9. To multiply a number by 5 and that product by 7 is the same as multiplying at once by what single number? 10. Multiply the following numbers mentally by 10:

85, 75, 9, 90, 7.3, 8.4, 16.28, .832. 11. Multiply the same numbers mentally by 100; by 1000; by 10000.

12. Divide the numbers of problem 10 mentally by 10; by 100; by 1000.

13. How inay any number be quickly multiplied by 1 with any number of zeros following it, by the use of the decimal point only?

14. How may any number be divided by 10, 100, 1000, 10000, etc., by using the decimal point only?

15. To multiply a number by 100 and then divide the product by 2 is the same as multiplying by what single number?

16. Make a rule for quickly multiplying a number by 50; by 25; by 500; by 250; by 331; by 3331 17. Find the products of the following pairs of numbers:

18. How do the multipliers of problem 17 compare? The multiplicands? How, then, must the products compare? 19. Find the products of the following:

The figures on the right of the decimal point in a number are called

decimal places.

20. How many decimal places are there in the product of (2) problem 19? Of (3)? Of (4)? Of (5)? Of (7)?

21. How many decimal places are there in both the multiplicand and the multiplier of the product of (2) problem 19? Of (3)? Of (4)? Of (5)? Of (7)?

22. What relation is there between the number of decimal places in the product and the number of decimal places in both the multiplicand and the multiplier?

Decimal numbers are multiplied precisely as are whole numbers; then the point must be so placed as to give as many decimal places in the product as there are in both the multiplicand and the multiplier.

§16. Division.

1. Write the quotients of the following from the one that is given without dividing:

2. In each part of problem 1 compare the number of decimal places in the quotient with the difference between the number in the dividend and the number in the divisor. What seems to be the law for pointing the quotient?

To divide decimal numbers, divide first as in whole numbers, then so point the quotient that the number of places in the quotient is the number of places in the dividend minus the number of places in the divisor.

NOTE: This usual way of pointing quotients in decimals is not convenient when there are more decimal figures in the divisor than in the dividend.

Another way that some prefer is to imagine the decimal point moved over the same number of places in the same direction in both dividend and divisor, until only one digit stands to the left of the decimal point in the divisor. Then according to what the digit is, ask: "About what is 1, 3, 1, 1, etc., of the modified dividend?" and place the point in the quotient to make it as near the correct answer to the question as possible.

3. Why may the decimal point be moved toward the right over 2, or 3, or 4, places in both dividend and divisor without changing the quotient? Toward the left over the same number of places?

§17. Multiplying When Some Digits of the Multiplier Are Factors of Others.

1. From twice a number, how may you obtain 4 times the number? From 4 times a number, how obtain 8 times the number?

2. Multiply (1) 19,279 by 842; (2) 15,423 × 328.

3. Observe the relations of the digits of each multiplier below, and shorten the work of multiplication, as above.

Any of the above ways may be used to test the correctness of products obtained in the usual way.

To check by casting out the nines, cast the nines out of the multiplicand, the multiplier, and the product. Multiply the excesses of the multiplicand and the multiplier. Cast the nines out of this product. If this last excess equals the excess of the original product, the work is probably correct.

ILLUSTRATION.-To check 6848 × 619 = 4238912, excess in 68488; excess in 619 excess in 56

is checked.

=

7; excess in 4238912

=

2, 7 X 8

=

56, 2. As excess in 4238912 and in 56 are equal, the work

A very useful check against blunders is to examine the facts of the problem and to decide about what the answer must be, before beginning to solve it.

§19. Original Problems.

Make and solve multiplication problems from the following facts, and check the work:

1. Silver is worth 56¢ an oz., 12 oz. to the pound.

2. Hay costs $23 per T.; oats 42¢ per bu. A horse is fed 162 lb. of hay and 10 bu. of oats a month. Bedding costs $2 a month.

9

3. Standard silver is pure silver and copper. A silver dollar weighs 412.5 grains. The total number of silver dollars coined on this basis was 378,166,769.

4. The number of grains in an ounce of silver is 480. The amount of silver bought under the Sherman Law by the United States government to coin into money was 168,674,682 ounces.

5. Light travels 186,600 mi. per second. It requires 498 seconds for light to reach the earth from the sun.

6. It reaches the earth from the moon in 175 seconds. 7. An oz. of pure gold is worth $20.67. There are 12 oz. in 1 lb. of gold.

8. An Alaskan miner can take away 200 lb. of gold from the mining district.

9. A cu. ft. of granite weighs 170 lb. A granite step measures 2′ × 8′.

10. The distance around the driving wheel of a locomotive engine is 22 ft. In going a certain distance the driving wheel turned 5280 times.

11. In the year 1901, 82,305,924 lb. of tea and 511,. 041,459 lb. of coffee were imported into the United States, Tea was worth 48¢ and coffee 26¢ per pound.

12. A prize-winning steer weighed 15.03 cwt. (1 cwt. 100 lb.) and sold for $9.00 per hundredweight.

=

13. The price of oats was 48¢ a bushel. A farmer raised 38 acres of oats, averaging 48 bushels per acre.

14. Pupils should prepare and solve problems based upon price lists obtained from the grocer and the butcher, or from the market reports of the daily papers.

§20. Division with Larger Numbers.

1. 5,128,672 ÷ 9272 =?

553

9272)5128672 46360 49267

46360

SOLUTION. The divisor, 9272, being too large to use readily, we first use a trial divisor. For the same reason, we select a trial dividend. 92, the trial divisor, is contained in 512, 5 times; but as the whole divisor contains four digits the partial dividend must be enlarged. 9272 is contained in 51,286, 5 times. The quotient figure 5 is of the same order as the last figure of the 29072 trial dividend, which is hundreds. We write 5 27816 in hundreds place in the quotient. Multiplying, 1256 the whole divisor by the quotient figure we have the product 46,360. Subtracting this product from the trial dividend, 4926 remains. 4926 hundreds 49,260 tens; and 49,260 tens + 7 tens 49,267 tens. Continue in the same

=

manner with each step that follows.

=

The last subtraction gives a remainder of 1256. This remainder must also be divided by the divisor 9272.

5,128,672 9272 = 5531259