12. Two cubical coal bins together hold 280 cubic feet of coal, and the sum of their lengths is 10 feet. Find the length of each bin.

13. There are two numbers whose difference is 4, and the difference of their cubes exceeds the difference of their squares by 276. Find them.

14. A number of two digits is equal to 4 times the sum of its digits, and the sum of the squares of the digits is 45. What is the number?

15. 168 feet of fence inclose a rectangular plot of land, and the area of the plot is 1440 square feet. Find the dimensions.

16. The sum of the squares of two numbers is 25, and the sum of their fourth powers is 337. What are the numbers?

17. The product of two numbers is 15 more than 5 times the larger number, and 6 less than 16 times the smaller number. Find the numbers.

18. In a number of two digits the units' digit is 3 times the tens' digit, and if the number is multiplied by the sum of the digits, the product is 52. Find the number.

19. If a number of two digits is multiplied by the tens' digit, the product is 96; but if the number is multiplied by the units' digit, the product is 64. Find the number.

20. The simple interest on $600 for a certain number of years at a certain rate is $120. If the time were 2 years shorter, and the rate 2% more, the interest would be $108. Find the time and the rate.

21. Two men working together can paint a house in 6 days, but one of them working alone would require 5 more days than the other if he worked alone. How many days would each working alone require for the task?

22. If the dimensions of a rectangle were each increased 2 feet, its area would be 165 square feet. If each were decreased 2 feet, the area would be 77 square feet. Find the dimensions of the rectangle.

23. If the sum of two numbers is added to their product, the result is 47. The sum of the squares of the numbers is 62 more than the sum of the two. What are the numbers?

24. The area of a rectangular field is 15 acres, and the perimeter is 200 rods. What are the dimensions of the field, and what is the length in feet of the diagonal?

25. The combined capacity of two cubical tanks is 637 cubic feet, and the sum of an edge of one and an edge of the other is 13 feet. Find the length of a diagonal of any face of each cube.

26. The sum of the areas of two square floors is 1300 square yards, and the combined length of the two perimeters is 600 feet. Find the area of each floor in square yards.

27. A web of cloth costs 50 cents more per yard than another, and each web costs $ 63. One web is 8 yards longer than the other. How many yards of cloth are there in each web?

28. The sum of the squares of the two digits of a number is 13. If the square of the units' digit is subtracted from the square of the tens' digit, and the remainder is divided by the sum of the digits, the quotient is 1. Find the number.

29. The sum of the squares of two numbers is increased by the sum of the numbers, and the result is 18. The difference of the squares of the numbers is increased by the difference of the numbers, and the result is 6. Find the numbers.

30. A certain floor having an area of 50 square feet can be covered with 360 rectangular tiles of a certain size; but if the masons use a tile 1 inch longer and 1 inch wider, the floor can be covered with 240 tiles. Find the sizes of the different tiles.

31. The sum of the volumes of two cubical blocks of stone is 2240 cubic inches, and the sum of an edge of one and an edge of the other is 20 inches. Find the volume and the total surface of each cube.

32. A field is 30 rods long and 20 rods wide, but its length is decreased and its width increased so that its area is 44 square rods greater. The change increases the perimeter by 2 rods. What amount is added to the length and subtracted from the width?

33. A bicyclist

starts on a 12-mile trip, intending to arrive at a certain time. After going 3 miles, he is delayed 15 minutes, and he must complete the journey at a rate 3 miles an hour faster in order to arrive on time. Find his original rate of speed.

34. If the difference of the squares of two numbers is divided by the smaller number the quotient and the remainder are each 4. If the difference of the squares of the numbers is divided by the greater number, the quotient and the remainder are each 3. What are the numbers?

35. A boatman rows 18 miles downstream in 6 hours less time than it takes him to return. If he were to double his ordinary rate, his rate downstream would be 10 miles an hour. Find his rate of rowing in still water, and the rate of the stream.

36. Find the sides of a rectangle whose area is unchanged if its length is increased by 4 feet and its breadth decreased by 3 feet; but which loses one third of its area if its length is increased by 16 feet and its breadth decreased by 10 feet.

37. Two bodies move toward each other from A and B and meet after 35 seconds. If it takes one 24 seconds longer than the other to move from A to B, how long does it take each to traverse the distance?

38. A and B start simultaneously from two towns to meet one another. A travels 1 mile an hour faster than B, and they meet in 4 hours. If B had increased his rate by 1 mile an hour and A had traveled at of his former pace, they would have met in 4 hours and 16 minutes. How far apart are the towns?

39. If the length of a certain rectangle is increased by 2 feet, and the width is decreased by 1 foot, the area of the rectangle will be unchanged. The area of the rectangle is the same as the area of a square whose side is 3 feet greater than the width of the rectangle. What are the dimensions of the rectangle?

40. A certain principal at a certain rate amounts to $1560 in one year at simple interest. If the principal were greater by $100 and the rate 1 times as great, the amount at the end of two years would be $1792. What is the principal and

what is the rate?

41. A rectangular piece of tin is made into an open box by cutting a 5-inch square from each corner and turning up the sides and ends. If a 3-inch square were cut from each corner, the box, made in the same way, would hold the same amount. The tin is 4 inches longer than it is wide. How much does the box hold?

42. A certain farm is rectangular in shape, and its length is four times its width. The farm cost as many dollars per acre as it is rods in length, and the number of dollars paid for it was four times the number of rods in its perimeter. Find the length and the width of the farm in rods, and the price paid for it.

43. Twenty persons contributed to the purchase price of a gift, one half of the whole amount being furnished in equal portions by the women of the party, and the other half by the men. Each man gave one dollar more than each woman, and a total of $48 was paid for the gift. Find how many men and how many women contributed to the cost, and the amount that each gave.

CHAPTER XXIV

RATIO. PROPORTION. VARIATION.

RATIO

349. If a and b are the measures of two magnitudes of the same kind, then the quotient of a divided by b is the ratio of a to b.

Ratios are expressed in the fractional form, %, or with the colon, a:b. Each form is read " a is to b."

350. In the ratio, a: b, the first term, a, is the antecedent, and the second term, b, is the consequent.

THE PROPERTIES OF RATIOS

351. The properties of ratios are the properties of fractions;

α

for the ratios,, m:n, (x+y): (x − y), etc., are fractions.

(a) THE MULTIPLICATION AND THE DIVISION OF THE
TERMS OF RATIOS

352. The value of a ratio is unchanged if both its terms are multiplied or divided by the same number.

353. A ratio is multiplied if its antecedent is multiplied, or if its consequent is divided, by a given number.

354. A ratio is divided if its antecedent is divided, or if its consequent is multiplied, by a given number.