1. From 9304206 there is taken a vay 327804. What is the remainder?

2. From 856262 there is taken away 76984. What is the remainder?

3. From 64231 there is taken away 5487. What is the remainder?

4. From 8534261 there is taken away 7986465. What is the remainder?

5. From 42764 there is taken away 3685. What is the remainder?

Ex. 26.

1. The larger of two numbers is 567804 and the smaller 56284. What is the difference?

2. The larger of two numbers is 75532 and the smaller 28967. What is the difference?

3. The larger of two numbers is 831244 and the smaller 79986. What is the difference?

4. The larger of two numbers is 7524125 and the smaller 6889. What is the difference?

5. The larger of two numbers is 94675 and the smaller 87659. What is the difference?

Mental Exercises in Addition and Subtraction.

19. The fundamental processes of addition and subtraction have been described under articles 13 and 14. One should, however, accustom himself to solve mentally such examples as are given under the following Exercises:

NOTE I. The pupil should practise upon combinations similar to those given in Ex. 27, until he can name results with a single mental effort.

NOTE 2. The examples under Ex. 28 can evidently be solved by an application of the first and the second principles of addition, and those under Ex. 29 by a similar application of the first and the second principles of subtractiou. Thus, to add 99 to 537, we add 100 and subtract 1 from the amount; to add 96 to 725, we add 100 and subtract 4 from the amount; and to add 104 to 233 we add 100 and add 4 to the amount. Following a similar principle, to subtract 99 from 875 we subtract 100 and add 1 to the remainder; to subtract 96 from 307 we subtract 100 and add 4 to the remainder; and to subtract 107 from 430 we subtract 100 and subtract 7 from the remainder. Name the results in the following exercises:

Ex. 30.

The following exercises will form a general review of notation, addition, and subtraction.

1. Find the sum of nine hundred thirty-four billion one hundred five thousand seven, and one hundred three thousandths four ten-thousandths; eight trillion two hundred five billion sixteen thousand, and seven hundred millionths; and two billion thirty-six million five hundred forty thousand, and two hundred eight thousandths nine hundred sixteen billionths seven ten-billionths.

2. Subtract eight trillion five million four, and five billionths six ten-billionths from three hundred nine trillion seven hundred eighty-four billion two hundred eighty-nine, and two hundred four millionths twenty-four hundred-millionths.

3. Find the sum of nine hundred forty-six trillion eight hundred thirty-four million sixteen, and eight hundred millionths thirty-three billionths seventy-six hundred-billionths; four hundred eight billion, and two thousandths seven hundred billionths; eighty-two trillion one, and four ten-trillionths; three hundred twenty-nine thousand eight, and two hundred forty thousandths fifteen millionths five billionths five hundred-billionths; and forty-six million nine hundred eightythree, and two hundred seven millionths six trillionths thirty-seven hundred-trillionths.

4. Subtract forty-eight billion one hundred thousand nine hundred thirty-four, and five hundred sixty-seven thousandths three hundred eight millionths forty-six billionths five hundred four trillionths from ninety-six billion five thousand four, and fourteen millionths seven ten-billionths.

5. Find the sum of two hundred four trillion nine million eleven, and fourteen thousandths five millionths seven hundred-billionths; eight billion four hundred million four thousand six, and nine hundred thousandths nine hundred-thousandths; seven hundred trillion nine hundred seven billion nine hundred forty-nine million one hundred, and five thousandths two hundred millionths sixty-five hundred-billionths.

6. Subtract three hundred four billion seven million eight hundred eleven, and seven millionths four hundred billionths from nine hundred billion eleven million eleven.

20. Problems in Addition and Subtraction.

In an arithmetical exercise, as the term is more commonly used, one is required simply to perform a certain operation. In a problem it is necessary first to determine by a course of reasoning what operation must be performed. The character of this reasoning in certain classes of problems is indicated by the following explanations:

66585947 2770217344

PROB. 3. A merchant sells during 23309004 4605918169 the year $125,000 worth of goods.

He receives from groceries $49,375.47, and from hardware $11,804.93. His remaining receipts are from dry-goods. What does he receive from dry-goods?

PROB. 4. Suppose that Venus and the Earth are on the same side of the Sun and in direct line with it. When so situated what is the least possible distance between them? (See note and diagram after Exercise 12.)

PROB. 5. Suppose that Uranus and Neptune are on opposite sides of the Sun and in direct line with it. When so situated what is the greatest possible distance between them?

EXPLANATIONS.

Prob. 1. By definition a difference is a number which must be added to a smaller number to produce a larger. Therefore, the larger number in the problem is the sum of 2874 and 9346, or 12220.

Prob. 2. From the definition of a difference it follows that a difference and a subtrahend equal a minuend. Therefore, a minuend may be thought of as a whole, and the subtrahend and a number equal to the difference as its two parts. It follows, therefore, that a given part, 937, must be subtracted from the given whole, 4256. 4256-937 = 3319. The answer to the problem, therefore, is 3319.