barrel must be worth of 5 dollars, which is 7 dollars. Hence, the following rule. RULE. Divide the value of the given quantity by the number expressing the quantity, the quotient will be the value of a unit of that quantity. 1. If 40 bushels of potatoes are worth 18 dollars, what is the value of 1 bushel? 2. If 12 yards of cloth are worth 21 dollars, what is the value of 1 yard? 3. A gentleman purchased a farm containing 3251 acres, for which he paid 10660 dollars; what did it cost per acre? Ans. 32 dollars. 4. A gentleman purchased a lot of land containing of an acre, for which he paid 28 dollars; what is the value of an acre? Ans. 32 dollars. PROBLEM VII. THE PRODUCT OF TWO NUMBERS OR FACTORS AND EITHER OF THEM BEING GIVEN, TO FIND THE OTHER.. If we analyze the questions in Problem 6, we shall perceive that the value of any given quantity is the product of the price of a unit multiplied by the number expressing the quantity. The area or superficial contents of a square or parallelogram is the product of its length multiplied by its breadth. Hence the following rule. RULE. Divide the given product by the given factor, the quotient will be the other factor. 1. The product of two numbers is 63, and one of them is 7; what is the other? 2. The product of two factors is 132, and one of them is 11; what is the other? 3. The floor of a school-room contains 2400 square feet; its length is 60 feet. What is its width ? Ans. 40 feet. 4. A certain lot of land contains 1650 square feet; its width is 22 feet. What is its length? Ans. 75 feet.. 5. A gentleman purchased a farm containing 120 acres ; its form is a right-angled parallelogram; its length is 160 rods. What is its width? Ans. 120 rods. 6. There is a board fence which is 5 feet in height, and contains 720 square feet; what is its length? 7. A farmer planted a piece of ground with potatoes in rows 3 feet apart; the hills in each row were also 3 feet apart; there were 120 rows and 18000 hills in the field. What was the number of hills in each row? PROBLEM VIII. THE PRODUCT OF THREE NUMBERS OR FAC TORS AND ANY TWO OF THEM BEING GIVEN, TO FIND THE OTHER. Suppose a stone pillar to be 3 feet in width, 2 feet in thickness, and 10 feet in length; what number of cubic feet does it contain? Multi Multiplying its width, 3 feet, by its thickness, 2 feet, the product is 6 square feet, the area of its end or base. plying this area 6 square feet, by its length 10 feet, the product is 60 cubic feet. Hence, it is plain, that if we divide the number of cubic feet in any regular solid body by the product of its width multiplied by its thickness, the quotient must be its length. Also, if the product of any three numbers or factors be divided by the product of any two of them, the quotient must be the third number or factor. Hence, the following rule. RULE. Divide the given product by the product of the two given numbers or factors, the quotient will be the third number or factor required. 1. Suppose a load of wood to be 8 feet long, and 4 feet wide; how high must it be to contain 1 cord or 128 cubic feet? 2. Suppose a pile of wood to be 8 feet high, and 4 feet wide; how long must it be to contain 4 cords or 512 cubic feet? 3. There is a stick of timber 1 foot 9 inches in width, 1 foot 6 inches in thickness; what must be its length to contain 1 ton? Ans. 19 feet. 4. The product of three numbers is 8250; the first is 40, the second is 16.5. What is the third? Ans. 12.5. 5. Suppose a carpenter wishes to construct a cubical box of sufficient capacity to contain 360 cubic feet; he has been directed to make it 12 feet in length, and 6 feet in width. What must be its depth? Ans. 5 feet. 6. There were 57600 cubic feet of earth dug from the cellar of a church; its length is 90 feet, and its depth 8 feet. What is its width? 7. In excavating a canal 2640000 squares of earth were removed; its width is 30 feet, and its depth 12 feet. What is its length? NOTE. A square of earth contains 216 cubic feet. PRACTICAL QUESTIONS IN DECIMAL FRACTIONS. Art. 124. 1. What is the sum of seventy-five hundred thousandths and ninety-five millions? What is their difference ? 2. The sum of two fractions is one hundred sixty-five ten-thousandths; the greater fraction is five thousandths. What is the smaller fraction? 3. The difference between two fractions is six hundred seventy-five thousandths; the smaller fraction is seventy-five thousandths. What is the greater fraction? 4. The difference between two fractions is three hundred and fifteen thousandths; the greater fraction is thirty-six hundredths. What is the smaller fraction? 5. The sum of two fractional numbers is 4.25; their difference is 2.75. What are the numbers? 6. The product of two fractional numbers is .015; one of the numbers is .06. What is the other? 7. If the price of a yard of broadcloth is $4.25, what is the value of .875 of a yard? 8. If .125 of a yard of broadcloth is worth .375 of a dollar, what is a yard of the same kind of cloth worth? 9. If 32.25 yards of shirting be worth $6.45, what is the value of 1 yard? 10. The floor of a school-room contains 983.125 square feet; its length is 32.5 feet. What is its width? 11. A gentleman purchased a lot of land containing 1868.625 square feet; it measures 24.75 feet in width. What is its length? 12. A gentleman purchased a farm for which he paid $3193.875; the price per acre was $25.50. What number of acres did the farm contain? 13. A log of mahogany contains 57.375 cubic feet; its length is 12.75 feet; its width is 2.25 feet. What is its depth? 14. There is a pile of wood which measures 5 cords; its length is 40 feet, its width is 3.75 feet. What is its height? 15. A carpenter was directed to make a bin of sufficient capacity to hold 75 bushels of grain. He was also directed to make it 8 feet in length inside, and 3.25 feet in width. What must be its depth? COMPARISON OF NUMBERS AND QUANTITIES. Art. 125. EVERY number is some proportional part of every other number, and every quantity is some proportional part of every other quantity of the same kind. We compare a less number with a greater to ascertain what part the less number is of the greater, and we compare the greater with the less to ascertain what part the greater is of the less. For a similar purpose, we compare a smaller quantity with a larger of the same kind, and the larger with the smaller. As fraction expresses every the part which its numerator is of its denominator, we can express the part which one of two given numbers is of the other by making that number which is called the part the numerator of a fraction, and the other the denominator; the fraction thus formed will express the required part. ILLUSTRATION. William has 5 apples and Henry has 10. It is plain that William has or as many as Henry, and it is equally plain that Henry has 12 or 2 times as many as William. What part of 8 shillings 3 pence is 2 shillings 9 pence? Ss. 3d. =99 pence. 2s. 9d. = 33 pence; and 33 pence is 33 or of 99 pence. = 625 What part of $1.625 is $4? $1.625 1625 mills. $4 = 4000 mills. 4000 mills is 4899 or 3 of 1625 mills. What part of 4.5 miles is 2.25 miles? 4.5= 450 hundredths. 2.25-225 hundredths. 225 hundredths is 225 or of 450 hundredths. What part of 47 is 23? 47=43. 23—13. 43 — 215. 13. And 117 is 113 of 25. From the above remarks and illustrations we derive the following RULE. If the two numbers or quantities are of different denominations, reduce them to the same. If the two fractions have different denominators, change them to fractions which shall have the least common denominator. Then write that number or numerator which the question requires to be a part of the other for the numerator of a fraction, and the other for the denominator; the fraction thus formed will express the part required. 1. What part of 9 pounds is 7 pounds? 3. What part of 17 acres is 15 acres? 5. What part of 2 shillings is 5 pence? 7. What part of 5 gallons is 3 quarts? 9. What part of 5 yards is 2.75 yards? 2. What part of 5 yards is 12 yards? 4. What part of 9 dollars is 16 dollars? 6. What part of 17 shillings is 3 pounds? 8. What part of 2 h. 30 m. is 1 h. 15 m.? 10. What part of of a mile is 11. What part of £12. 15s. 6d. is £9. 12s. 3d. 2qrs.? Ans. 1. 12. What part of 500 dollars is 25 dollars and 15 cents? 13. What part of 5 tons is 15 cwt. 3 qrs. 503 Ans. Too 14 pounds? Ans. 1. qr. 2 na. ? Ans. 골룸. 14. What part of 35 yards 3 qrs. is 15 yds. 1 15. What part of 5 hhds. is 3 hhds. 17 gals. 3 qts. 1 pt.? Ans. 181. 16. What part of 10 dollars is 15 dollars and 75 cents? Ans.. 17. What part of 42 miles is 17 miles 5 furlongs ? Ans. z. 18. What part of 125 acres is 175 acres 2 roods? 19. What part of of is of 7? 20. What part of 252 is of 12? 21. What part of 17 hours 20 minutes is 21 Ans.. Ans.. Ans. . hours? Ans. §. Ans. 184. 22. What part of 75.25 acres is 25.5 acres? 23. What part of a barrel of flour can I purchase with 5 dollars, when the price is $7.375 a barrel? Ans.. 24. When flour is worth 6 dollars a barrel, what part of a barrel can be purchased with 2.25 dollars? Ans. of a barrel. 25. When wheat is of a dollar a bushel, what part of a bushel can you buy with of a dollar? Ans. 26. What part of a ton of iron can you purchase with 27ğ dollars, when the price of a ton is 95 dollars? Ans. of a ton. 27. When coal is worth 7 dollars a ton, what part of a ton can you purchase with 2.125 dollars? Ans. 17 of a ton. 28. A owns 425 acres of land, B owns 375 acres. What part as much land does B own as A? Ans. 183. 503 |