers
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PRACTICE PROBLEMS
21. Express 2.5 cubic feet as cubic inches 22.
Express 123,500 cubic inches as cubic yards 23.
Express 1,260 cm3 as dm3 24. Express 1.268 cubic
meters as cubic centimeters 25. Express 13.5
pints as gallons 26. Express 4.6 cubic feet as
cubic inches 27. Express 18.2 ft/min as mi/hr 28.
Express 1.2 short tons/cubic foot as pounds/cubic
inch
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PROBLEM ANSWER KEY