Summary of Definitions.

Exact Divisor. A number contained without a remainder in a

given number.

Common Divisor. A number contained without a remainder in two or more given numbers.

Greatest Common Divisor. The greatest number contained without a remainder in two or more given numbers.

Multiple. A number that will contain a given number without a remainder.

Common Multiple. A number that will contain two or more given numbers without a remainder.

Least Common Multiple. The least number that will contain two or more given numbers without a remainder.

Fraction. One or more equal parts of a unit.

An indicated division.

Decimal. A fraction in which the unit is divided into a number of parts denoted by a power of 10.

An expression of division in which the divisor is indicated by so placing a period, or decimal point, in the dividend that each figure at its right will represent a 0 of the divisor.

Common Fraction. A fraction in which the unit is divided into a number of parts not denoted by a power of 10.

An expression of division in which the dividend is written above and the divisor below a horizontal line.

Proper Fraction. A fraction in which the indicated operation cannot be performed.

Improper Fraction. A fraction in which the iudicated operation can be performed.

Mixed Number. A whole number and a fraction considered as one number.

Reduction of Fractions. Changing the forms of fractions without changing their value.

Reciprocal of a Fraction. The fraction with its terms inverted. Compound Fraction. A term applied to two or more fractions connected by the word 'of'.

Complex Fraction. A fraction with a fraction for a denominator, or numerator, or both.

Finite Decimal. The exact equivalent of some common fraction whose denominator contains no other factor than 2 or 5. Infinite Decimal. The approximate equivalent of a common fraction whose denominator contains some other factor than or 5.

Review Questions.

What numbers are divisible by 2? by 5? by 4? by 25? by 8? by 125? Explain the principle that governs the divisibility of a number by each of the preceding numbers.

What numbers are divisible by 3? by 9? by 11? Explain the principles governing such divisibility.

If a number is divisible by 5 and by 6, is it necessarily divisible by 30? Why?

If a number is divisible by 4 and by 6, is it necessarily divisible by 24? Why not ?

Define a divisor; a common divisor; a greatest common divisor. Find the greatest common divisor of 35, 48, and 63. Give a rule for finding the greatest common divisor of two or more numbers. Find the least common multiple of 15 and 25; of 156 and 360. Give a rule for finding the least common multiple of two numbers. Find the least common multiple of 36, 56, and 64; of 128, 320, 144, and 4Sɔ. Give a rule for finding the least common multiple of more than two numbers.

Define cancellation. Reduce to its simplest form the expression (48 X 7 X5 X5 X 150) ÷ (35 × 3 × 30X8×5). Give a rule for simplifying an expression by cancellation.

Define a fraction; a numerator; a denominator; a proper fraction; an improper fraction; a mixed number; a compound fraction; a complex fraction; a common denominator; a least common denominator; the reciprocal of a fraction; reduction of fractions. Give in written form an illustration of each.

128

Reduce to its lowest terms. Give a rule for reducing a fraction to its lowest terms.

{+} = ? }1⁄2-12=? Give a rule for adding fractions having a common denominator; for subtracting.

What fractions with a common denominator are equivalent to and? Give a rule for reducing fractions to equivalent fractions having a common denominator.

?? Give a rule for adding fractions not having a common denominator; for subtracting.

3+4 ? 4-23

for subtracting.

=? Give a rule for adding mixed numbers;

Explain the multiplication of by 4; of by 3. Give a rule for multiplying a fraction by an integer.

Explain the multiplication of 8 by 4; of 8 by . Give a rule for multiplying an integer by a fraction.

Explain the multiplication of by . Give a rule for multiplying a fraction by a fraction.

Explain the multiplication of 53 by 4; of 8 by 63. Give a rule for multiplying when the multiplier or the multiplicand is a mixed number.

Explain the division of & by 4; of by 5. Give a rule för dividing a fraction by an integer.

Explain the division of 8 by ; of 8 by 5. Give a rule for dividing an integer by a fraction.

Explain the division of 3 by 2. Give a rule for dividing when both dividend and divisor are fractions.

Explain the multiplication of 53 by 63. Give a rule for multiplying when both multiplier and multiplicand are mixed numbers. Multiply 95} by 341. Give a rule for such multiplications. Multiply 79 by 791; 9

cations.

by 97. Give a rule for such multipli

Multiply 24 by 1511; 3242 by 896. Give a rule for such multiplications.

Divide 2 by 32; 7 by 23. Give a rule for dividing by a mixed number.

Explain the division of 43 by 3. Give a rule for dividing a mixed number by an integer.

Give a rule for

Explain the reduction of 21 to an improper fraction. Give a rule for reducing a mixed number to an improper fraction. Explain the reduction of to a mixed number. reducing a mixed number to an improper fraction. Write a compound fraction. Simplify it.

fying a compound fraction.

Give a rule for simpli

Write a complex fraction. Simplify it. Give a rule for simplifying a complex fraction.

15

Can be reduced to an exact decimal? 773? 13? 74? 275? What fractions alone can be reduced to exact decimals?

Reduce to

a decimal 13; 17. How many o's must be added to the numerator of 3 that it may be changed to an exact decimal? to the numerator of? to the numerator of 15 ? to the numerator of any fraction whose denominaor contains only 2's and 5's.

What common fraction is eqnivalent to .5? to .05? to .25? to .005. ? to .785? to .0005? to .0384? Give a rule for changing a decimal to a common fraction. Change each of the preceding decimals to a common fraction in its lowest terms.

Explain the multiplication of .075 by .34. Give a rule for multiplication of decimals.

Explain the division of 8.3925 by .45. Give a rule for division of decimals.

Explain the plan of paying for milk and cream at the Claremont Creamery.

Complete and read the table on pages 137 and 138.

Multiply, without writing the partial products, 964 by 33; 4967 by 43. Give a rule for such multiplications.

Explain in full the process of proving a multiplication by eliminating 9's; by eliminating 11's. Explain the process of proving a division by eliminating 9's; by eliminating 11's

Explain the obtaining of approximate results in multiplication of decimals; in division of decimals.

Explain the converting of factors into more convenient factors; the multiplying of 63 by 67, 54 by 56, etc.; the process of dividing in long division without writing the partial dividends.

Give the six Natural Divisions in the order of the area of each; of the population of each; of the density of population of each. Explain each problem on page 153.

Review Exercises and Problems.

NOTE. Make all exercises of your own construction as difficult as possible, so as to thoroughly test your knowledge of the principles underlying them.

1. Write in figures a number of five integral and four decimal periods. Write the same number in words, writing the decimal portion first in separate periods and secondly by the common method.

2. Write in words, by both methods, a number of four integral and four decimal periods. Write the same number in figures.

3. Find the area of all the states west of the Mississippi River; find the population of these states. Find the area of the states east of the Mississippi River; find the population of these states. Find the density of population of the first group of states; of the second group. What would be the population of the first group if its density of population were that of the second? What would be the population of the second group if its density of population were that of the first?

4. Find the difference between the areas of each of the two following divisions: Hawaii and Connecticut; Italy aud Colorado; Belgium and New Hampshire; Philippine Islands and Arizona; Japan and California; Scotland and South Carolina; Texas and Austria-Hungary; England and New York.

5. Using multiplicands of ten figures, multiply by short methods by 25; by 31; by 125; by 163; by 333; by six other convenient parts of some power of ten.

6. Using a dividend of ten figures, divide by 250; by 121; by 13; by six other convenient parts of some power of ten. Express each result both as a complete quotient and as an integral quotient with a remainder.

7. Using multiplicands of six figures, multiply without performing any addition of partial products, by 56; by 72; by 64; by 240; by 360. Divide dividends of 8 figures by the same numbers, expressing each result both as a complete quotient and as an integral quotient with a remainder.

8. Using factors with 0's at their right, perform six exercises in multiplication. Using divisors ending with 0's, per form six exercises in division. Express each result both as a quotient and as an integral quotient with a remainder.

9. Using only two multipliers, multiply multiplicands of six figures by 568; by 497; by 246; by 832. Using only three multipliers, multiply multiplicands of six figures by 128328; by 756112; by 27963.

10. Without multiplying by any figure of the multiplier, multiply a multiplicand of six figures by 98; by 997; by 999; by 498; by 9975.

11. Without writing down either of the partial products, multiply by 11 a multiplicand of six figures; of eight figures; of fifteen figures.

12. Make and solve five exercises in the greatest common divisor; five in the least common multiple of two numbers; five in the least common multiple of more than two numbers.

13. Write five proper fractions and reduce them to their lowest terms. Write five improper fractions and reduce them to mixed numbers. Write five mixed numbers and re