Units, Tens, and Higher Orders.

32. Case I. When no term of the subtrahend is greater than the term in the same order of the minuend.

Example. From 796 subtract 354.

Illustration. Suppose that Mr. Jones has seven sacks of money, each containing one hundred silver dollars, nine rolls of ten dollars each, and six dollars lying loose on his table, as represented in the following picture, and that Mr. Smith calls to collect 354

dollars. The pupil will readily understand that Mr. Jones has only to give Mr. Smith four of the single pieces, five of the ten dollar rolls, and three of the sacks containing one hundred dollars each, and that he will then have 442 dollars remaining.

Note. The learner should practice himself in such illustrations till he is familiar with them. He will thus surely learn the significance of the processes of arithmetic.

In the same way, tell how you would take 325 from 697 buttons, supposing that you had 6 cards having 100 buttons sewed on to each, 9 cards with 10 buttons on each, and 7 loose buttons. Show how the remainder would be found by work on the slate.

19-32. From 879 take 213, 425, 263, 34, 728, 658, 870, 457, 23, 43, 654, 222, 333, 400.

33-46. Take 432 from each of the following numbers: 543, 733, 645, 987, 655, 438, 679, 542, 632, 777, 989, 656, 686, 567.

Note. In the process of subtraction there are two modes of reckoning. For instance, in subtracting 5 from 9, some say, "5 from 9 leaves 4"; others, "5 and 4 are 9"; both writing the 4 as it is spoken, or better, as it comes into the mind. The results are the same, but the latter wording is recommended in practice for many reasons, one of which is that it is less liable to error. It should not be introduced, however, till the former is well understood by the learner.

Example. From 789 subtract 435.

789

435

354

Wording.-5 and 4 are 9, 3 and 5 are 8, 4 and 3 are 7. Only the results printed in heavy type should be spoken, and these should be written as uttered.

This is called the "making up method." Besides being less liable to error, other advantages will be seen in Case II. See Ex. 68, p. 44, also Shorter Method in Long Division, p. 90.

33. Case II. When any term of the subtrahend is greater than the term in the same order of the minuend.

Example. From 442 subtract 136.

Illustration. After Mr. Jones had paid Mr. Smith, he had $442 left, out of which he is paying Mr. Brown $136. But since he has only two loose dollars, he is here represented as having

taken ten dollars from one of the rolls, and put them with the two, thus making twelve single pieces. From these he has put forward six pieces. He has shoved forward also the three remaining ten dollar rolls, and one sack containing a hundred dollars. Thus he has dollars left.

The process is represented in figures as follows:

Note. The pupil does not understand the foregoing illustrations if he can not go farther and explain the case, where he has to "borrow," both from the tens and from the hundreds; or, where there are no tens, perhaps also no units.

67. Explain how you could most conveniently take 585 pencils from the number of pencils represented above.

68. In the same way explain how you would proceed to take 378 match-sticks from the number represented below.

According to the "making up method" mentioned in note at foot of p. 41, the wording in the process of subtracting 378 from 604 would be as follows:

604

378

226

Eight and 6 are 14, carry one to 7, 8 and 2 are 10, carry 1 to 3, 4 and 2 are 6, the numbers represented in heavy type being the only ones spoken aloud. These should be written while they are being pronounced.

69-78. Subtract 328 from each of the following numbers: 442, 560, 643, 751, 962, 876, 777, 691, 564, 886.

79-86. Find the difference between 435 and each of the following numbers: 561, 872, 960, 253, 864, 950, 762, 341.

Try to solve these examples orally before using the slate.

107. 255-15= 115. 256-150= 123. 378-108- 131. 457-259=