ANALYSIS OF PROBLEMS Directions PROBLEM: To find the altitude of a rectangle whose base is 8 in. and whose area is 152 sq. in. 1. Read carefully, noting that the base and the area of a rectangle are given, and that the altitude is required. 2. Plan: Recall that the area of a rectangle is the product of the numbers expressing its length and breadth. Hence 152 is 8 times the required number and must be divided by 8. 19 3. Computation. 8)152 4. Test. 8 X 19 152. In the solution of every problem there are four main parts: a. Read the problem; note carefully what is given and what is required. b. Plan the work; determine how to find what is required from what is given. c. Make the computations as planned. d. Test the result. 1. Read and determine what is given and what is required: A train runs from New York to Cleveland, 624 mi., in 16 hr. ; find the speed in miles per hour. 2. Plan the work to be done in solving Exercise 1. 3. Make the computation. 16)624 4. What is the test? 5. Read: The area of a triangle is 176 sq. ft.; its altitude is 11 ft.; what is its base? 6. Plan the solution. Thus, in this SUGGESTION.-When the calculation contains more than one process, it is better to indicate all of them in the plan. problem: 1. 2 × 176 the altitude times the base. times the base. The base = 352 ÷ 11 = = = -. 2. 352 = 11 PLANNING PROBLEMS Plan the solution; do not make the computation: 93 1. How many feet per minute is a train moving when traveling 42 mi. per hour? 2. A train left Cincinnati at 8:15 A. M. and arrived at St. Paul, 702 miles distant, at 2:57 A. M. the next day; find its average speed in miles per hour. 3. The base of a rectangle is 16 in., its area is 3 sq. ft.; find the altitude in inches. 4. Show how to find the perimeter of the figure. 5. Draw a figure like this. Draw a line from A to B; show how to find the area of the figure. 6. A man earns $75 a month. 17 ft. 36 ft. 8 ft. B 25 ft. 12 ft. He spends $10 per month for room rent; $3.50 per week for board, $125 per year for clothing and other expenses; in how many years will he save $1,000 at this rate? 7. 40 ft. of wire weighed 1 lb.; what was the weight of 31 mi. of this wire? 8. A, B, and C own a store; A owns 4 of it and B owns as much as A; what part does C own? 9. The base of a rectangle is 17 in.; its perimeter is 48; what is its altitude? 10. A 100-acre farm contains 4 lots. Three of the lots contain 75 acres, 20 acres, and 75 acres respectively; how many acres does the fourth lot contain? NOTE. -Hereafter the planning of a problem should be made a distinct and important feature of its solution. In very simple problems, the plan and the work need not be separated, but any uncertainty as to how a problem is to be worked shows clearly that a separate, correct plan is necessary. NEVER WORK AT RANDOM. 94 SOLVING PROBLEMS 1. of the distance from Detroit to Chicago is 190 mi.; what is the whole distance? PLAN. 1. of the distance = of 190 mi. 2. 3, or the whole distance, = 3 × 95 mi. = mi. 2. A dressmaker paid $1.40 for of a yard of velvet; what was the price per yard? PLAN. 1. yd. cost of $1.40 Plan and solve: $ = $—. 3. When 13 ounces of cinnamon cost $.65, what is the cost of cinnamon per pound? 3 4. In 1900, of the population of the United States was 10 millions. Find the population. 5. $24 is of the cost of a wagon; what is its cost? What is of its cost? 8 15 6. of a ton of coal costs $7.35; what is the cost of a ton? Of of a ton? 7. of the number of passengers on a street-car is 12; how many are there in the car? If of the whole number are women, how many are women? 8. of the population of Chicago in 1900 was 500,000; what was the population? 20 9.3 of the population of Pittsburg in 1900 was 48,000; what was the population? What was of it? 10. of the total newspaper product in the United States in 1900 was 60,000 tons; what was the total product? II. At an entertainment of the audience sat on the main floor and 360 in the gallery; how many people were there in the audience? 12. A society of 300 men march in a Fourth of July procession. They march 4 abreast in ranks 6 ft. apart; how long is the procession which they form? PROBLEMS-LAND 95 1. How many miles are there in the distance around the block of land as shown in the picture? How many rods are there in this distance (1 mi. = 320 rods)? The land is 2. What is the shape of the Cass farm? divided into fourths, eighths, and sixteenths. How many rods are there in its length? In its breadth? 3. How many rods of line-fence are there between the Cass farm and the French farm? Between the Cass farm and the Rich farm? 4. Make and answer 5 other questions about the fences. 5. What part of the square mile does the Cass farm occupy? The Rich farm? The Ross farm? The Smith farm? A square mile of land contains 640 acres and is called by surveyors a section. 6. How many acres are there in the French farm? In the Smith farm? In the Burns farm? 7. Make and solve 3 similar problems. 8. How many acres are there in a quarter-section? Which of the above farms occupy a quarter-section? 9. From the dimensions in rods of the Smith farm find its area in square rods. How many acres does it contain? How many square rods are there in an acre? 10. What farms occupy the east half (E) of the section? Which the north half (N)? Which the S W 1? 96 PROBLEMS-LAND MEASURE In surveying the newer States, the government surveyors divided the land as nearly as possible into townships 6 miles square. The townships are slightly narrower at the top than at the bottom on account of the curvature of the earth. Each township is divided into 36 parts called sections and these are recorded by number as shown in the figure. The township is regarded as square for all computations. 1. How many square miles are there in a township? 2. In every township the government set apart section number 16 for school purposes; how many acres is this? In a locality where such land sold for $25 an acre, what was the gift worth? 3. Section 31 is the one shown on p. 95; what part of the area of a township is this section? How many acres are there in a township? 4. How many acres are there in a row of sections in a township? In a column? 5. What will be the cost of a line-fence between section 9 and section 10 at $1.75 a rod? 6. If half of a township is timber land worth $10 an acre and the rest is farming land worth $25 an acre, what is the value of the township? 7. Make and solve 3 other problems about land. |