CHAPTER XVI.

PERCENTAGE AND ITS APPLICATIONS.

Art. 332. Percentage is the name of that department of Arithmetic which treats of computing by hundredths.

A certain percentage, or per cent. of a quantity, is so many hundredths of that quantity. Thus, 6 per cent. of a person's income is 6 hundredths of it.

NOTE. The term per cent. is a contraction of the Latin per centum, which signifies by the hundred.

Art. 333. The essential terms connected with operations in percentage are rate per cent., base, and percentage. The rate per cent. is the number of hundredths.

The base of percentage is that number, of which a certain per cent. is computed.

The percentage is the quantity found by computing a certain per cent. of the base.

The non-essential terms, sometimes used, are amount and difference.

The amount is the sum of the base and percentage. The difference is the difference of the base and percentage.

Art. 334. The notation of rate per cent. is by three methods, namely, by common fractions, decimals, and the sign %. Thus, 6 per cent. may be written 8, or .06, or 6%. The sign % is read "per cent." Thus, 5% is read "Five per cent."

The sign is generally used in mere written expressions, and the decimal and fractional methods in calculations.

Art. 335. From the definition of per cent. it follows that
One per cent. of a quantity is 100 of it.

Less than one per cent. of a quantity is less than 100 of it.
One hundred per cent of a quantity is that quantity.

Two hundred per cent. of a quantity is twice that quantity, &c.

NOTE. The expressions one-half per cent., one-fourth per cent., &c., mean one-half of one per cent., one-fourth of one per cent., &c.

EXERCISES.

Write, in three ways, 1 per cent.; 4 per cent.; 7 per cent.; 10 per cent.; 12 per cent.; 25 per cent.; 50 per cent.; 3 per cent.; 30 per cent.

Write in pure and mixed decimal form 61 %.

Ans. .0625; .061.

Write in like manner 121%; 183 %; 31%; 37%; %; 1%; %; %; 7%; %; 5%; 87%; §%; 62%; %%. Write, in three ways, 125 per cent. Ans. 125; 1.25; 125%. Write in like manner 100 per cent.; 110 per cent.; 250 per cent.; 375 per cent.; 411 per cent.; 1000 per cent.; 650 per cent.

CALCULATIONS IN PERCENTAGE.

CASE I.

Art. 336. To find any percentage of a number.

Ex. 1. Find 7% of $125.

Ans. $8.75.

FIRST METHOD.-$125 ÷ 100-$1.25; $1.25 X7 $8.75. SECOND METHOD.-$125 X .07 $8.75.

BY PROPORTION.-100: 7::$125: $8.75. Or, 1:.07 :: $125: $8.75.

Rule.-Multiply one-hundredth of the number by the num ber of hundredths. Or,

Multiply the number by the rate expressed decimally. Or,
Find that part of the number which the rate is of 100.

NOTE.-By the last method it is often best to use lowest terms. Thus, 5% is %; 10% is o; 61% is ; 20% is ; 25% is ; 50% is ; &c.

17. A man having 1250 bu. of wheat, sold 23% of it; how much did he sell?

18. My salary is $1200; If I pay 18% for board, 71% for clothing, 1% for travelling expenses, 33 % for books, and 81% for incidentals, what are my yearly expenses?

CASE II.

Ans. $468.

Art. 337. To find the rate per cent. which one number is of another.

Ex. 1. What per cent. of $20 is $2.50?

Ans. 12%.

FIRST METHOD.-One per cent. of $20 is $0.20, and $2.50 is as many per cent. of $20, as $0.20 is contained times in $2.50, that is, 121%.

SECOND METHOD.-Since $20 is 100% of itself, and $2.50 is of $20, $2.50 is of 100%, or 12% of $20.

18

BY PROPORTION.-$20 : $2.50 :: 100: 121⁄2, or $20 : $2.50 :: 1:.125.

Rule.-Divide the percentage by 1% of the base. Or,
Find that part of 100 which the percentage is of the base.

Art. 338. To find the base from the rate and percentage. Ex. 1. Three dollars is 6% of what?

Ans. $50.

FIRST METHOD.—$3 ÷ .06 = $50. Or, $3 ÷ 180 0.50 X 100 $50.

=

ANALYSIS.—If 18 of a quantity is $3, then 5 of that quantity is of $3, or $3, and 188, or the whole of the quantity, must be 100 times $3, or $30, which equals $50.

SECOND METHOD.-100% ÷ 6% = 163: 163 X $3 = $50. ANALYSIS. Since $3 is 6% of a quantity, that quantity must be as many times $3 as 100% contains 6%, that is, 16 times $3, or $50.

BY PROPORTION.-6%: 100% :: $3:$50.

Rule.—Find as many times the percentage as 100 contains the rate. Or,

Divide the percentage by the rate expressed as hundredths.

Ans. $1325.

7. I spent $198.75, which was 15% of my salary; what

was my salary?

8. $10.24 is % of how much?

9. $875 is 216% of what?

10. I sold 48 sheep, which was 5% of my flock; how many

had I at first, and how many had I left?

Ans. At first, 900. Left, 852.

CASE IV.

Art. 339. To find the base from the rate and amount, or from the rate and difference.

Ex. 1. What number, increased 25%, equals $50?

Ans. $40.

METHOD INDICATED.-$50 ÷ 1.25 $40. Or, $50 ÷

ANALYSIS. Since $50 equals 188 +25%, or 188, of the number, that is, 1.25 times the number, 1 of the number is 13 of $50, or $0.40, and 188 of the number is 100 times $0.40, or $40.

Ex. 2. What number, diminished 25%, equals $30?

METHOD INDICATED.-$30 ÷ .75 = $40.

Ans. $40.

75

= $40. Or, $30 ÷ 100

ANALYSIS. Since $30 equals 188 — 25%, or 75%, of the number, To of the number is of $30, or $0.40, and 188 of the number is 100 times $0.40, or $40.

Rules.-I. Divide the amount by the sum of 1 and the rate expressed as hundredths.

II. Divide the difference by the difference between 1 and the rate expressed as hundredths.

EXAMPLES FOR PRACTICE.

3. What number increased by 18% of itself equals 382.32 ? Ans. 324. 4. What number diminished by 18% of itself equals 265.68?

5. $3.60 is 33% less than how much?

6. $5.40 is 331% more than what?

Ans. 324. Ans. $5.40.

Ans. $4.05.

7. A merchant during the year gained 12% on his capital, and found he was worth $5180; what was his capital?

8. $59.50 is 163% greater than what? 9. $2382 is 3% less than what?

Ans. $51. Ans. $2400.

10. After spending 35% of my money, I have $206; how much had I at first?

11. What number increased by 500% of itself equals 5?