PRIN. 1.-When the dividend and divisor have a common denominator, divide the numerator of the dividend by the numerator of the divisor.

PRIN. 2.-Any fraction having unity for its numerator, is contained in unity as many times as there are units in its denominator; if the fraction has 2 for its numerator, it is contained one-half as many times; if it has 3 for its numerator, one-third as many times; etc.

PRIN. 3.-Two fractions, each having unity for its numerator, are divided by making the denominator of the divisor the dividend, and that of the dividend the divisor, regarding these numbers as integers and performing the division.

For,

=

2; 1÷1=3; 1÷1=4; 1÷1= 5.

f÷1=4; 1÷1=0; 4÷÷1=8;

÷÷=10. }÷}=1; }÷}=}=1}; }÷1=1=2; 1÷}=}=21.

PRIN. 4.-If the dividend is increased, the quotient is increased just as many times; if the dividend is diminished, the quotient is diminished in like manner.

COR. To divide any number, either integral or fractional, by a fraction, invert the divisor and then multiply the dividend by it.

30. What is the difference between two fractions whose numerators are each 1 and whose denominators are 5 and 7? Ans..

REM.-The numerator will be the difference and the denominator the product of the denominators.

11. Multiply 31, 51, 63, † of †, and 71⁄2. Ans. 355.

23. Divide of of off by off of P. Ans. 1.

Reduce to a simple fraction,

1. Twelve and five tenths.

2. Five hundredths.

3. Five thousandths.

4. Five hundred, and five hundredths.

5. Five thousand, and five thousandths.

6. Fifty thousand, and five ten-thousandths.

7. Three hundred and seventy-five thousandths. 8. Two thousand three hundred and seventy-five tenthousandths.

9. Thirty-two thousand three hundred and seventyfive hundred-thousandths.

10. Two hundred and seventy-five hundred-thousandths.

11. Seventy-five hundred-thousandths.

12. Five hundred thousandths.

Read the following:

13. .01, 2.07, 11.11, 13.0013, 50.501, .0016. 14. 33.00033, 119.019, 3.000003, 4.00044.

Add the following numbers:

1. Fifty-five thousand, and twelve thousandths; twentyone hundred, and twenty-one hundredths; fifty-two, and two hundred and seventy-five thousandths.

Ans. 57152.497.

2. Twenty-one, and twenty-one hundredths; two hundred and ten, and two hundred and ten thousandths; three thousand four hundred and fifty-one, and three thousand four hundred and fifty-one ten-thousandths. Ans. 3682.7651.

3. Forty-four, and four tenths; four hundred, and four hundredths; four thousand, and four thousandths; forty thousand, and four ten-thousandths; four hundred thousand, and four hundred thousandths; four million, and four millionths. Ans. 4444444.444444.

4. From four million, and four millionths take four thousand, and four thousandths. Ans. 3995999.996004.