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XIV

NOTES ON CHAPTER ELEVEN

With this chapter begins the regular work in percentage, and it is important that the pupils obtain, as soon as possible, a correct idea of what is meant by the term per cent.

Many of the various subdivisions of this topic found in some books, are taken up only incidentally, while others are omitted altogether, the aim being to give the pupils a foundation upon which they can subsequently build, rather than to scatter their energies over too great a diversity of subjects.

860. The reduction of a common fraction to a per cent, consists in changing the former to a decimal of two places. In reducing 1 to a decimal, the result is .5, or 5 tenths; in changing 1 to an equivalent per cent, the result is 50 per cent, or 50 hundredths. In reducing to a decimal, the answer is given in three places, .125, or 125 thousandths; in changing it to a per cent, the division is stopped at the second place, and the remainder written as a fraction, 124 per cent, or 121 hundredths.

The denominator of a per cent being always the same, 100, the comparative value of several per cents is known at sight. To compare ş and gas common fractions, they must be changed to 25 and 44 ; if a further comparison is to be made between these and y, a new common denominator must be employed, and the fractions reduced to do izo, and 120 Changing the fractions to decimals, 625 thousandths, 6 tenths, and 58} hundredths, simplifies the comparison; but it is still easier to determine their relative value when they are expressed as 622 per cent, 60 per cent, and 581 per cent.

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The teacher must not be discouraged if the pupil fails to grasp at once the full meaning of percentage. Definitions will not help materially; much practice in working examples is necessary to give the knowledge desired.

863. Many children find it difficult to distinguish between 1% and 50%. If the former is read in the business way, of one per cent, it may make the distinction plainer.

864. Per cents being generally given in two figures, scholars hesitate to give the correct answers: 300%, 250%, 125%, 1633%, 420%, 910%.

865. While pupils will find 331% of 81 cows, by dividing 81 by 3, they should understand that they are really multiplying 81 by 33} hundredths, or 81 by j. In 4, 6% of 150, or 16 of 150, may be obtained by multiplying 150 by 6 and cutting off two ciphers; or by dividing 150 by 100, obtaining 17, and multiplying this quotient by 6; or by reducing 6% to %, and finding

o of 150. In 9, the pupil should find 1% of $ 640 and take onehalf of the result.

The scholars should be permitted to use their own method of solving these problems, the different analyses given by the pupils furnishing their class-mates an opportunity to select a simpler method.

866. Although every pupil may not be able to determine at once the shortest way of calculating a given example, no one should be allowed to work 3, by multiplying by 33}. When the multiplication by { has been performed, the answer has been obtained, except as to the location of the decimal point, and the waste of time in multiplying by 3, repeating this product, and adding three columns should not be tolerated. No fault should be found with the average pupil for failing to recognize in 1, that 63% is is; or that in 12, 31% is zo. The general method should be to multiply by the figures given to represent the per cent; except in such cases as 121%, 169%, 25%, 331%, 374%, 50%, and possibly a few others. $1240

o Where the given numbers are used, they should be

1 made the multipliers and expressed as hun- 2 09 o dredths. Nothing is gained in 5, by reducing *

X.042 $1.55 the { to a decimal; although in 13, writing 51%, .055, might make it easier for some.

868. The rule generally given of finding the percentage, by multiplying by the rate expressed as hundredths, is here modified to the extent of using the common fraction to express hundredths, instead of the decimal, as being more in conformity with early algebraic methods.

The teacher that prefers to ascertain the base or the rate by the older arithmetical method, will omit 30–41.

30. 100 X 65 = 12.06Ans. 35. x + = 132; ete. 31. 138 = 26 ; ete. 37. c- = 78 ; etc. 32. (of n=1 Ans. =&o Ans. 33. * = 42; ete. 39. 160 = 10; etc. 34. x+ eAns. 800 of <= soo Ans.

41. 30o = 23 ; etc. 42. Let x=rate. Then of 65 = 26, or o * 65 = 26.

In an equation containing quantities to be multiplied, the multiplication should be performed before the equation is cleared

65 x 90 on 18 2 – 26. of fractions. This equation becomes

cs 100 it being immaterial whether canceling be done or not.

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43. After a little experience with this class of examples, the equation may be written at once, in the order in which the terms are given :

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45.

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While the algebraic method is of no advantage to the bright scholar, it makes the employment of rules unnecessary in the case of the ordinary pupil. 48. xx 54 x1=11x ** 100 * 1 2000 49. 11: = 44.

**200 50. x .

51. 100

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Divide the cost of the oats by 30¢ to find the number of bushels.

68. Assessed value = of $ 48000 = $32000. Taxes on 32 thousand dollars = $18.50 x 32.

869. In giving answers to these and to all other exercises, no “ guessing " should be allowed. The pupil should be permitted to obtain the correct result in his own way — that is, no inflexible rule should be given him to follow — but he should be able to get the answer, using the algebraic method if that seems to him the easiest, as it may be in some instances.

The examples are not arranged by “cases,” so that each will have to be understood before it can be worked.

The careless pupil will probably give the wrong answer to 13; saying 6, instead of 600; he will be likely too, in 14, to use the larger number as a divisor, and to obtain 449% instead of 225%. These mistakes are less likely to occur if he uses equations 3= * and =201. Even those scholars that have solved

200 ana 100 in their arithmetic work of the lower grades, examples similar to 15, will have new light thrown on their method by using the equation, x + = 20. In mental work, however, the first term should be made 5,, to reduce it in size, so that it can be more easily remembered. 24 is simplified by changing the fractions to whole numbers - . is what per cent of 12, 9 is what per cent of 10 — before beginning to calculate the rate. In 25, 17 and 63 become { and 2., 3 and 40, 9 and 40.

870. 1-5 can be worked by the pupils without any explanation ; 6-20 present more difficulty. The beginner in algebra

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