Read and give approximate results; then solve, using the shortest methods of solution: 43. Father bought 75 cows at $871 each, and sold them at $100 each. How much did he gain? 44. Margaret bought 15 lb. of tea at $.40 a pound, 16 lb. of rice at $.12 a pound, and 16 lb. of sugar at $.061 a pound. How much change should she receive from $15? 45. From a piece of cloth containing 40 yd. three dress patterns of 12 yd. each were sold. How much was the remainder worth at $1.25 a yard? 46. Mother bought 6 lb. of codfish at 2 lb. for $.25, 12 lb. of starch at 3 lb. for $.25, and 6 lb. of coffee at 3 lb. for $1. Find the amount of her bill and the change she received from $5. 47. Make and solve two problems involving multiplication by parts of a dollar, and two involving division by parts of a dollar. EQUATIONS A statement of equality between two equal quantities is called an equation; as, 5 × 6 = 30; 1 ft. 4 in. = 16 in. A parenthesis ( ) is sometimes used to group expressions. Operations within a parenthesis must be performed first. Thus, 4 + (7 × 5) = 4 + 35 = 39; while (4 +7) × 5 = 11 × 5 = 55. 1. (7 × 5) + 6 = ? 8. (483) × 6=? 9. (10 × 6) (5 × 3) = ? 10. (8 x 9) (3 × 4) = ? 11. (5 × 7)÷(8 − 3) = ? 12. 9 x 7 = ? +56 13. (168)+9=7+? 14. (244)-3=6÷? I. Before you try to solve a problem you must find out exactly what it means. That is, you must consider: a. What facts are stated or implied in the problem. b. What kind of answer the question asks for. c. By what steps the required answer can be found from the given facts. II. The most important habit to acquire is accuracy. A wrong answer is worthless. Always test your work. Also make a mental estimate of the answer. 1. How much cheaper is it to buy a barrel of flour (196 lb.) for $12 than at 8 a pound? John answered $144.80. He had pointed off wrong in multiplying. If he had first formed a mental estimate, he could not have made such a mistake. Thus, 196 lb. = nearly 200 lb. 200 × 8% 1600 ¢ = $16. $16 $12 $4. The exact answer found by the written work is $3.68. 2. A man gave a check of $10,000 in payment of two bills of $4998 and $3993. How much change should he get? = Mental estimate: The bills amounted to about $5000+ $4000, or $9000; change, about $1000. The exact answer obtained by the written work is $1009. Tell by mental estimates whether : 3. 20 x 23 is more or less than 46. 6. $25.40 ÷ 8 is more or less than $.03. 8. 13 × 7 is more or less than 7. 9. Find the cost of 37 yd. of ribbon at $.23 a yard. Written work: 37 = 31; 31 × $.23 = $.891. Test: 3 × $-23 = $.69; of $.23 = $.20}; $.69 + $.20} = $.89 III. The second essential is rapidity. To secure rapidity, always choose the shortest method of work where several methods are possible. It is sometimes well to indicate the necessary operations before performing any of them. Then the work may often be shortened by cancellation. 10. If a man earns $18 in 6 days, how much, at the same rate, does he earn in 28 days? = $3; 28 × $3 SOLUTION. $18÷ 6 = $84. This method is called unitary analysis, because the earning for the unit, 1 day, is first found. 11. If a man earns $ 18 in 6 days, how much, at the same rate, does he earn in 24 days? SOLUTION. 24 da. = 4 × 6 da.; 4 × $18 = $72. This is called the method of comparison, because the answer is found by comparing 6 da. directly with 24 da. 12. If a man earns $18 in 6 days, how much, at the same rate, does he earn in 10 days? 13. How many cubic yards of earth are removed in digging a ditch 54 ft., long 31 ft. wide, and 2 ft. deep? It is often convenient to indicate a solution in the form of an equation. (See p. 231.) 14. A boy paid $1.40 a hundred for newspapers and sold them at $.02 apiece. How much did he gain (a) on 100 papers? (b) on 1 paper? SOLUTION. (a) Gain on 100 papers = (100 x $.02) — $1.40 = $.60. (b) Gain on 1 paper = $.60 100 = $.006. 1. The cost of providing a 2 weeks' outing for 100 girl scouts was $650. Find the cost per week for each girl. 2. How much gasoline will be saved by driving 5400 miles in a car that averages 27 miles per gallon instead of in one that averages 18 miles per gallon? 3. During one week 226,997 thrift stamps were sold. Find their value, at 25 each. 4. Find the value, at $4.18 each, of 13,799 war savings stamps sold in the same week. 5. Make and solve a problem about the amount you save by buying three articles at a bargain sale. 6. A wage increase of $2,500,000 was granted to 15,000 street car employees. What was the average for each? 7. The pay of a conductor was increased from $.39 an hour to $.50 an hour. How much did the increase amount to in a week consisting of six 8-hour days? 8. Mrs. Hunt received $100 a month for household expenses. Her account one month showed that she spent $20 for rent, $27.50 for food, $9.50 for light, heat, and laundry, $16.50 for clothing, and $16.75 for other items. How much did she save that month? 9. It took 4000 hours of work to complete an airplane. How long did it take 80 men working 8 hours a day to do the work? 10. The distance covered by the air mail service in 78 hours was 5304 miles. What was the average speed per hour? 11. Make and solve a problem suggested by some game that you play. The heat provided for our bodies by our food is measured in heat units. The following is part of a Bill of Fare : 12. Alice ordered beefsteak and apricots. Find the cost and the number of heat units. 13. Margaret ordered soup, potatoes, and custard. Find the cost and the number of heat units. 14. How much less did Margaret spend than Alice, and how many more heat units did her food provide? Compare the following two meals as to cost and heat units: 15. Beef cutlet, potatoes, and ice cream. 16. Bean soup, omelette, and cup custard. 17. The receipts of a railway for 365 days were $119,685.23, and the expenses, $96,478.02. Find the average daily profits. NOTE. Find averages to nearest cent, that is, call $.574, $.57; call 3.575, and $.576, $.58; etc. 18. A ticket agent sold in one day: 450 tickets @ $1.50; 380 tickets @ $1.00; 520 tickets @ $.75; and 310 tickets @ $.50. Find the total amount of his sales. 19. Find the average amount of each subscription for each loan: |