a dollar represent cents. Since a mill is of a cent, or I of a dollar, the third figure represents mills.

Thus, $25.16 is read twenty-five dollars and sixteen cents; $25.168 is read twenty-five dollars sixteen cents and eight mills.

SYMBOLS OF AGGREGATION.

178. The vinculum—, parenthesis ́), brackets [], and brace {} are called symbols of aggregation, and are used to include numbers which are to be considered together; thus, 13×8-3, or 13 × (8-3), shows that 3 is to be taken from 8 before multiplying by 13.

When the vinculum or parenthesis is not used, we have 13x8-3 = 104-3 = 101.

179. In any series of numbers connected by the signs +, -, x, and ÷, the operations indicated by the signs must be performed in order from left to right, except that no addition or subtraction may be performed if a sign of multiplication or division follows the number on the right of a sign of addition or subtraction until the indicated multiplication or division has been performed. In all cases the sign of multiplication takes the precedence, the reason being that when two or more numbers or expressions are connected by the sign of multiplication the numbers thus connected are regarded as factors of the product indicated, and not as separate numbers.

EXAMPLE.-What is the value of 4 X 24-8+17?

SOLUTION.-Performing the operations in order from left to right, 4× 24 = 96; 96—8 = 88; 88+17= 105. Ans.

180. EXAMPLE.-What is the value of the following expression: 1,296 12+160-22 × 3 = ?

SOLUTION. 1,296 12 108; 108+ 160 = 268; here we cannot subtract 22 from 268 because the sign of multiplication follows 22; hence, multiplying 22 by 31, we get 77, and 268-77 = 191. Ans.

Had the above expression been written 1,296÷12+160 -22×31÷7+25, it would have been necessary to have divided 22×3 by 7 before subtracting, and the final result would have been 22 × 3 = 77; 77 ÷ 7 = 11; 268-11 = 257; 25725282. Ans. In other words, it is necessary to perform all the indicated multiplication or division included between the signs + and −, or — and +, before adding of subtracting. Also, had the expression been written 1,296 12+160-241÷7×31+25, it would have been necessary to have multiplied 3 by 7 before dividing 24, since the sign of multiplication takes the precedence, and the final result would have been 34×7 = 241; 24÷24 = 1; 268 −1 = 267; 267 +25 = 292. Ans.

It likewise follows that if a succession of multiplication and division signs occur, the indicated operations must not be performed in order, from left to right—the multiplication must be performed first. Thus, 24×3÷4×2÷9×5 = {. Ans. In order to obtain the same result that would be obtained by performing the indicated operations in order, from left to right, symbols of aggregation must be used. Thus, by using two vinculums the last expression becomes 24X3÷4X2÷9×5 = 20, the same result that would be obtained by performing the indicated operations in order, from left to right.

EXAMPLES FOR PRACTICE.

181. Find the values of the following expressions:

ARITHMETIC.

(SECTION 4.)

PERCENTAGE.

1. Percentage is the process of calculating by hun. dredths.

2. The term per cent. is an abbreviation of the Latin words per centum, which mean by the hundred. A certain per cent. of a number is the number of hundredths of that number which is indicated by the number of units in the per cent. Thus, 6 per cent. of 125 is 125 XT80 = 7.5; 25 per cent. of 80 is 80 x 25 = 20; 43 per cent. of 432 pounds is

3. The sign of per cent. is %, and is read per cent. Thus, 6% is read six per cent.; 121% is read twelve and onehalf per cent., etc.

When expressing the per cent. of a number to use in calculations, it is customary to express it decimally instead of fractionally. Thus, instead of expressing 6%, 25%, and 43% as 1, 2, and 3, it is usual to express them as .06, .25, and .43.

25

The following table will show how any per cent. can be expressed either as a decimal or as a fraction:

For notice of copyright, see page immediately following the title page.

4. The names of the different elements used in percentage are: the base, the rate per cent., the percentage, the amount, and the difference.

5. The base is the number on which the per cent. is computed.

6. The rate is the number of hundredths of the base to be taken.

7. The percentage is the part, or number of hundredths, of the base indicated by the rate; or, the percentage is the result obtained by multiplying the base by the rate. Thus, when it is stated that 7% of $25 is $1.75, $25 is the base, 7% is the rate, and $1.75 is the percentage.

8. The amount is the sum of the base and percentage. The difference is the remainder obtained by subtracting the percentage from the base.

9.

Thus, if a man has $180, and he earns 6% more, he will have altogether $180 +$180 ×.06, or $180 + $10.80 $190.80. Here $180 is the base; 6%, the rate; $10.80, the percentage; and $190.80, the amount.

Again, if an engine of 125 horsepower uses 16% of it in overcoming friction and other resistances, the amount left for obtaining useful work is 125 – 125×.16 = 125 - 20 = 105 horsepower. Here 125 is the base; 16%, the rate; 20, the percentage; and 105, the difference.

Hence,

10. From the foregoing it is evident that to find the percentage, the base must be multiplied by the rate. the following

Rule.-To find the percentage, multiply the base by the rate expressed decimally.

EXAMPLE.-Out of a lot of 300 bushels of apples 76% were sold. How many bushels were sold?

SOLUTION.

76%, the rate, expressed decimally, is .76; the base is 300; hence, the number of bushels sold, or the percentage, is, by the above rule,

300 X.76 =228 bushels. Ans.

Expressing the rule as a

=

Formula, percentage base rate.

11. When the percentage and rate are given, the base may be found by dividing the percentage by the rate. For, suppose that 12 is 6%, or 186, of some number; then 1%, or To, of the number, is 12÷6, or 2. Consequently, if 2 = 1%, or, 100%, or 188 = 2×100 =200. But, since the same result may be arrived at by dividing 12 by .06, for 12.06 = 200, it follows that:

Rule.- When the percentage and rate are given, to find the base, divide the percentage by the rate expressed decimally.

Formula, base = percentage÷rate.

EXAMPLE.-Bought a certain number of bushels of apples and sold 76% of them. If I sold 228 bushels, how many bushels did I buy? SOLUTION. -Here 228 is the percentage, and 76%, or .76, is the rate; hence, applying the rule,

228.76 = 300 bushels. Ans.

12. When the base and percentage are given, to find the rate, the rate may be found, expressed decimally, by dividing the percentage by the base. For, suppose that it is desired to find what per cent. 12 is of 200. 1% of 200 is 200 X.01 = 2. Now, if 1% is 2, 12 is evidently as many per cent. as the number of times that 2 is contained in 12, or 12 ÷ 2 = 6%. But the same result may be obtained by dividing 12, the percentage, by 200, the base, since 12÷200 =.06 = 6%. Hence,

Rule. When the percentage and base are given, to find the rate, divide the percentage by the base, and the result will be the rate expressed decimally.

Formula, rate = percentage÷base.

EXAMPLE.-Bought 300 bushels of apples and sold 228 bushels. What per cent. of the total number of bushels was sold?

SOLUTION.-Here 300 is the base and 228 is the percentage; hence, applying rule,

rate 228 300 = .76 = 76%. Ans.

EXAMPLE.-What per cent. of 875 is 25?

SOLUTION.-Here 875 is the base, and 25 is the percentage; hence. applying rule,

25 ÷ 875 = .029 = 29%. Ans.