Work:YardsFeetInches108103381245610466Solution: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.
Work:YardsFeetInches108103381245610466
Work:
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 is used in measuring lines and distances.
There are two systems in use in the United States, theEnglish Systemand theFrench System. The English system is the one commonly used, while the French system is used in making scientific measurements. (See under Metric System.)
Architects, carpenters, and mechanics frequently write ′ for foot, and ′′ for inch. Thus 8′ 7′′ means 8 feet 7 inches.
Other measures of length are:
The knot is used in measuring distances at sea. It is equivalent to 1 minute of longitude at the equator.
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.
The following measures of length are also used:
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.
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
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.
Facts about Wall Paper:
(1) Wall paper in this country is1⁄2yard 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 thenumber of yards needed, approximately.
(5) Divide the number of yards by 8, and the result is thenumber 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:
Facts about Carpets:
(1) Carpets are usually3⁄4yard 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 21⁄4feet 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.
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 621⁄2rods wide?
80 × 621⁄2= 5000 square rods; 5000 ÷ 160 = 311⁄4acres.
Ans.311⁄4acres.
(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
Table of Surveyors’ Square Measure
Texas Land Measure(Also used in Mexico, New Mexico, Arizona, and California)
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 = 331⁄3inches1,900.8 varas = 1 mile.
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.
One cubic foot
One cubic foot
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 thediagramis 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
Acordof 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 acord foot. 8 cord feet = 1 cord.
Aperchof stone or masonry is 161⁄2feet long, 11⁄2feet thick, and 1 foot high, and contains 243⁄4cubic feet.
Acubic yardof earth is considered aload.
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 621⁄2pounds or 1000 ounces avoirdupois.
By actual measurements, it has been found that a bushel, dry measure, contains about 11⁄4cubic feet. This makes it easy to estimate about how many bushels any bin will hold.
Example: An open tank made of iron1⁄4inch thick, is 4 feet long, 2 feet 6 inches broad, and 2 feet deep, outside measurement. Assuming that iron weighs 7.8 times as much as water, find the weight of the tank.
The external volume of the tank = 2 × 21⁄2× 4 cubic feet = 20 cubic feet.
Since the iron is1⁄4inch thick, the inside length is1⁄2inch less than the outside, the inside breadth is1⁄2inch less than the outside, and the inside depth is1⁄4inch less than the outside.
Therefore the interior volume
=291⁄2× 471⁄2× 233⁄4cubic inches
=59 × 95 × 9516cubic inches
=3327911⁄16cubic inches
Therefore, volume of iron in the tank
=20 cubic feet - 3327911⁄16cubic inches
=12805⁄16cubic inches.
But 1 cubic foot of iron weighs as much as 7.8 cubic feet of water,i. e., 7.8 × 1000 ounces, or 7800 ounces.
∴Weight of tank =12805⁄16× 78001728 × 16pounds.
∴Weight of tank= 361.199 pounds,Ans.
Example: A wood pile is 8 feet high and 40 feet long. The sticks are 4 feet long. How many cords in it?
Solution: Being 8 feet high, it is 2 cords high. 40 feet in length equal 5 cords in length. Hence, the pile contains 2 × 5 cords, or 10 cords.
To estimate a bin:
(1) Find the number of cubic feet in the bin.
(2) Divide the number of cubic feet by 11⁄4.
(3) The result is the number of bushels.
Example: How many bushels will a bin hold, if its inside measurements are, length 20 feet, width 12 feet, depth 8 feet?
Solution: The number of cubic feet in a bin is 8 × 12 × 20, or 1920.
If 1 bushel contains 11⁄4cubic feet, in 1920 cubic feet there are as many bushels as 11⁄4is contained times in 1920, or 1536.
Work:
8 × 12 × 20 = 1920
1920 ÷ 11⁄4= 1536.
The work may be indicated in this way as well:—
8 × 12 × 20 ×4⁄5= 1536.
To get the number of heaped bushels of corn in the ear in a crib:
(1) Multiply the length of the crib in inches by the width in inches.
(2) Multiply the product obtained, by the height of the corn in the crib in inches.
(3) Divide the result by 2748.
Example: How much corn in the ear can I put into a crib 12 feet wide, 20 feet long, and 10 feet deep?
Solution: The number of cubic inches in the crib is 144 × 240 × 120, or 4,147,200.
Since 2748 cubic inches hold 1 bushel, 4,147,200 cubic inches hold as many bushels as 2748 is contained times in 4,147,200, or 1509+ bushels.
Work:
144 × 240 × 1202748= 1509+.
Measures used in telling the extent of room in vessels are calledmeasures of capacity.
There are two kinds of capacity measures, dry measures and liquid measures.
Dry measures are used to measure grain, seeds, and the like.
Liquid measures are used to measure water, milk, oils, etc.
Liquid measures
Liquid measures
A pint, quart, or gallon, dry measure, is more than the same quantity, liquid measure; for a quart, dry measure, is1⁄32of a bushel, or1⁄32of 2150.4 cubic inches, which is about 671⁄5cubic inches, while a quart liquid measure is1⁄4of 231 cubic inches, or 573⁄4cubic inches.
In determining the capacity of cisterns, reservoirs, etc., 311⁄2gallons are considered a barrel (bbl.), and 2 barrels, or 63 gallons, a hogshead (hhd.). In commerce, however, the barrel and hogshead are not fixed measures.
Casks of large size, called tierces, pipes, butts, tuns, etc., do not hold any fixed quantity. Their capacity is usually marked upon them.
The standard gallon of the United States contains 231 cubic inches, and will hold a little over 81⁄3pounds of distilled water. The imperial gallon, now adopted by Great Britain, contains 277.274 cubic inches, or 10 pounds of distilled water, temperature 62 degrees Fahrenheit, the barometer standing at 30 inches.
These measures are used in mixing medicines.
A minim is about 1 drop.
A common Winchester bushel (the standard of the United States) contains 2150.42 cubic inches.
A dry quart contains 67.2 cubic inches.
A liquid quart contains 57.75 cubic inches.
Example1: Reduce 5 bushels 2 pecks 4 quarts 1 pint to pints.