ALGEBRAIC SOLUTION: Let x denote the number of pounds in the weight of the heavier mass. Let y denote the number of pounds in the weight of the lighter mass. Then, x + y denotes the number of pounds in the combined y denotes the number of pounds in the additional weight, = 4. Solve the following problems by both the algebraic and the arithmetical method, and state which of the solutions is the shorter: 4. A man cut for me 3 times as many ash trees as oak trees, and as many hickory trees as ash trees and oak trees together. In all he cut 32 trees. How many of each kind did he cut? 5. A man bought 4 times as many 2¢ stamps as 5¢ stamps, and as many 10¢ stamps as 2¢ stamps. For all he paid $4.95. How many of each kind did he buy? NOTE. Let x denote the number of 5¢ stamps purchased. 6. The side of a rectangular field is twice its breadth and the distance around the field is 240 rd. How many acres does the field contain? 7. I bought a number of books one day, 4 times as many the next day, and 3 books the third day. In all I bought 33 books. How many did I buy each day? 8. A newsboy sold a certain number of papers on Wednesday, twice as many on Thursday, and 3 more on Friday than on Thursday. In all he sold 48 papers. How many did he sell each day? NOTE. Let x denote the number of papers sold on Wednesday. 9. Texas lacks 17,470 sq. mi. of being large enough to make 5 states the size of Illinois. How large is Illinois if Texas contains 265,780 square miles? 10. Maude has 78¢, which lacks 18¢ of being 6 times as much money as James has. How many cents has James? 11. It is twice as far from Chicago to Niles by a certain route as from Niles to Albion, and from Chicago to Albion is 190 mi. How far it is from Chicago to Niles? 12. Rochester, N. Y., is 17 mi. more than twice as far from Chicago as is Detroit. Rochester is 587 mi. from Chicago. How far is Detroit from Chicago? 13. Niagara Falls is 8 mi. less than 9 times as far from Chicago as is Michigan City. Niagara Falls is 514 mi. from Chicago. How far is it from Chicago to Michigan City? 14. It is 4 times as far, less 16 mi., from Chicago to N. Y. City as from Chicago to Ann Arbor. If it is 976 mi. from Chicago to N. Y. City, how far is it to Ann Arbor? 15. Champaign is 5 mi. more than as far from Chicago as is Cairo. If it is 128 mi. from Chicago to Champaign, how far is it from Chicago to Cairo? 16. A father and his son together earn $75 a month, and the father earns 4 times as much as the son. How much does each earn? 17. William has 3 times as many marbles as Joseph, and both together have 24. How many marbles has each? 18. A line 32 in. long was divided into 2 parts, one of which is 8 in. longer than the other. How long is each part? 19. A tree 75 ft. high was broken by the wind so that the part standing was 15 ft. longer than the part broken off. How long was each part? 20. The area of a rectangular field is 6400 sq. rd. and its length is 160 rd. What is its width? 21. The area of a rectangular sidewalk is 120 sq. rd. and the width is 14 yd. How long is the walk? 22. A man received $180 for 72 days' work. What was his daily wage? 23. Find the side of a square field equal in area to a rectangle 80 rd. long by 20 rd. wide. 24. A man received $3000 for 80 A. of land. How much did he receive per acre? 25. The circumference of a circle is 2200 rd. What is the radius? (Equation: 6r = 2200.) = 26. The area of a circle is 2014 m.2. How long is the radius? The circumference? 27. Find the radius and the circumference of a circle whose area is 1 square inch. $135. Formal Work. Find the value of the unknown quantity, x or y, in 22. In the following equations W denotes the weight of the rope, or cable, in lb. per yd.; L the working load in tons; S, the breaking load in tons, and D, the diameter of the rope, or cable, in inches. For hemp cable, W = .577D2; L For tarred hemp S = 1.480D2. For manila rope, W cable, W = 1.036D2; L = = = .109D2; S = .654D2. .247D2; .765D2; L = .329D2; S = 1.877D2. For iron wire rope, W = 3.847D2; L = 2.862D2; S = Find the weight per yd., the working load and the breaking load of a hemp cable of 1" diameter; of 21" diameter. 23. Solve similar problems for tarred hemp rope; for manila rope. 24. Find W, L, and S for an iron wire rope for which D = 11"; D = 21". 25. For the same values of D, find W, L, and S for steel wire rope. §136. Equations Containing Two Unknown Numbers. In these problems two numbers will be unknown. The facts of each problem will furnish two equations. From these facts write both equations. Then change one or both equations, or combine them, in accordance with Principles I-V, pp. 255-258. 1. The sum of two numbers is 10 and their difference is 4. Find the numbers. Add Equations (1) and (2) member to member and write Which Principle enables us to get Equation (4) from Equation (3)? Putting this value, 7, of x, in place of x in Equation (1), we have (5) 7+ y = 10 Subtracting 7 from both sides of Equation (5), we get By which Principle do we have the right to subtract 7 from both sides of Equation (5) and then write the two differences equal? The two desired numbers are then 7 and 3. N.B.-The problem shows how to find any two numbers when their sum and their difference are known. 2. The sum of two numbers is 18 and their difference is 6. Find the numbers. 3. A piece of wire 8 ft. long is cut into two pieces, one of which is 2 ft. longer than the other. How long is each piece? 4. A 16-oz. mixture of acid and water contains 9 oz. more water than acid. What is the weight of each part of the mixture? 5. In a school of 40 children there are 12 more girls than boys. How many girls are in the school? How many boys? |