Denominate Numbers
Simple denominate numbers.—When we speak of measures, whether they are of money, extension, time, or weight, we use terms like 5 dollars, 4 yards, 3 hours, or 10 pounds to express the quantity we are talking about.
Sometimes we use two or more terms or names to express the measure, as 3 hours, 15 minutes, 10 seconds; 4 gallons, 3 quarts, 1 pint. These are compound denominate numbers.
The chief differences between compound numbers and simple numbers is, that with the exceptions of United States money, and the metric system of weights and measures, the denominations of compound numbers do not increase or decrease by the scale of ten.
Reduction.—Reduction of Compound Numbers is the process of changing them from one denomination to another without altering their value.
Reduction Descending is changing the denomination of a number to another that is lower, as: 2 hours = 120 minutes; 2 feet = 24 inches.
Reduction Ascending is changing the denomination of a number to another that is higher, as: 120 minutes = 2 hours; 24 inches = 2 feet.
Rules for Addition of Denominate Numbers
First.—Write the names of the different units to be used in addition, placing them in a horizontal row, the largest to the left.
Next.—Write the numbers of each unit to be added, below the names of the units, each in its proper place.
Then.—Add and place each sum below the column added.
Example: Add 7 hours 15 minutes 30 seconds, 9 hours 30 minutes 40 seconds, and 11 hours 40 minutes 32 seconds.
Work:
| hours | minutes | seconds |
|---|---|---|
| 7 | 15 | 30 |
| 9 | 30 | 40 |
| 11 | 40 | 32 |
| 28 | 26 | 42 |
Explanation: 32 seconds + 40 seconds + 30 seconds = 102 seconds. But, 102 seconds = 1 minute 42 seconds. Write the 42 below and carry the 1 minute. 1 minute (carried) + 15 minutes + 30 minutes + 40 minutes = 86 minutes. But, 86 minutes = 1 hour 26 minutes. Write the 26 and carry the 1 hour. 1 hour + 11 hours + 9 hours + 7 hours = 28 hours. Result = 28 hours 26 minutes 42 seconds.
Subtraction of Denominate Quantities
Example: Subtract 6 tons 12 cwt. 9 pounds 10 ounces from 15 tons 7 cwt. 13 pounds 9 ounces.
Work:
| Tons | Cwt. | Pounds | Ounces |
|---|---|---|---|
| 15 | 7 | 13 | 9 |
| 6 | 12 | 9 | 10 |
| 8 | 15 | 3 | 15 |
Explanation: (1) Place as in addition of denominate quantities. 10 ounces cannot be taken from 9 ounces, so we must take 1 pound from the 13 pounds and add it to the nine ounces. 16 ounces + 9 ounces = 25 ounces. 25 - 10 = 15. Write the 15 below.
(2) Now there are only 12 pounds left to take the 9 from. 12 - 9 = 3. Write the 3 below.
(3) 12 is larger than 7. 1 ton + 7 cwt. = 27 cwt. 27 - 12 = 15. Write the 15 below.
(4) 14 - 6 = 8. Write the 8 below.
(5) Result = 8 tons 15 cwt. 3 pounds 15 ounces.
Multiplication of Denominate Quantities
Example: Multiply 21 yards 2 feet 11 inches by 6.
Work:
| Yards | Feet | Inches |
|---|---|---|
| 21 | 2 | 11 |
| 6 | ||
| 131 | 2 | 6 |
Explanation: (1) 6 × 11 inches = 66 inches = 5 feet 6 inches. Write the 6 below and carry the 5.
(2) 6 × 2 feet = 12 feet. 12 feet + 5 feet (carried) = 17 feet, or 5 yards 2 feet. Write the 2 below and carry the 5.
(3) 6 × 21 yards = 126 yards. 126 yards + 5 yards = 131 yards.
(4) Result = 131 yards 2 feet 6 inches.
Division of Denominate Quantities
Problem: Divide 3 years 9 months 4 days by 12.
Work
| Years | Months | Days | Hours | ||
|---|---|---|---|---|---|
| 12 | ) | 3 | 9 | 4 | 0 |
| 0 | 3 | 22 | 20 | ||
Explanation: (1) We cannot divide 3 by 12, so we reduce 3 years to months. 3 years = 36 months. 36 months + 9 months = 45 months. 45 ÷ 12 = 3, and a remainder 9. Write the 3 and carry the remainder 9.
(2) 9 months (carried) = 270 days. 270 days + 4 days = 274 days. 274 ÷ 12 = 22, and a remainder 10. Write the 22 and carry the 10.
(3) 10 days = 240 hours. 240 ÷ 12 = 20. Write the 20.
(4) Result = 3 months 22 days 20 hours.
Reduction Ascending
Rules: 1. Divide the given denomination by the number which will reduce it to the next higher denomination. Divide the quotient in the same manner, and continue the operation until the entire quantity is reduced.
2. To the last quotient annex the several remainders in their proper order. The result will be the answer.
Example: Reduce 201458 inches to higher denominations.
| Work: | Solution: | |||
|---|---|---|---|---|
| 12 | 201458 inches | 201458 inches = 16788 feet 2 inches. | ||
| 3 | 16788 feet 2 inches | 16788 feet 2 inches = 5596 yards 2 inches. | ||
| 5 | 1⁄2 | 5596 yards | 5596 yards 2 inches = 1017 rods 2 yards 1 foot 8 inches. | |
| 2 | 2 | |||
| 11 | 11192 half yards | |||
| 320 | 1017 rods | 5 half yards | 1017 rods 2 yards 1 foot 8 inches = 3 miles 57 rods 2 yards 1 foot 8 inches. | |
| 3 miles 57 rods | | | |||
| 2 yds. 1 ft. 6 in. | ||||
201458 inches = 3 miles 57 rods 2 yards 1 foot 8 inches.
Reduction Descending
Rules: 1. Write the given quantity in the order of its denominations, beginning with the highest, and supply vacant denominations with ciphers.
2. Multiply the highest denomination by the number which will reduce it to the next lower denomination, and add to the product the units of the lower denomination, if there be any.
3. Proceed in the same manner until the entire quantity is reduced to the required denomination.
Example: Reduce 10 yards 8 feet 10 inches to inches.
Work:
| Yards | Feet | Inches |
|---|---|---|
| 10 | 8 | 10 |
| 3 | ||
| 38 | ||
| 12 | ||
| 456 | ||
| 10 | ||
| 466 | ||
Solution:
10 yards = 10 × 3 feet = 30 feet. 30 feet and 8 feet are 38 feet.
38 feet = 38 × 12 inches, or 456 inches. 456 inches + 10 inches = 466 inches.
Note.—To prove the above work, use reduction ascending, beginning with the result.
Long or Linear Measure
Long or linear measure is used in measuring lines and distances.
There are two systems in use in the United States, the English System and the French System. The English system is the one commonly used, while the French system is used in making scientific measurements. (See under Metric System.)
Table of Long Measure
| 12 inches (in.) | = | 1 foot (ft.) |
| 3 feet | = | 1 yard (yd.) |
| 51⁄2 yards, or 161⁄2 feet | = | 1 rod (rd.) |
| 320 rods, or 5280 feet | = | 1 mile (mi.) |
| 1760 yards | = | 1 mile |
| mi. | rd. | yd. | ft. | in. | ||||
| 1 | = | 320 | = | 1760 | = | 5280 | = | 63360 |
Architects, carpenters, and mechanics frequently write ′ for foot, and ′′ for inch. Thus 8′ 7′′ means 8 feet 7 inches.
Other measures of length are:
| 1 hand | = | 4 in. | Used in measuring the height of horses. |
| 1 fathom | = | 6 ft. | Used in measuring depths at sea. |
| 1 knot, nautical or geographical mile | = | 1.15262⁄3 miles or 6086 feet. | |
The knot is used in measuring distances at sea. It is equivalent to 1 minute of longitude at the equator.
Surveyors’ Linear Measure
| 7.92 | inches | = | 1 link (l.) |
| 25 | links | = | 1 rod (rd.) |
| 4 | rods or 100 links | = | 1 chain (ch.) |
| 80 | chains | = | 1 mile (mi.) |
| mi. | ch. | rd. | l. | in. | ||||
| 1 | = | 80 | = | 320 | = | 8000 | = | 63360 |
The linear unit commonly employed by surveyors is Gunter’s chain, which is 4 rods or 66 feet.
An engineers’ Chain, used by civil engineers, is 100 feet long, and consists of 100 links.
Measures of Length
The following measures of length are also used:
| 3 barleycorns | = | 1 inch. Used by shoemakers. | |||||
| 4 inches | = | 1 hand. Used to measure the height of horses. | |||||
| 6 feet | = | 1 fathom. Used to measure depths at sea. | |||||
| 3 feet | = | 1 pace. | - | Used in pacing distances. | |||
| 5 paces | = | 1 rod. | |||||
| 8 furlongs | = | 1 mile. | |||||
| 1.15 statute miles | = | 1 geographical, or nautical mile. | |||||
| 3 geographical miles | = | 1 league. | |||||
| 60 geographical miles | - | = | 1 degree | - | of Latitude on a Meridian, or of Longitude on the Equator. | ||
| 69.16 statute miles | |||||||
The length of a degree of latitude varies. 69.16 miles is the average length, and is that adopted by the United States Coast Survey.
The standard unit of length is identical with the imperial yard of Great Britain.
The standard yard, under William IV., was declared to be fixed by dividing a pendulum which vibrates seconds in a vacuum, at the level of the sea, at 62 degrees Fahrenheit, in the latitude of London, into 391,393 equal parts, and taking 360,000 of these parts for the yard.
The following denominations also occur: The span = 9 inches; 1 common cubit (the distance from the elbow to the end of the middle finger) = 18 inches; 1 sacred cubit = 21.888 inches.
Surface Measures
Square Measure, used in measuring surfaces, such as cloth, ceilings, floors, etc.; paving, glazing, and stone-cutting, by the square foot; roofing, flooring, and slating by the square of 100 feet.
A surface has two dimensions, length and breadth.
A square is a figure that has four equal sides and four right angles.
The unit of measure for surfaces is a square, each of whose sides is a linear unit. Thus, a square inch is a square, each of whose sides is one inch long; a square foot is a square, each of whose sides is one foot long, etc.
The area of a square is the product of two of its sides. Thus, the area of a surface 3 feet square is 3 × 3 = 9 square feet.
Hence, to find the area of a rectangle:
Rule.—Multiply the length by the breadth expressed in units of the same denomination.
As the area of a rectangle is found by taking the product of the numbers representing its length and breadth, it is evident that if the area be divided by either of those numbers, the quotient will be the other number. Hence, to find either side of a rectangle when its area and the other side are given:
Rule.—Divide the area by the given side. The quotient will be the required side.
Table of Square Measure
| 144 | square inches (sq. in.) | = | 1 square foot (sq. ft.) |
| 9 | square feet | = | 1 square yard (sq. yd.) |
| 301⁄4 | square yards | = | 1 square rod (sq. rd.) |
| 160 | square rods | = | 1 acre (A.) |
| 640 | acres | = | 1 square mile (sq. mi.) |
Sq. ′ and sq. ′′ are frequently used for square foot and square inch. Thus, 15 sq.′ 6 sq.′′ means 15 square feet 6 square inches.
A square is 100 square feet. It is used in measuring roofing.
Practical Application of Square Measure
Papering
Facts about Wall Paper:
(1) Wall paper in this country is 1⁄2 yard wide, and comes in rolls 8 yards long, or in double rolls, 16 yards long.
(2) It is sold by the roll only.
(3) Bordering is sold by the linear yard.
(4) Make liberal allowances for waste in matching figures.
(5) If the border is wide, the strips need not extend to the ceiling.
Rules for Measuring:
(1) Measure the distance around the room in feet.
(2) Deduct the width of doors and windows.
(3) Divide the difference by 11⁄2, and the quotient will be the number of strips needed.
(4) Multiply the number of strips by the number of yards in a strip, and the product is the number of yards needed, approximately.
(5) Divide the number of yards by 8, and the result is the number of single rolls needed.
Example: A room of ordinary height, 16 feet by 24 feet, has three windows and 2 doors, each 4 feet wide. How many rolls of paper are needed to paper the sides?
Solution:
| Distance around the room | = | 80 feet |
| Width of doors and windows | = | 20 feet |
| After deducting for doors and windows | 60 feet | |
| 60 ÷ 7 = 84⁄7, or 9 double rolls. | ||
Carpeting
Facts about Carpets:
(1) Carpets are usually 3⁄4 yard wide and are sold by the linear yard.
(2) Always draw a diagram of the floor or stairs to be covered.
(3) The number of yards required depends on which way the strips run—whether lengthwise or across the room. Sometimes by running the strips lengthwise, there is less waste in matching the pattern.
(4) The part cut off in matching patterns is charged to the purchaser.
Rules for Estimating:
The number of yards required will be the number of yards in a strip (including the waste for matching), multiplied by the number of strips.
Example: What is the cost of carpeting a room 16 feet by 24 feet at 85c per yard? The carpet is 21⁄4 feet wide and the strips run lengthwise.
Solution:
16 ÷ 21⁄4 = 71⁄9. Hence, I must buy 8 strips.
24 ÷ 3 = 8, which is the number of yards in a strip.
8 × 8 yards = 64 yards.
64 yards will cost 64 × 85c, or $54.40.
To this must be added the cost of sewing, the laying of the carpet, and the waste in matching the pattern.
Land Measure
Rule.—To find the number of acres in a tract of land, divide the number of square rods by 160, or number of square chains by 10.
Example: (1) How many square rods, also acres, in a field 80 rods long and 621⁄2 rods wide?
80 × 621⁄2 = 5000 square rods; 5000 ÷ 160 = 311⁄4 acres.
Ans. 311⁄4 acres.
(2) In tract, 79 chains 84 links (79.84 chains) by 41 chains 25 links (41.25 chains)?
79.84 × 41.25 = 3293.4 square chains; 3293.4 ÷10 = 329.34 acres. Ans. 329.34 acres.
Table showing one side of a Square Tract or Lot containing
| 1 | acre | = | 208 | .7 | feet | = | 43,560 | square feet | |
| 1 | 1⁄2 | acres | = | 255 | .6 | feet | = | 65,340 | square feet |
| 2 | acres | = | 295 | .2 | feet | = | 87,120 | square feet | |
| 2 | 1⁄2 | acres | = | 330 | feet | = | 108,900 | square feet | |
| 3 | acres | = | 361 | .5 | feet | = | 130,680 | square feet | |
| 5 | acres | = | 466 | .7 | feet | = | 217,800 | square feet | |
| 10 | acres | = | 660 | feet | = | 435,600 | square feet | ||
| 1⁄10 | acre | = | 66 | feet | = | 4,356 | square feet | ||
| 1⁄8 | acre | = | 73 | .8 | feet | = | 5,445 | square feet | |
| 1⁄6 | acre | = | 85 | .2 | feet | = | 7,260 | square feet | |
| 1⁄4 | acre | = | 104 | .4 | feet | = | 10,890 | square feet | |
| 1⁄3 | acre | = | 120 | .5 | feet | = | 14,520 | square feet | |
| 1⁄2 | acre | = | 147 | .6 | feet | = | 21,780 | square feet | |
| 3⁄4 | acre | = | 180 | .8 | feet | = | 32,670 | square feet |
Table of Surveyors’ Square Measure
| 2721⁄2 | square feet = 1 square rod |
| 16 | square rods = 1 square chain |
| 160 | square rods, or 10 square chains = 1 acre |
| 640 | acres = 1 square mile, or section |
| 36 | square miles, or 36 sections = 1 township |
Texas Land Measure
(Also used in Mexico, New Mexico, Arizona, and California)
| 26,000,000 | square varas (square of | 5,099 | varas) | = | 1 league and 1 labor | = | 4,605 | .5 | acres | ||
| 1,000,000 | square varas (square of | 1,000 | varas) | = | 1 labor | = | 177 | .136 | acres | ||
| 25,000,000 | square varas (square of | 5,000 | varas) | = | 1 league | = | 4,428 | .4 | acres | ||
| 12,500,000 | square varas (square of | 3,535 | .5 | varas) | = | 1⁄2 league | = | 2,214 | .2 | acres | |
| 8,333,333 | square varas (square of | 2,886 | .7 | varas) | = | 1⁄3 league | = | 1,476 | .13 | acres | |
| 6,250,000 | square varas (square of | 2,500 | varas) | = | 1⁄4 league | = | 1,107 | .1 | acres | ||
| 7,225,600 | square varas (square of | 2,688 | varas) | = | 1,280 | acres | |||||
| 3,612,800 | square varas (square of | 1,900 | .8 | varas) | = | 1 section | = | 640 | acres | ||
| 1,806,400 | square varas (square of | 1,344 | varas) | = | 1⁄2 section | = | 320 | acres | |||
| 903,200 | square varas (square of | 950 | .44 | varas) | = | 1⁄4 section | = | 160 | acres | ||
| 451,600 | square varas (square of | 672 | varas) | = | 1⁄8 section | = | 80 | acres | |||
| 225,800 | square varas (square of | 475 | varas) | = | 1⁄16 section | = | 40 | acres | |||
| 5,645 | .376 | square varas (square of | 75 | .137 | varas) | = | 4,840 square yards | = | 1 | acre | |
To find the number of acres in any number of square varas, multiply the latter by 177 (or to be more exact, by 1771⁄8), and cut off six decimals.
1 vara = 331⁄3 inches
1,900.8 varas = 1 mile.