1. The area of a perimeter is 34 feet.

rectangle.

EXERCISE 122.

rectangle is 60 square feet, and its Find the length and breadth of the

2. The area of a rectangle is 108 square feet. If the length and breadth of a rectangle are each increased by 3 feet, the area will be 180 square feet. Find the length and breadth of the rectangle.

3. If the length and breadth of a rectangular plot are each increased by 10 feet, the area will be increased by 400 square feet. But if the length and breadth are each diminished by 5 feet, the area will be 75 square feet. Find the length and breadth of the plot.

4. The area of a rectangle is 168 square feet, and the length of its diagonal is 25 feet. Find the length and breadth of the rectangle.

5. The diagonal of a rectangle is 25 inches. If the rectangle were 4 inches shorter and 8 inches wider, the diagonal would still be 25 inches. Find the area of the rectangle.

6. A rectangular field, containing 180 square rods, is surrounded by a road 1 rod wide. The area of the road is 58 square rods. Find the dimensions of the field.

7. Two square gardens have a total surface of 2137 square yards. A rectangular piece of land whose dimensions are respectively equal to the sides of the two squares, will have 1093 square yards less than the two gardens united. What are the sides of the two squares?

8. The sum of two numbers is 22, and the difference of their squares is 44. Find the numbers.

9. The difference of two numbers is 6, and their product exceeds their sum by 39. Find the numbers.

10. The sum of two numbers is equal to the difference of their squares, and the product of the numbers exceeds twice their sum by 2. Find the numbers.

11. The sum of two numbers is 20, and the sum of their cubes is 2060. Find the numbers.

12. The difference of two numbers is 5, and the difference of their cubes exceeds the difference of their squares by 1290. Find the numbers.

13. A number is formed of two digits. The sum of the squares of the digits is 58. If twelve times the units' digit is subtracted from the number, the order of the digits will be reversed.

Find the number.

14. A number is formed of three digits, the third digit being twice the sum of the other two. The first digit plus the product of the other two digits is 25. If 180 is added to the number, the order of the first and second digits will be reversed. Find the number.

15. There are two numbers formed of the same two digits in reverse order. The sum of the numbers is 33 times the difference of the two digits, and the difference of the squares of the numbers is 4752. Find the numbers.

16. The sum of the numerator and denominator of a certain fraction is 5; and if the numerator and denominator are each increased by 3, the value of the fraction will be increased by . Find the fraction.

17. The fore wheel of a carriage turns in a mile 132 times more than the hind wheel; but if the circumferences were each increased by 2 feet, it would turn only 88 times Find the circumference of each.

more.

CHAPTER XXI.

RATIO, PROPORTION, AND VARIATION.

316. The relative magnitude of two numbers is called their ratio, when expressed by the fraction which the first is of the second.

Thus, the ratio of 6 to 3 is indicated by the fraction, which is sometimes written 6:3.

317. The first term of a ratio is called the antecedent, and the second term the consequent. When the antecedent is equal to the consequent, the ratio is called a ratio of equality; when the antecedent is greater than the consequent, the ratio is called a ratio of greater inequality; when less, a ratio of less inequality.

318. When the antecedent and consequent are interchanged, the resulting ratio is called the inverse of the given ratio.

Thus, the ratio 3:6 is the inverse of the ratio 6: 3.

319. The ratio of two quantities that can be expressed in integers in terms of a common unit is equal to the ratio of the two numbers by which they are expressed.

Thus, the ratio of $9 to $11 is equal to the ratio of 9:11; and the ratio of a line 23 inches long to a line 3 inches long, when both are expressed in terms of a unit of an inch long, is equal to the ratio of 32:45.

320. Two quantities different in kind can have no ratio, for then one cannot be a fraction of the other.