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quantity which is of the For the 2d term write

RULE. For the 3rd term, write that same denomination as that to be found. the greater of the other two quantities when the 4th term is to be greater than the 3d; or the less, when the 4th term is to be less than the 3d. Then divide the product of the 2d and 3d terms by the 1st; the quotient will be the required 4th term.

If the 1st and 2d terms are quantities of the same kind, but of different denominations, they must be reduced to the same denomination.

If the 3d term is a compound denominate number, it must be reduced to the lowest given denomination.

PROBLEMS.

1. If 6 paces equal 5 yds., how many paces in 100 yds.? Ans. 120 paces.

2. If 1 kilo. equals 2 lbs, how many lbs. in 50 kilos.? Ans. 110 lbs.

3. If 1 mm. equals in., how many inches in 75 mm.? Ans. 3 in.

4. If 1 cm. equals .3937 in., how many inches in 15 cm.? Ans. 5.91 in.

5. If 1 m. equals 39.37 in., how many meters in 1,000 yds.? Ans. 914.4 m.

6. If 1 Km. equals mi., how many yards in 5 Km.? Ans. 5,500 yds.

CHAPTER VII

PERCENTAGE

Per cent. means by the hundred.

The sign of per cent. is %, which is read "per cent." Thus: 4% is read "4 per cent.," and means to, or .04.

Percentage is the result obtained by taking a given per cent., or so many hundredths of a given number.

The given per cent., or number of hundredths taken, is called the rate.

The number on which the percentage is estimated is called the base.

The base plus the percentage is called the amount.

The relation between the base, rate, and percentage is such that, any two of them being given, the third can be found. Three cases may arise.

CASE I.-GIVEN THE BASE AND RATE, REQUIRED THE PER

CENTAGE.

RULE.-Multiply the base by the rate expressed decimally, the product will be the percentage.

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CASE II.-GIVEN THE BASE AND PERCENTAGE, REQUIRED

THE RATE.

RULE.-Divide the percentage by the base, the quotient will be the rate.

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2. 95 is what % of 1,900? Ans. 5%.
3. 2 is what % of 15? Ans. 131%.

4. 5.12 is what % of 640? Ans. %.

CASE III.-GIVEN THE RATE AND PERCENTAGE, REQUIRED

THE BASE.

RULE.-Divide the percentage by the rate expressed decimally, the quotient will be the base.

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2. 3.80 is 5% of what? Ans. 76.
3. 19.20 is % of what? Ans. 3,200.
4. 189.8 is 104% of what? Ans. 182.5.

CHAPTER VIII

POWERS AND ROOTS

A power of a quantity is either the quantity itself or some product of the quantity by itself. The quantity so multiplied is called the root of the power.

Powers are named according to the number of times the quantity is multiplied; this is indicated by a small figure called an exponent, written to the right of and above the quantity. Thus 21 is the first power of 2; 22=2×2=4, is the second power, or square, of 2; 23=2X2X2=8, the third power, or cube of 2; and so on.

The exponent 1 is ordinarily omitted, and when no exponent is written 1 is understood.

Roots are named according to the number of the times they enter a given power as a factor; this is indicated by what is called the radical sign, √, or by a fractional exponent.

27, the third, or When no index is

When the radical sign is used, a small figure, called an index, is placed over the sign to show the name of the root if any other than the second, or square, root is to be indicated. Thus: √4 indicates the square root of 4; cube, root of 27; 32, the fifth root of 32. written, the index 2 is always understood. When a fractional exponent is used, the numerator indicates a power; the denominator, a root. Thus, 4indicates the first power and square root of 4; 8 the second power of 8 and the

cube root of the result.

When a root of a quantity can be found exactly, the latter is called a perfect power of this root; when it can not be found exactly, an imperfect power. Thus: 8 is a perfect third power

whose cube root is 2; an imperfect second power, since its square root is 2.8284271+.

SQUARE ROOT.

The first ten numbers and their squares are

Numbers: 1 2 3 4 5 6 7 8 9 10
Squares: 1 4 9 16 25 36 49 64 81 100

From this it is evident that the square root of any perfect square, expressed by two figures, will be expressed by a single figure.

CASE I.--TO FIND THE SQUARE ROOT OF ANY WHOLE NUMBER OR DECIMAL.

RULE.-Separate the number into periods of two figures each, commencing at units or at the decimal point.

Find the greatest perfect square in the first period on the left; place its root on the right, like a quotient in division; then subtract the square from the period and to the remainder bring down the next period for a dividend.

Double the root figure found and place it on the left for a trial divisor; find how often this is contained in the dividend, exclusive of the right-hand figure, and annex the quotient to the root and to the divisor; multiply the completed divisor by the quotient, subtract the product from the dividend, and to the remainder bring down the next period as before.

Double the whole root found for a new trial divisor, and proceed as before until all the periods have been brought down.

If any trial divisor be greater than its dividend, the corresponding figure of the root will be 0.

If the product of a trial divisor by a figure of the root be greater than the corresponding dividend, the figure of the root is too large.

If the number is not a perfect square, there will be a remainder after all the periods have been brought down. In this

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