II. COMPOUND NUMBERS Denominate Number. A concrete number that denotes measure is called a denominate number. Thus, $8 and 6 bu. 2 pk. are denominate numbers, but 7 men and 8 chairs are not. Compound Number. A denominate number in which two or more units are expressed is called a compound number. For example, 7 pk. 4 qt. and 2 mi. 8 rd. 5 ft. are compound numbers. Reduction. We may express 4 ft. 3 in. as 51 in., or as 41 ft. Such a change from one form of a denominate number to another without changing the value is called reduction. Illustrative Examples. 1. Reduce 8 ft. to inches. We therefore have to multiply ft. by 102, which gives 102 ft. Dividing 102 by 12, we have 8, and so the result is 8 ft. Reduction to a lower denomination was formerly called reduction descending, while reduction to a higher denomination was called reduction ascending. Terms and definitions like these have little value, the subject of compound numbers having lost its former importance. We speak of 2 ft. 3 in. instead of 24 ft., but we rarely use other such denominations together. Thus we usually speak of 2 lb. 4 oz. as 24 lb. Table of Length. The following is the table of length: 12 inches (in.) = 1 foot (ft.) 3 feet1 yard (yd.) 5 yards, or 16 feet = 1 rod (rd.) 320 rods, or 5280 feet = 1 mile (mi.) The teacher may call attention to such additional facts as these: The hand (4 in.) is used in measuring the height of horses at the shoulder; sailors use the fathom (6 ft.) for measuring depths, and the knot (nautical mile 12 common, or statute, miles = 20 6080 ft.) for distances at sea. Such facts need not be learned. = 1. If you live REVIEW OF TABLE OF LENGTH mi. from school, how many feet must you walk in coming to school? 2. In Ex. 1 how many yards must you walk? 11. If you buy 33 yd. of velvet, how many inches of velvet do you buy? 12. A field is 40 rd. long. Express the length in feet; in yards; as a fraction of a mile. 13. The distance between two railway stations is known to be 9 mi. Express the distance in miles and feet. 14. If a horse is 16 hands high, and a hand is 4 in., what is the height of the horse in inches? in feet and inches? Table of Square Measure. The following is the table of square measure: 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet 1 square yard (sq. yd.) = 30 square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) 640 acres = 1 square mile (sq. mi.) Carpenters, architects, and mechanics often write 8" for 8 in., and 5' for 5 ft. They also use sq." and sq.' for square inches and square feet. In this book all of these forms are used. If a square is 1 in. square, and its area is 1 on a side, it is 1 in. sq. in. If a rectangle is 4 in. long and 2 in. wide, its area is 4 x 2 sq. in., or 8 sq. 1 square inch 1 sq. in. in. 1 in. The pupils may express this in either of these two forms: 1 in. In practical work we simply multiply 4 by 2, thinking, "There are as many square inches as 2 x 4, or 8 sq. in." REVIEW OF SQUARE MEASURES 1. Find the number of square inches in the top of a desk the area of which is 31 sq. ft. 2. The school platform has an area of 31 sq. yd. What is its area in square feet? 3. A flower bed has an area of 2 sq. rd. What is its area in square yards? in square feet? 4. Find the number of square feet in 7 sq. yd.; in 1 sq. rd. MEASURING AND DRAWING 1. Measure this page of the arithmetic. Draw a plan of the page to the scale. As the pupil probably learned in primary arithmetic, to draw to the scale of 1 in. to 100 rd., we represent 100 rd. by 1 in. To draw to the scale, we represent 12 in. by 1 in. This should be explained at the blackboard. 2. Measure the top of your desk. Draw a plan to the scale of 1 in. to 1 ft. 3. Draw a plan of the top of your desk to the scale 4. Measure the length and width of the schoolroom. Draw a plan and write below it the scale that you have used. 5. Measure the size of the paper upon which you make your drawings. Draw a plan to the scale. 6. Measure the size of any other book you are studying. Draw a plan of a page to the scale 7. A drawing has been made of a box cover. The cover is 14 in. long and the drawing is 31 in. long. What scale was used? 8. A drawing of a floor 24 ft. long and 20 ft. wide is 6 in. long and 5 in. wide. What scale was used? 9. A drawing is made of a rectangle. The drawing has an area of 6 sq. in. If a scale twice as large were used, what would be the area of the drawing? Draw two plans of the rectangle, so as to show that the answer is correct. 10. A drawing is made of a lamp 18 in. high, and the drawing is 31 in. high. What scale was used? The teacher should have the pupils make measurements of desks, doors, blackboards, and other objects in the schoolroom and make scale drawings from these measurements. The pupils should gradually come to know the meaning of the working drawings used by the carpenter, the builder, and the manufacturer. Table of Cubic Measure. The following is the table of cubic measure: 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) A perch of stone or masonry is usually considered as 1 rd. long, 1 ft. high, and 11⁄2 ft. thick, and it contains 243 cu. ft. It varies, however, in different parts of the country. A cubic yard of earth is considered a load. If a block is 1 in. on an edge, and is in the form of a cube, it is a 1-inch cube, and the volume is 1 cu. in., as in A. If a block is 1 in. square at the base and 3 in. high, the volume is 3 cu. in., as in B. If a block is 3 in. square at the base, the base contains 3 x 3 sq. in., or 9 sq. in. If the block is 3 in. high, the volume is 3 x 9 cu. in., or 27 cu. in., as in C. If a block is 4 in. long, 3 in. wide, and 2 in. high, the volume is 4 x 3 x 2 cu. in. 24 cu. in. REVIEW OF CUBIC MEASURES 1. If your schoolroom is 30 ft. long, 20 ft. wide, and 12 ft. high, how many cubic feet does it contain? 2. The ordinary brick is 2 in. thick, twice as wide as thick, and twice as long as wide. How many cubic inches in a brick? Find the volumes of boxes of these dimensions: 5. 34 in., 28 in., 16 in. 3. 71 in., 14 in., 18 in. 4. 4 in., 8 in., 10 in. 6. 64 in., 36 in., 18 in. |
