1. PR. The front wheel of a carriage makes 6 more revolutions than the hind wheel in traveling 360 feet. But if the circumference of each wheel were 3 feet greater, the front wheel would make only 4 revolutions more than the hind wheel in traveling the same distance as before. What are the circumferences of the two wheels?

Let x stand for the number of feet in the circumference of front wheel, and y for the number of feet in the circumference of hind wheel. Then in traveling 360 feet the front wheel makes

If 3 feet were added to the circumference of each wheel, the

Whence x = 12, the circumference of the front wheel, and y= 15, the circumference of the hind wheel.

EXERCISES IX.

1. The square of one number increased by ten times a second number is 84, and is equal to the square of the second number increased by ten times the first.

2. The sum of two numbers is 20, and the sum of the square of the one diminished by 13 and the square of the other increased by 13 is 272. What are the numbers?

3. Find two numbers such that their difference added to the difference of their squares shall be 150, and their sum added to the sum of their squares shall be 330.

4. Find two numbers whose sum is equal to their product and also to the difference of their squares.

5. The sum of the fourth powers of two numbers is 1921, and the sum of their squares is 61. What are the numbers?

6. If a number of two digits be multiplied by its tens' digit, the product will be 390. If the digits be interchanged and the resulting number be multiplied by its tens' digit, the product will be 280. What is the number?

7. If a number of two digits be divided by the product of its digits, the quotient will be 2. If 27 be added to the number, the sum will be equal to the number obtained by interchanging the digits. What is the number?

8. The product of the two digits of a number is equal to one-half of the number. If the number be subtracted from the number obtained by interchanging the digits, the remainder will be equal to three-halves of the product of the digits of the number. What is the number?

9. If the difference of the squares of two numbers be divided by the first number, the quotient and the remainder will each be 5. If the difference of the squares be divided by the second number, the quotient will be 13 and the remainder 1. What are the numbers?

10. The sum of the three digits of a number is 9. If the digits be written in reverse order, the resulting number will exceed the original number by 396. The square of the middle digit exceeds the product of the first and the third digit by 4. What is the number?

11. A rectangular field is 119 yards long and 19 yards wide. How many yards must be added to its width and how many yards must be taken from its length, in order that its area may remain the same while its perimeter is increased by 24 yards?

12. The floor of a room contains 30 square yards, one wall contains 21 square yards, and an adjacent wall contains 13 square yards. What are the dimensions of the room?

13. A merchant bought a number of pieces of cloth of two different kinds. He bought of each kind as many pieces and paid for each yard half as many dollars as that kind contained yards. He bought altogether 19 pieces and paid for them $ 921.50. How many pieces of each kind did he buy?

14. The diagonal of a rectangle is 20 feet. If the length of one side be increased by 14 feet and the length of the other side be diminished by 2 feet, the diagonal will be increased by 123 feet. What are the lengths of the sides of the rectangle?

15. A certain number of coins can be arranged in the form of one square, and also in the form of two squares. In the first arrangement each side of the square contains 29 coins, and in the second arrangement one square contains 41 more coins than the other. How many coins are there in a side of each square of the second arrangement ?

16. A piece of cloth after being wet shrinks in length by one-eighth and in breadth by one-sixteenth. The piece contains after shrinking 3.68 fewer square yards than before shrinking, and the length and breadth together shrink 1.7 yards. What was the length and breadth of the piece?

17. A merchant paid $125 for two kinds of goods. He sold the one kind for $91 and the other for $36. He thereby gained as much per cent on the first kind as he lost on the second. How much did he pay for each kind?

18. Two workmen can do a piece of work in 6 days. How long will it take each of them to do the work, if it takes one 5 days longer than the other?

19. Two men, A and B, receive different wages. A earns $42, and B $40. If A had received B's wages a day, and B had received A's wages, they would have earned together $4 more. How many days does each work, if A works 8 days more than B, and what wages does each receive?

20. In 8 hours workmen remove a pile of stones from one place to another. Had there been 8 more workmen, and had each one carried 5 pounds less at each trip, they would have completed the work in 7 hours. Had there been 8 fewer workmen and had each one carried 11 pounds more at each trip, they would have completed the work in 9 hours. How many workmen were there and how many pounds did each one carry at every trip?

21. A man has two square fields in which he wishes to plant trees, the outer rows to be distant from the edges by half the distance between the rows. If he plants the trees in the first field 21 yards apart, and in the second field 2} yards apart, he will need 11,412 trees. But if he plants the trees in the first field 2 yards apart, and in the second field 3 yards apart, he will need only 569 trees. How long is a side of each of the fields?

22. A tank can be filled by one pipe and emptied by another. If, when the tank is half full of water, both pipes be left open 12 hours, the tank will be emptied. If the pipes be made smaller, so that it will take the one pipe one hour longer to fill the tank and the other one hour longer to empty it, the tank, when half full of water, will then be emptied in 15 hours. In what time will the empty tank be filled by the one pipe, and the full tank be emptied by the other?

CHAPTER XXV.

RATIO, PROPORTION, AND VARIATION.

§ 1. RATIO.

1. The Ratio of one number to another is the relation be tween the numbers which is expressed by the quotient of the first divided by the second.

E.g., the ratio of 6 to 4 is expressed by 4,

=

The ratio of one number to another is frequently expressed by placing a colon between them; as 5:7.

The first number in a ratio is called the First Term, or the Antecedent of the ratio, and the second number the Second Term, or the Consequent of the ratio.

Thus, in the ratio a: b, a is the first term, and b the second.

2. Since, by definition, a ratio is a fraction, all the properties of fractions are true of ratios; as a: b = ma: mb.

3. The definition given in Art. 1 has reference to the ratio of one number to another. But it is frequently necessary to compare concrete quantities, as the length of one line with the length of another line, etc.

If two concrete quantities of the same kind can be expressed by two rational numbers in terms of the same unit, then the ratio of the one quantity to the other is defined as the ratio of the one number to the other.

E.g., the ratio of 2 yards to 14 yards is 2: 14,

Observe that by this

quantities is a number.

21 35

==

14 16

definition the ratio of two concrete Also that the quantities to be com

pared must be of the same kind. Dollars cannot be compared with pounds, etc.

4. If two concrete quantities cannot be expressed by two rational numbers in terms of the same unit, they are said to be Incommensurable one to the other.

Thus, if the lengths of the two sides of a right triangle be equal, the length of the hypothenuse cannot be expressed by a rational number in terms of a side as a unit, or any fraction of a side as a unit.

=

If a side be taken as the unit, the hypothenuse is expressed by √2, an irrational number. And the ratio of the hypothenuse to a side is √2:1, √2, a number comprised in the number system. In the following article we will prove that the ratio of any two incommensurable quantities can be expressed as a number comprised in the number system.

5. Let P and Q be two incommensurable quantities. We assume that the ratio P: Q is greater than the ratio P': Q, wherein P' is less than P and is commensurable with Q, and that the ratio P: Q is less than the ratio P": Q, wherein P" is greater than P and is commensurable with Q. Let us take 19 Q as the unit. Then we can find two consecutive integral multiples of Q, which are therefore commensurable with Q, Q be these multiples. The m m + 1

n

n

between which P lies. Let

n

m+ 1 Qand ratios of these multiples to Q are respectively

the hypothesis

n

and

n

Then by

lies, have the properties (i.) and (ii.), Art. 6, Ch. XVIII. They therefore define an irrational number.

7. Which is the greater ratio, a +2b: a + b or a + 3b: a + 2 b?

6x+2y= 10, what is the value of the ratio x : y?

8. If

3 x

y

α

1

6.

to

?

х 3

(x-3)2