Part 219

3-1/3 is 1/30 6-1/4 is 1/16 8-1/3 is 1/12 12-1/2 is 1/8 16-2/3 is 1/6 18-3/4 is 3/16 20 is 1/5 25 is 1/4 31-1/4 is 5/16 33-1/3 is 1/3 37-1/2 is 3/8 40 is 2/5 41-2/3 is 5/12 43-3/4 is 7/16 50 is 1/2 56-1/4 is 9/16 58-1/3 is 7/12 60 is 3/5 62-1/2 is 5/8 66-2/3 is 2/3 68-3/4 is 11/16 75 is 3/4 80 is 4/5 81-1/4 is 13/16 83-1/3 is 5/6 87-1/2 is 7/8 91-2/3 is 11/12 93-3/4 is 15/16

This table embodies all the aliquot parts of 100 and their equivalent fractions which are generally used in practical calculations.

PROBLEMS IN GRAIN, STOCK, COTTON, COAL, HAY, LUMBER, ETC.

To find the value of articles sold by the unit, hundred or thousand.

RULE.--Multiply the quantity by the price, or vice versa, and point off the proper number of decimal places in the result.

Find the cost of a bale (518 pounds) of cotton at 7-3/8c per pound.

518 × .07 = 36.26 „ .00-3/8 = 1.94-1/4 --------- _Ans._ $38.20-1/4

At 7c (.07) per pound, 518 pounds cost $36.26; at 3/8c, $1.94-1/4. For 3/8 of 518, multiply by 3, and divide product by 8.

Find cost of a lot of hogs, weighing 8740 pounds, at $4.35 per hundredweight.

87.40 4.35 -------- 380.1900

The price being $4.35 per 100 pounds and as in 8740 pounds there are 87.40 hundredweight, four decimal places are pointed off. _Ans._ $380.19.

Find the cost of 2864 feet of lumber, at $17-1/4 per 1000 feet.

Price being dollars per 1000, point off three places. (2.864 × 17-1/4 = 49.404.) _Ans._ $49.40.

To find the value of articles sold by the ton (2000 pounds).

RULE.--Multiply the weight by the price and take half of the product.

Find the cost of 2680 pounds of hay, at $11-1/2 per ton.

Point off three places, when price is dollars; five if dollars and cents. (2680 × 11-1/2 = 30820; 30820 ÷ 2 = 15.410.) _Ans._ $15.41.

When the long ton of 2240 pounds is used.

RULE.--Multiply the weight by the price and divide the product by 2.240.

Find the cost of 4800 pounds coal, at $6-3/4 per long ton. (4800 × 6-3/4) ÷ 2.24 = $14.46, _Ans._

To find the cost of grain, when the price per bushel and weight is given.

RULE.--Reduce the weight to bushels, and multiply by the price.

Find the cost of 3570 pounds of shelled corn, at 36c per bushel.

56)3570(63.75 bu. .36 ------ _Ans._ $22.9500

To reduce pounds of shelled corn to bushels, divide by 56. At 36c per bushel, 63.75 bushels come to $22.95.

Find cost of 2900 pounds of wheat, at 57c per bushel.

To reduce pounds of wheat to bushels divide by 60. 2900 ÷ 60 = 48-1/3 bushels; 48-1/3 × .57 = $27.55, _Ans._

In computing the value of grain, the operation can often be abbreviated by cancellation.

RULE.--Write the weight and price per bushel, on the right of a vertical line, and the number of pounds to the bushel on the left. Then cancel common factors, as explained above.

Find the cost of 3230 bushels of wheat, at 72c per bushel.

60 ¦ 3230 ¦ 72 12 323 × 12 = 38.76

Here we cancel the 0’s on both sides; then, 6 and 72, which leaves 323 and 12. Their product being the answer.

At 28c per bushel, what will 4080 pounds of oats cost?

32 ¦ 4080 510 4 ¦ 28 7 ------ _Ans._ $35.70

Oats, 32 pounds to the bushel. See table, page 861. Cancel 32 and 4080, then, 4 and 28, leaving the factors 510 and 7.

Other short cuts for computing cost of merchandise, produce, etc.

Find cost of 26-1/2 dozen eggs, at 18-1/2c a dozen.

26 × 18 = 4.68 1/2 of 44 = .22 --------- 1/2 × 1/2 = 1/4 4.90-1/4 _Ans._ $4.90.

When both fractions are 1/2. To product of the whole numbers, add 1/2 of their sum, and annex 1/4 to answer.

Of 53-3/4 pounds of butter, at 28-3/4c per pound.

53 × .28 = 14.84 3/4 of 81 = .60-3/4 --------- 3/4 × 3/4 = 9/16 15.45 _Ans._ $15.45-9/16.

To the product of the whole numbers, add 3/4 of their sum, plus the square of 3/4.

Of 13-1/4 yards of flannel, at 31-1/4c per yard.

13 × .31 = 4.03 + .11 = 4.14, _Ans._

To 4.03 add .11, 1/4 of 44 (13 + 31). The 1/16 (1/4 × 1/4) is disregarded.

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 2 | 2 yards 1 foot 8 inches. -------+------------------------- 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 |3 miles 57 rods || inches. +----------------/\------- | 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: SOLUTION: Yards Feet Inches 10 8 10 10 yards = 10 × 3 feet = 30 feet. 30 feet and 8 3 feet are 38 feet. --- 38 38 feet = 38 × 12 inches, or 456 inches. 456 12 inches + 10 inches = 466 inches. --- 456 10 --- 466

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.) 5-1/2 yards, or 16-1/2 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 = 1.1526-2/3 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} {of Latitude on a Meridian, } = 1 degree {or of Longitude 69.16 statute miles } {on the Equator.

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.) 30-1/4 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 1-1/2, 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 = 8-4/7, 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 2-1/4 feet wide and the strips run lengthwise.

SOLUTION:

16 ÷ 2-1/4 = 7-1/9. 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 62-1/2 rods wide?

80 × 62-1/2 = 5000 square rods; 5000 ÷ 160 = 31-1/4 acres.

_Ans._ 31-1/4 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

272-1/2 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 177-1/8), and cut off six decimals.

1 vara = 33-1/3 inches

1,900.8 varas = 1 mile.

THE MEASURE OF SOLIDS, OR CUBIC MEASURE

Just as the rectangle is the chief surface considered in arithmetic, so the rectangular solid is the chief solid body.

A rectangular solid is bounded by six rectangular surfaces, each opposite pair of rectangles being equal and parallel to each other.

A rectangular solid thus has three dimensions--length, breadth, and thickness.

If the length, breadth, and thickness are all equal to one another, the solid is called a cube. Hence, a cubic foot, the unit of volume, is a solid body whose length, breadth, and thickness are each a linear foot. Similarly, a cubic inch measures one linear inch in length, breadth, and thickness; and a cubic yard measures one linear yard in length, breadth, and thickness.

The number of cubic feet (or inches, or yards) in the volume of a rectangular solid is equal to the number of linear feet (or inches, or yards) in the length, multiplied by the number of linear feet (or inches, or yards) in the breadth, multiplied by the number of linear feet (or inches, or yards) in the thickness.

This is usually abbreviated into

Length × breadth × thickness = volume, or cubic content.

For example, suppose the solid in the diagram is 10 feet in length, 8 feet in breadth, and 5 feet in thickness. It is clear that the solid can be cut into five slices, each 1 foot thick, by planes parallel to the bottom. But, the bottom contains 10×8 square feet and above each square foot there is a cubic foot. Thus, each slice contains 10×8 cubic feet. Therefore, since there are five slices, the whole solid contains 10×8×5, or 400 cubic feet.

Since length × breadth × thickness = cubic content, it follows that, if we know any three of these four quantities, we can find the fourth.

The student should remember that

(a) A cubic foot of water weighs 1000 ounces (avoirdupois) approximately.

(b) A gallon of pure water weighs 10 pounds (avoirdupois).

We have thus a relation between weight, capacity, and cubic content.

TABLE OF CUBIC MEASURE

1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) 128 cubic feet = 1 cord (C.)

Cubic Cubic Cubic Yard Feet Inches 1 = 27 = 46656

A _cord_ of wood or stone is a pile 8 feet long, 4 feet wide, and 4 feet high.

A pile of wood 4 feet high, 4 feet wide and 1 foot long makes a _cord foot_. 8 cord feet = 1 cord.

A _perch_ of stone or masonry is 16-1/2 feet long, 1-1/2 feet thick, and 1 foot high, and contains 24-3/4 cubic feet.

A _cubic yard_ of earth is considered a _load_.

Brick work is commonly estimated by the thousand bricks.

Bricklayers, masons, and joiners commonly make a deduction of one half the space occupied by windows and doors in the walls of buildings.

In computing the contents of walls, masons and bricklayers multiply the entire distance around on the outside of the wall by the height and thickness. The corners are thus measured twice.

A cubic foot of distilled water at the maximum density, at the level of the sea, and the barometer at 30 inches, weighs 62-1/2 pounds or 1000 ounces avoirdupois.

By actual measurements, it has been found that a bushel, dry measure, contains about 1-1/4 cubic feet. This makes it easy to estimate about how many bushels any bin will hold.