QUESTIONS ON ARITHMETIC.

PART I.

What is arithmetic? Define unit; number; integer. What is an abstract number? A concrete number? A simple number? A compound number? Define fraction; problem; rule; sign. What is notation? What is numeration? Name the letters that are used in Roman notation. How many figures are used to express numbers? Which figures are called significant, and why?

Define addition; subtraction; multiplication; division.

What is United States money? What other name is given to it? Of what metals are the United States coins made? Name the gold coins; the silver coins. How are dollars distinguished from cents when written in figures?

What is the meaning of reduction? How are fractions named? Define numerator; denominator; proper fraction. What is a mixed number? An improper fraction? What form of fraction is sometimes called complex? When is a fraction said to be in its lowest terms? When is one number said to measure another? What is necessary before adding or subtracting fractions?

What is cancellation? What is a factor of a number? Define prime number; prime factors of a number. Define common divisor of two or more numbers. Define greatest common divisor of two or more numbers. How is the greatest common divisor used? Define multiple; common multiple; least common multiple. How is the least common multiple used?

What is a bill? What is an account? What is the meaning of creditor? What is the meaning of debtor? When a debtor makes a payment, under what heading is the item placed? When a debtor receives value, under what heading is the item placed?

What is a denominate number? Name and define the two kinds. Name the different measures and weights, and give their uses.

Define decimal fraction, and show how it is written. Define mixed decimal; complex decimal. How many decimal places in any product? How many decimal places in any quotient? What is a denominate fraction? Give an example.

RULES AND TABLES.

PART I.

SIMPLE NUMBERS.

NOTATION.

BEGIN at the left hand, and write the figures belonging to the highest period.

Write the hundreds, tens, and units of each period in their order, putting a cipher in the place of any order that is omitted.

NUMERATION.

Point off the number into periods of three figures each, beginning at the right hand.

Beginning at the left hand, read each period separately, giving the name to each period except the last.

ADDITION.

Write the numbers to be added so that all figures of the same order shall stand in the same column.

Add all the figures in each column separately, beginning at the right, and place the right-hand figure of each sum under the column added, carrying the remaining figure or figures, if any, to the next column.

SUBTRACTION.

Write the less number under the greater so that figures of the same order shall stand in the same column,

Begin at the right, and subtract each figure of the subtrahend from the figure above it, and write the difference beneath.

If any figure of the subtrahend is greater than the figure above it, increase the upper figure by 10 before subtracting, and then add 1 to the next figure of the subtrahend and proceed as before.

MULTIPLICATION.

Write the multiplier under the multiplicand so that units of the same order shall stand in the same column.

Begin at the right, and multiply each figure of the multiplicand by the unit figure of the multiplier, placing the righthand figure of each product under the figure multiplied, and adding the left-hand figure or figures, if any, to the next product.

If the multiplier consists of more than one figure, proceed with each figure in the same manner as with the first, placing the right-hand figure of the first product under that figure of the multiplier which is used to produce it; the sum of the partial products is the required product.

DIVISION.

Write the divisor at the left of the dividend, with a line between them.

Find how many times the divisor is contained in the least number of figures at the left hand of the dividend that will contain it, and write the result for the first figure of the quotient.

Multiply the divisor by this quotient figure, and subtract the product from those figures of the dividend in which it was contained. Annex to the remainder the next figure of the dividend for a new dividend, and divide as before.

Proceed in the same manner till all the figures of the dividend have been used.

If any new dividend does not contain the divisor, write a cipher in the quotient and annex the next figure of the dividend.

UNITED STATES MONEY.

REDUCTION.

CASE I.

To change dollars to cents, multiply by 100. To change cents to mills, multiply by 10.

To change dollars and cents to cents, or dollars, cents, and mills to mills, remove the decimal point and the sign for doi lars. To change dollars and cents, or cents, to mills, remove the decimal point and the sign as before, and annex a cipher.

CASE II.

To change cents to dollars, divide by 100. To change mills to dollars, divide by 1000.

ADDITION.

Add as in simple numbers, placing dollars under dollars, cents under cents, and mills under mills. Write the decima! point in the amount between the dollars and cents.

SUBTRACTION.

Subtract as in simple numbers, placing denominations of the same kind under each other. Write the decimal point in the remainder between the dollars and cents.

MULTIPLICATION.

Multiply as in simple numbers, and place the decimal point in the product between the dollars and cents.

DIVISION.

Divide as in simple numbers, and place the decimal point in the quotient between the dollars and cents. When there are dollars and cents in both dividend and divisor, change both to the same denomination before performing the division.

COMMON FRACTIONS.

REDUCTION.

CASE I.

DIVIDE both numerator and denominator by any number that will measure them, until they are prime to each other.

CASE II.

Divide the required denominator by the given denominator, and multiply the terms of the given fraction by the quotient.

CASE III.

Multiply the whole number by the denominator, and if there is a numerator add it to the product, and place the denominator under the result.

CASE IV.

Divide the numerator by the denominator.

CASE V.

Multiply both terms of each fraction by the denominators of all the other fractions.

ADDITION.

When fractions have the same denominator, add the numerators and place the common denominator under the sum. When they have different denominators, change them to a common denominator, and add as before.

SUBTRACTION.

When the fractions have the same denominator, find the difference of the numerators and place it over the common denominator. When they have different denominators, change the fractions to a common denominator before subtracting the

numerators.