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CHECKING ADDITION BY CASTING OUT THE NINES

To cast out the nines from a number add its digits and whenever the sum equals or exceeds 9, drop 9 and continue adding the next digits to what remains, dropping 9 whenever the sum equals or exceeds 9. The last remainder is called the excess.

ILLUSTRATION.-Cast the 9's out of 647,255.

Beginning on the left, 6+4=10; drop 9, giving the remainder 1. 1+7+2=10, drop 9. 1+5+5=11, drop 9. The last remainder is 2 and this is the excess of 647,255.

To check addition by casting out nines, cast the nines out of the addends and the sum. Then cast out the nines from the excesses of the addends. If this last excess of the excesses equals the excess of the sum, the work is probably correct.

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7. Add the following and check by casting out the nines:

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1. What number added to 8 gives 14?

Think thus: 2 added to 8 gives 10; and 4 more makes 14. 2 and 4 (or 6) added to 8 gives 14. Or, one may say directly, 14 minus 8, or 8 from 14, 6.

2. What number added to 18 gives 24? 34? 44? 54?

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Subtraction is the way of finding the difference between two numbers,

or of taking from a number a part of itself.

The number to be subtracted is the subtrahend.

The number from which to subtract is the minuend.

The result is the difference, or remainder.

4. Find the differences of the following pairs of num

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Write the numbers to be subtracted in a column, so that the decimal points are in a vertical line. Write zeros enough in the minuend to make the same number of decimal places as there are in the subtrahend. Then subtract as with whole numbers.

CHECKING SUBTRACTION

Check subtraction by adding the difference to the subtrahend and noticing whether the sum equals the minuend.

Subtraction may be checked also by casting out the nines.

All that is necessary is to cast out the nines from the minuend, the subtrahend and the remainder. If the difference between the excesses of the minuend and the remainder equals the excess of the remainder, the work is probably correct.

ILLUSTRATIONS

Check by casting out nines. Check by addition.

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5. Subtract the following and check in both ways:

(1) 8700 (2) 58.00 (3) 1062.700 (4) 9036.0900 (5) 681.7060

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§11. Problems in Addition and Subtraction.

1. The published daily circulation of a city newspaper

from week to week is here given:

FIRST WEEK SECOND WEEK THIRD WEEK FOURTH WEEK FIFTH WEEK

DAY COPIES DAY COPIES

DAY COPIES DAY COPIES DAY COPIES

1..255,572

2..303,062

8..305,725

9..303,489

3..323,513 10..304,991

15..304,636 22..304,836 29..303,383 16..302,173 23..304,698 30..302,005 17..302,650 24..310,870 4..312,724 11..304,746 18..304,255 25..299,780 5..306,009 12..305,515 19..302,942 26..301,455 6..297,918 13..299,095 20..297,684 27..298,446 Total

Total

Total

Total

Total

Total for mo.

Find the weekly totals and the total for the month.

2. In the year 1891 the United States imported 94,628,119 lb. of coffee, which was 54,262,757 lb. less than was imported during the two previous years. How many pounds were imported during the two previous years?

3. The United States imported 79,192,253 lb. of tea in 1889; 83,494,956 lb. in 1890, and 82,395,924 lb. in 1891. How many pounds of tea were imported in the three years?

4. In 1889, $27,024,551 worth of molasses was imported into the United States; in 1890, $31,497,243 worth, and in 1891, $32,659,172 worth. How many dollars' worth was imported during the three years?

§12. Areas of States and Territories.

Such problems may be selected from the following list as have a special geographical interest to each school.

From the facts on page 13, solve the following:

1. Find the total area of the North Atlantic division of states; of the South Atlantic division. Check the work.

2. Which is the larger, and by how much, the North or South Atlantic division? Check the work.

3. Find the total area of the North Central division; of the South Central division. Check the work.

4. Find the total area of the Western division. Check the work.

5. Which is the larger, and by how much, the Western or the North Atlantic division? The Western or the South Atlantic division? The Western or the South Central division? Check the work.

6. Find the total area of the United States without the outlying territory. Check the work.

7. Find the total area of the outlying territory of the United States. Check the work.

8. Find the total area of the United States including the outlying territory. Check the work.

9. Compare the area of the section you live in with that of the other section, or sections of the same division. 10. Solve other problems on the table of page 13.

AREAS IN SQUARE MILES OF STATES AND TERRITORIES
GEOGRAPHICALLY GROUPED

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