multiplication only by the tens figure increased by 1.

Thus 27 × 29 810-27 = 783; 65 × 49 = 3250 -65 = 3185.

(5) All involving a factor whose tens and units are expressed by the same figure, require but one multiplication.

Thus 28 x 22: +344 = : 3784.

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56056616; 86 × 44 = 3446

These five cases include no less than 2919, or more than 70 per cent. of all the possible combinations. The contractions given in paragraphs 41 and 45, even if those in § 41 be confined to numbers differing by ten or less, will apply to 55 of the combinations not already provided for, and will leave at most 1121, or about one-fourth of the whole number of possible combinations which require multiplication by both figures of the multiplier. But these 1121 are the very combinations least frequently employed. It is safe to say that not one-tenth of the combinations occurring in daily business require full multiplication.

Where such multiplication is necessary, how difficult is it, under our system of arrangement? Let the pupil approach it by three steps.

(1) Learn to add instantly and from the left any two numbers of two figures each. § 23.

Thus 45 +44 89; 55 + 47 = 102.

(2) Learn to multiply instantly and from the left two figures by one figure. § 32.

Thus 15 x 3 = 45. 26 × 4 104. 39 × 6 = 234.

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(3) Learn to combine these two operations. § 34.

These are but three steps, and three easy steps, if properly taught; but he who takes them strides over the field of mental calculation with sevenleague boots. This thoroughly learned, all the rest is easy, and the pupil can fill invoices like the following as fast as his pen can write the figures:

44 yds. sheeting at 11 c. (4.84 + .29)

$5.13

(The mental processes are indicated in parenthesis). To show more fully how these principles are applied, as well as to afford practice for the pupil, problems are added with full solutions.

1. What cost 32 tons of hay at $21 a ton?

32 × 2o = 61o; + 32 × 1 = $672. Ans.

2. How much would a merchant receive for 54 yards of cloth at 32 c. a yard?

54 ×.3° = 162o; +54×.02 = $17.28. Ans.

3. If I buy 81 lbs. of tea at 58 c. a pound, what I will it cost me?

81 x 5° 40°; +81 x.08 $46.98. Ans.

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4. Required the cost of 36 horses at $92 each. $3312. Ans.

5. What must be paid for 63dozen eggs at 26 cts. a dozen ?

$16.38. Ans.

6. What cost 55 yds. of cloth at 34 cts. a yard? 34x.5° 17°°; + 1.7°= $18.70. Ans.

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70

7. What cost 33 bushels of potatoes, at 46 cts. a bushel?

80

.46 × 3° 13.8°; +13=$15.18.

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Ans.

8. What cost 88 lbs. of coffee, at 28 cts. a pound? .28 × 8° 22°; +2.2= $24.64. Ans.

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24

9. If a locomotive travels 44 miles an hour, how far will it travel in 17 hours?

68

17 × 4° = 6°°; + 748 min. Ans.

10. What sum of money equally divided among 99 men, will give them 66 cts. each ?

.66 × 9° 59°; + 5" $65.34. Ans.

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GOODRICH'S ART OF COMPUTATION.

PART FIRST.

DEFINITIONS.

1. A Unit is a single thing.

As a dollar, an apple, or the simply abstract number one. 2. An Integer or Whole Number, is either a unit or a collection of units.

As one, two, eleven, three thousand.

3. A Fraction is one or more equal parts of a unit. It is commonly expressed by two integers, placed the one over the other with a straight line between. The integer below the line is called the Denominator, because it denominates or names the number of parts into which the unit is divided. The number above the line is called the Numerator, because it numerates or expresses the number of parts taken.

Thus in To, read five one-thousandths, the denominator 1000 indicates that the unit is divided into 1000 equal parts, and the numerator, 5, shows that 5 of these parts are taken.

When the denominator is 10, 100, 1000, etc., it is often omitted and indicated by prefixing a point, called the Decimal Point, to the numerator, which must be so written, by prefixing o's if necessary, as

14

to contain as many figures as there are O's in the denominator. Fractions so expressed are called Decimal Fractions.

Thus Tu may be written .005; first prefixing to the 5 two O's, that it may contain three figures, and then prefixing the decimal point. To read a decimal fraction, or to write it as a common fraction, take the significant figures for the numerator, and 1 with as many O's annexed as it contains figures altogether, for the denominator. Thus .0176 would be read or written

176

10000

4. A Number is either an integer, a fraction, or an integer and fraction combined. In the last case, it is called a Mixed Number.

Thus 11,,.06 and 5 are all numbers.

5. Computation is combining, separating or comparing numbers. These processes are usually spoken of as Synthesis, Analysis and Comparison. 6. Synthesis includes

(a) Addition, or finding a number equal in value to two or more numbers taken together. The sign of addition is +.

Thus 2+24. 5+6+8=19. 44+56 100.

(b) Multiplication, a special case of addition, where the numbers added are all equal. It is performed by finding the sum of as many times one number as there are units in the other.

Multiplicand, meaning the number multiplied; Multiplier, meaning the number which multiplies; Factor, meaning an agent, applying to either multipli