done. No. 5 may be made out in the form here shown or similar to the one given in Art. 546. See Art. 642 for a bill for goods bought at different times; or use the heading given in this article. 754. What has been said about percentage in Art. 746, is applicable to this topic. Such children as hear their parents talk of savings-banks, etc., know sufficient about interest for the purposes of this chapter. No rules should be given. 756. The pupils should deduce their own rule for calculating the area of a right-angled triangle. 758. In Art. 653 the pupils have been taught to multiply 18% by 6 in one line; in Art. 654, they have learned how to divide 18 by 2, which is the same as finding of 18, so that nothing new is here presented. 763-764. Although these examples are not strictly practical, they are useful in giving the pupils the facility necessary to perform readily operations involving fractions or decimals. While it is not necessary to work them all, the scholars should by this time have acquired such expertness in the fundamental operations as to be able to obtain the results in a very short time. 765. See Arithmetic, Art. 591. XII NOTES ON CHAPTER NINE The technical terms used in denominate number work should now be regularly employed by teacher and pupil, but set definitions should not be memorized. The scholars should be required to arrange their work properly, and to perform the various operations with as few figures as are consistent with accuracy.. 767. In reducing 16 gal. 1 qt. to quarts, the pupil should write 65 qt. at once. He multiplies by 4, saying 4 sixes are 24, and 1 are 25 - writing the 5, etc. In reducing 31 gal. to quarts, the work should occupy but a single line. See Arithmetic, Art. 653. 770. No special rule should be given in Nos. 33, 34, and 35 for the reduction of a fractional or a decimal denominate unit. 773. A pupil should be permitted to work such examples as No. 2 in his own way. They do not occur frequently enough in practice to make it advisable to give them special treatment; but the teacher should suggest, as in other exercises, the advisability of shortening the work by indicating operations and cancelling. Thus, 12 6. 750 lb.=75%, T.=& T.; $5×53=$26.871, or $26.88. Ans. = 7. No. of tons $18.76+$5-3.752; .752 T. (.752 × 2000) lb. 1504 lb. Ans. 3 T. 1504 lb. = = 8. 7 T. 296 lb.=14296 lb.; ($35.74 ÷ 14296) × 18748 = Ans. 9. 9 T. 1568 lb. 19568 lb.; $48.92 19568 = cost per lb. $73.11 ÷ ($48.92 ÷ 19568), or ($73.11 × 19568) ÷ $48.92= number of pounds. Reduce to tons, etc. 774. By this time, the pupils should know how to add compound numbers, so that the chief duty of the teacher should be to see that the operation is not spun out too much. A scholar of this grade should not find the total number of ounces in 1 by adding each column separately; he should say 27, 36, 39 oz., or 2 lb. 7 oz., without writing anything but the 7 oz., which is put in its proper column and 2 lb. carried. In 4, the addition of the units' column of minutes gives a sum of 15. Since minutes are changed to hours by dividing by 60, which ends in a cipher, the units' figure of the remainder will be 5, so that this figure may be written in the total. Carrying one, the sum of the tens' column is 11, which contains 6 once with a remainder of 5. This is written in its place, making 55 minutes, and 1 hour is carried. The two columns of hours are added in one operation-21, 38, 43, or 1 day 19 hours. 6 should be treated in the same way, no side work being permitted. In 7, the pounds are reduced to tons by dividing by 2000, so that the sum of the units', tens', and hundreds' columns of pounds may be written in the total, the sum of the thousands' column being divided by 2 to reduce to tons. 775. Nothing should be written but the results. In 27, the addition of 1 ton to 1552 lb. will change only one figure of the latter, and this change can be carried in the head. In 29, 320 rods should be added to 15 rods mentally and 24 rods deducted from the sum, only the answer being written. 779. In dividing 5 bu. by 4, 79, the answer is not to be given as 1 bu.; the division should be continued through pecks. The result in 88 should contain weeks, days, hours, and minutes. 784. While these drills seem somewhat difficult for mental work, they should not be too severe for children that have been studying arithmetic for over five years, especially if the previous drills have been faithfully attended to. The ability of many children to handle numbers seems to decrease after the fourth school year, the greater portion of the subsequent instruction being given to new topics to the neglect of continued practice in the fundamental processes. The conscientious teacher should remember that the bulk of the mathematical work of most of her scholars after they leave school will not extend much beyond what has been learned in the first four years. The ability to handle at sight or mentally such numbers as are here given, will be of use to the scholars in various ways. The average pupil attends to only one figure at a time; and he is frequently unable, after a simple addition or multiplication, to see that his answer is very far astray. Practice with such drills as these, and in the sight approximations, will enable him to test his work in such a way as to detect any very serious error. Scholars find it easier to add or subtract such numbers as 163, 8610, etc., when they are read "one, sixty-three;" "eighty-six, ten;" etc. Following the order in which the figures are read seems the most natural way in mental work. When a pupil is asked to find the sum of 163 and 137, he is less likely to make mistakes if he proceeds in this way: 263, 293, 300; adding to the first number — 163—100, 30, and 7 in the order in which the figures are repeated to him. 786. In multiplying 21 by 15, 41 by 14, etc., the scholar generally finds it easier to commence with the tens: 15 twenties are 300, 15 ones are 15-315; 14 forties are 560, and 14 are 574. 48 × 16 becomes of 48 hundred; 32 × 371 of 32 hundred, etc. 787. These exercises present rather more difficulty, and are probably not so useful, qn the whole, as the others. For this reason, they should be employed as sight work chiefly. 788. In 13×5, multiply 13 first by 5, and then, obtaining 65+3, or 683. In dividing 24 by 23, reduce both to thirds - 72 thirds 8 thirds 72÷8 = 9. = 790. The teacher should not neglect such addition exercises as are scattered throughout the book. 791. It happens occasionally in multiplying by a mixed number, that the units' figure of the integer and the numerator of the fraction are the same. In such a case, a few figures will be saved by following the method given in the text-book, instead of writing again the product by 3 as shown above. 4846 X 3 5)14538 2907 14538 etc. 2761 × 999 2761000 792. The product by 100 may be placed above the number, if desired. In multiplying by 1000, the multiplicand is subtracted from 1000 times itself. To find the product of 9832 by 990, multiply by 99, and annex a cipher to the result. Taking one-fourth of 268400 gives the answer to 21. 2758239 Ans. 800. The pupils should find for themselves in 5 the number of square inches in a square foot, etc. A drawing is asked in the first part of 14, so that children will see that the dimensions are not 4×6. The short method of finding the area of the fence in 15, by multiplying 900 by 10, should not be given yet: the scholars should be permitted, for the present, to calculate the area of one part at a time. In 16, it is suggested that the area of the walk be ascertained by subtracting from the whole area (250 × 200) sq. ft., the area of the part left for the garden (230 × 180) sq. ft.; but the scholars should be encouraged to calculate the surface of the walk in another way, such as by taking the two ends as measuring each 250 ft. by 10 ft., and the sides as 180 ft. each by 10 ft. The number of square feet in the sidewalk of 17 will be (270×220)-(250x 200); or (270 × 10) + (270 × 10)+(200 |