MULTIPLICATION.

455. In the multiplication of duodecimals, the product of two dimensions is area or surface, and the product of three dimensions is solidity or volume. (344, 349.)

WRITTEN EXERCISES.

456. 1. Multiply 9 ft. 8' by 4 ft. 7'.

OPERATION.

9 ft.

8'

4 ft. y'

ANALYSIS.-Begin at the right. 8′ × 7′56′′ = 4' 8". Write the 8" one place to the right, reserving the 4' to add to the next product. Then 9 ft. x 7' = 63′; 63′ + 4′ = 67′ = 5 ft. 7', 5 ft. 7' 8" which write in the places of feet and primes. 38 ft. 8' Next multiply by 4 feet; 8' x 4 ft. = 32' = 2 ft. 8'. Write the 8' in the place of primes, reserving the 2 ft. to add to the next product. 36 ft.; 36 ft.+ 2 ft. =38 ft., which write in the place of feet. Adding the partial products, the sum equals 44 ft. 3' 8", the product required.

44 ft.

3' 8"

Then 9 ft. x 4 ft.

=

2. How many square feet in 4 boards, each 12 ft. 9' long, and 1 ft. 4' wide?

RULE.-I. Write the terms of the multiplier under the corresponding terms of the multiplicand.

II. Multiply each term of the multiplicand by each term of the multiplier, beginning with the lowest order of units in each. Reduce each product to higher denominations when possible, and write in their proper places. The sum of the partial products will be the product required.

3. Multiply 10 ft. 6' 4" by 5 ft. 3' 8".

4. Find the area of a floor 14 ft. 8' wide and 16 ft. 5' long. 5. What are the solid contents of a block of marble 6 ft. 10' long, 4 ft. 3' wide, and 1 ft. 9' thick?

9. DUODECIMALS.

DENOMINATE NUMBERS-CONTINUED.

8. LONGITUDE AND TIME.

1. How PER

FORMED.

2. UPON WHAT PRINCIPLES BASED.

2. First Meridian.

1. Product of two dimensions.

5. MULTIPLICATION. 2. Product of three dimensions.

3. Rule, I, II.

MEASUREMENTS

So..

458. Measurements involve a practical application of the Weights and Measures to various operations required in the mechanic arts, and to the common business of life.

RECTANGULAR SURFACES.*

459. A Rectangle is a plane figure bounded by four sides, having all its angles right angles.

It has two dimensions-length and breadth.

Rectangle.

When all its sides are equal, it is called a Square.

460. The Area of a rectangle is the surface included within the lines which bound it, and is expressed by the number of times it contains a given unit of measure.

461. The Unit of Measure for surfaces is a square each side of which is a unit of some known length.

Thus, the unit of square inches is 1 square inch; of square feet, 1 square foot; of square yards, 1 square yard, etc.

* Measurements of plane figures requiring a knowledge of Involution and Evo. lution are treated at the close of this book under the head of "Mensuration."

The diagram represents a square yard, each side of which is 1 yd.

or 3 ft. long, and the whole 18 divided into square feet, 1 sq. ft. being the Unit of Measure. In one row there are 3 sq. ft., in 3 rows there are 3 times 3 sq. ft., or 9 sq. ft. Hence the area of 1 sq. yd. is 9 sq. ft.

So the area of a rectangle formed by 2 adjacent rows, is expressed by 3 sq. ft. × 2, or 6 sq. ft.

462. To find the area of a rectangle:

RULE. Find the product of the numbers denoting the length and breadth, expressed in the same denomination; the result is the area.

463. To find either dimension of a rectangle : RULE.-Divide the area by one dimension; the quotient is the other.

464. Artificers compute their work by linear, superficial or square, and cubic measures.

1. Glazing and stone-cutting are estimated by the square foot. 2. Plastering, paving, ceiling, etc., commonly by the square foot, or the square yard.

3. Roofing, flooring, partitioning, slating, etc., generally by the square of 100 square feet; sometimes by the square foot, or square yard.

4. One thousand shingles, averaging 4 in. wide, and laid 5 in. to the weather, are estimated to cover a square.

5. Bricklaying is generally estimated by the thousand bricks.

WRITTEN EXERCISES.

465. 1. How many square feet in a floor 27 ft. long and 21 ft. wide? How many square yards?

2. How many feet wide is a hall that is 26 ft. long and contains 195 square feet?

3. What is the length of a lawn that contains 305 sq. yd. and is 45 ft. wide?

Find the area of rectangles having the following

dimensions :

4. 12 yards square.

5. 18 yd. 2 ft. square.

6. 184 rd. by 201⁄2 rd.

7. 5 ch. 14 1. by 6 ch. 25 1.

8. 25 ft. 6 in. by 16 ft. 9 in.

9. 14 yd. 1 ft. 10 in. square.

The area and one dimension given, find the other dimension of the following rectangles :

10. Area 3741 square feet, length 20 ft. 6 in. 11. Area 5 A. 41 P., width 7 chains 25 links. 12. Area 180 sq. yd. 4 sq. ft., width 9 yd. 2 ft. 13. How many square yards in the sides of a room 16 ft. long, 12 ft. 6 in. wide, and 9 ft. 3 in. high?

14. How many planks 12 ft. long and 10 in. wide will be required to floor a room which is 24 ft. by 20 ft.? Find the number of yards in length of carpeting required for rooms of the following dimensions:

15. For a room 24 ft. by 16 ft. 6 in.; carpet 1 yd. wide. 16. For a room 52 ft. by 35 ft.; carpet 2 ft. 4 in. wide. 17. For a room 28 ft. by 23 ft. 9 in.; carpet 30 in. wide. 18. For a room 27 ft. 3 in. by 22 ft. 6 in.; carpet 2 ft6 in. wide.

Find the cost of carpeting rooms, their dimensions, and the width and price of carpet being as follows:

19. Floor, 34 ft. by 18 ft. 6 in., carpet 2 ft. wide, at $.94 a yard.

20. Floor, 30 ft. 3 in. by 22 ft., carpet yd. wide, at $1.08 a yard.

21. Floor, 18 ft. by 16.4 ft., carpet yd. wide, at $21 a yard.