2. If 26 yards of cloth cost $39, what will 8 yards cost?

Since the answer is to be in dollars, we make 39 the third term. 8 yards will cost less than 26 yards. Since the fourth term is to be less than the third, the second term must be less than the first.

To solve problems by proportion :

Make that number the third term which is of the same kind as the required answer.

Of the two remaining terms, write the larger for the second term if the answer is to be greater than the third term, and the smaller for the second term if the answer is to be less than the third term.

Find the missing term, canceling when possible.

Proportion is simply another form of analysis, usually much shorter than the ordinary form. It is used largely in solving problems in geometry, physics, and chemistry.

3. A water meter registers 378 cubic feet in 27 days. At this rate, how many cubic feet will it register in 60 days?

4. The speeds of two automobiles are to each other as 8 to 15. If the rate of the slower is 20 miles per hour. what is the rate of the faster?

5. If 15 men can pave a street in 36 days, how many men must be employed to pave it in 18 days?

6. If a farmer receives $40 for a pile of wood containing 6 cords, how much ought he to receive for 15 cords?

7. Mr. A's income from an investment of $630 was $30. Mr. B's income in the same transaction was $50. Find his investment.

8. A farmer harvested 66 bushels of corn from a 4-acre field. How many bushels ought he to harvest from a 17-acre field?

9. How long ought a loan of $750 to be kept to balance a loan of $300 for 10 days?

10. A man owning

of his share at $27,400.

of of the building?

of a building estimates the value

At this rate, what is the value

11. If a farmer's daughter exchanges 15 dozen eggs for 20 yards of cloth, how many dozen will be required to pay for 3 yards?

12. A camping party of 6 persons has provisions sufficient for 15 days. If 3 more persons join the party unexpectedly, how long can the outing last?

13. What is the height of a flagstaff casting a shadow 125 feet long, if a stick 10 feet long casts a shadow 6 feet long at the same time?

14. By applying the principle of proportion, find the height of various objects, such as the school building, telegraph poles, trees, etc.

POWERS AND ROOTS

42 means that 4 is to be used twice as a factor; 48 that 4 is to be used three times as a factor.

A product obtained by using a number as a factor a specified number of times is a power.

The product obtained by using the number as a factor twice is the second power; by using the number as a factor three times, the third power; and so on.

The second power of a number is its square; the third .power, its cube.

The figure placed at the right and a little above the number is the exponent; it tells the power to be found.

1. Tell the second power, or square, of each of the numbers from 1 to 25. Of 30; 40; 50; 60; 70; 80; 90; 100.

2. Tell the third power, or cube, of each of the numbers from 1 to 12. Of 20; 30; 40; 50.

(3)2 means that is to be used as a factor twice. Thus,

One of the equal factors of a power is the root of the power.

One of the two equal factors of a number is the square root of the number; one of the three equal factors is the cube root. Thus, the square root of 64 is 8; the cube root

of 64 is 4.

A root is indicated by the sign √, called the radical sign. Thus, V81 means the square root of 81; $64 means

the cube root of 64.

A small figure, called the index, written in the angle of the radical sign tells the root to be found. The sign without any index figure means the square root.

From the above table it will be seen that the square of any whole number of one figure consists of one or two figures; the square of any whole number of two figures consists of three or four figures; the square of any whole number of three figures consists of five or six figures, and so on. That is, the square of any whole number consists of twice as many figures, or one less than twice as many figures, as in the number itself. Hence, the number of figures in the square root of any number may be deternined by beginning at units and separating the number

into periods of two figures each. Thus, the square root of 6724 consists of two figures: the square root of 17956 consists of three figures.

2. How many figures in the square root of 729? 2209? 11,664? 108,900? 640,000 ?

3. Find the square root of 1024.

32

1024 9 62 124

124

(1) Beginning at units, separate the number into periods of two figures each.

(2) Find the greatest square in the left-hand period (10). The greatest square is 9. Write its root (3) over the period, and write the square (9) under the period.

(3) Subtract 9 from 10 and bring down the next period. (4) Double the root already found and write the result at the left of the new dividend. 3 x 2 = 6. This is the trial divisor. Omitting the right-hand figure (4) of the dividend, find how many times the trial divisor is contained in 12. 12÷6=2. Write 2 in the root, and also at the right of the trial divisor. 62 is the true divisor.

(5) Multiply the true divisor (62) by the last root figure (2). Since there is no remainder, the square root of 1024 is 32.

To find the square root of a number:

(1) Beginning at units, separate the number into periods of two figures each.

(2) Find the greatest square in the left-hand period and write its root above it.

(3) Subtract this square from the period and bring down the next period.

(4) Double the root already found.

Omitting the last

figure of the dividend, divide the dividend by this number. Write the quotient in the root, and also at the right of the trial divisor.

(5) Multiply the true divisor by this last root figure.