DUODECIMALS.

Art. 122. DUODECIMALS are fractions of a square or cubic foot, and have their origin from the multiplication of one compound number by another, in finding the area of surfaces, and the cubical contents of solid bodies.

In measuring surfaces and solids, the dimensions are usually taken in feet and inches, feet being integers or whole numbers, and inches 12ths of a foot; hence it is plain that the multiplication of feet and inches by feet and inches is multiplying integers and 12ths by integers and 12ths. Then it follows, that, when feet in length are multiplied by feet in width, the product is square feet. When feet in length are multiplied by inches in width, or when inches in width are multiplied by feet in length, the product is 12ths of a square foot. When inches in length are multiplied by inches in width, the product is 144ths of a square foot or square inches.

It is also obvious, that, when square feet are multiplied by feet in length, the product is cubic feet. When square feet are multiplied by inches in length, or when 12ths of a square foot are multiplied by feet in length, the product is 12ths of a cubic foot. When 12ths of a square foot are multiplied by inches or 12ths of a foot in length, the product is 144ths of a cubic foot. When 144ths of a square foot or square inches, are multiplied by inches or 12ths of a foot in length, the product is 1728ths of a cubic foot or cubic inches.

ILLUSTRATION. What number of cubic feet are there in a granite pillar 3 feet 9 inches in width, 2 feet 3 inches in thickness, and 12 feet 6 inches in length?

Ans. 105++114+1728=10511⁄2 cubic feet.

Since the fractions of a foot decrease uniformly in a twelvefold ratio, we may write the numerators only, if we distinguish each of thern by some mark. 12ths of a foot are usually distinguished by an accent, thus, ('); 144ths, thus, ("); 1728ths, thus, (''' ). 12ths of a foot are called

primes; 144ths, seconds; and 1728ths, thirds.

Omitting the denominators, the operation will appear as follows.

Ans. Ft. 105 5′ 7′′ 6"" =

105 cubic feet. From the foregoing remarks and illustrations we derive this

RULE. Write the several denominations of the multiplier under the corresponding denominations of the multiplicand. Multiply the several denominations of the multiplicand by each of the denominations in the multiplier, in succession, beginning with the lowest, and write the first term or lowest denomination of each partial product directly under its multiplier. Find the sum of the partial products as in addition of compound numbers, their sum will be the total product.

1. What number of square feet are there in a board 12 feet in length, and 6 inches in width?

3. How many cubic feet are there in a box 5 feet in length, 4 feet in width, and 3 feet 6 inches in depth?

2. How many square feet are there in a floor 20 feet in length, and 15 feet 6 inches in width?

4. What number of cubic feet are there in a stick of timber 12 feet 9 inches in length, and 2 feet square?

5. What number of square feet are there in a floor 16 feet 6 inches long, and 12 feet 8 inches wide?

Ans. 209 square feet. 6. How many square feet are there in a board 17 feet

6 inches in length, and 1 foot 7 inches in width?

Ans. 27 sq. ft. 8′ 6′′-2717 sq. ft.

=

PROBLEMS.

Art. 123. A PROBLEM is a question proposed, in which two or more numbers or terms are given, to find one or more numbers or terms answering the conditions of the question. PROBLEM I. THE SUM OF TWO NUMBERS AND ONE OF

THEM BEING GIVEN, TO FIND THE OTHER.

The sum of two numbers is 20, and one of them is 12; what is the other?

12-8, the

If we take the given number 12 from the given sum 20, the number left must be the other; thus, 20other number. Hence, the following rule.

RULE. Subtract the given number from the given sum, the remainder will be the number required.

1. James gave 75 cents for an arithmetic and slate; the price of the arithmetic was 50 cents. What was the price of the slate?

2. A farmer paid 110 dollars for a yoke of oxen and a cow; the cow was worth 30 dollars. What was the value of the oxen?

3. Bell-metal is composed of copper and tin; the great bell at Moscow weighs 432000 pounds, and contains 345600 pounds of copper. What number of pounds of tin does it

contain?

Ans. 86400 lbs. of tin.

PROBLEM II. THE DIFFERENCE BETWEEN TWO NUMBERS AND THE SMALLER NUMBER BEING GIVEN, TO FIND THE GREATER NUMBER.

Suppose the difference between two numbers to be 25, and the smaller number to be 50; what is the greater number?

Adding the difference 25 to the smaller number 50, their sum is 75, the greater number. Hence, the following rule. RULE. Add the difference to the smaller number, their sum will be the greater number.

1. Henry paid 25 cents for his breakfast, which was 15 cents less than he paid for his dinner. How many cents did he pay for his dinner?

2. William paid 35 cents more for his cap than he paid for his shoes; his shoes cost 90 cents. What number of cents did he pay for his cap?

3. Suppose the difference between two numbers to be 4750, and the smaller number to be 7250; what is the greater number?

Ans. 12000.

PROBLEM III. THE DIFFERENCE BETWEEN TWO NUMBERS AND THE GREATER NUMBER BEING GIVEN, TO FIND THE SMALLER NUMBER.

Suppose the difference between two numbers to be 30, and the greater number to be 120; what is the smaller number?

If we take the difference 30 from the greater number 120, the remainder is 90, the smaller number. Hence, the following rule.

RULE. Subtract the difference from the greater number, the remainder will be the smaller number.

1. Sarah performs 350 ques- 2. Mary has studied 500 pages tions in arithmetic each week; of history; Eliza has studied 125 Caroline performs 75 questions pages less than Mary. What numless than Sarah. How many ques- ber of pages of history has Eliza tions does Caroline perform each studied? week?

3. Suppose the difference between two numbers to be 12650, and the greater number to be 24225; what is the smaller number? Ans. 11575.

PROBLEM IV. THE SUM AND DIFFERENCE OF TWO NUMBERS BEING GIVEN, TO FIND THE NUMBERS.

If, to the sum of two numbers, we add their difference, the amount will be twice the greater number. Take the numbers 60 and 40; their sum is 100, and their difference is 20, and 100+20=120, which is twice 60, the greater number.

Again; if, from the sum of two numbers, we subtract their difference, the remainder will be twice the smaller number; thus, 100-20-80, which is twice 40, the smaller number. Hence, the following rules.

RULE I. Add the difference of the two numbers to their sum, divide the amount by 2, the quotient will be the greater number; then subtract the difference from the greater number, the remainder will be the smaller number.

RULE II. Subtract the difference of the two numbers from their sum, divide the remainder by 2, the quotient will be the smaller number; then add the difference to the smaller number, their sum will be the greater number.

1. A gentleman paid 325 dollars for a horse and chaise; the chaise cost 75 dollars more than the horse. What was the cost of each?

2. The salaries of two teachers, A and B, amount to 2700 dollars a year; A receives 300 dollars more than B. What is the salary of each?

3. A gentleman left an estate amounting to 18500 dollars, to be divided between his wife and daughter; the wife was to have 3750 dollars more than the daughter. What number of dollars did each receive?

Ans. Wife received $11125. Daughter received $7375.

PROBLEM V. THE PRICE OF A UNIT OF ANY QUANTITY

BEING GIVEN, TO FIND THE VALUE OF THE GIVEN QUANTITY. ALSO, TO FIND THE VALUE OF ANY PART OF A UNIT OF THAT QUANTITY.

Suppose the price of a barrel of flour to be 7 dollars, what is the value of 5 barrels? The value of 5 barrels must be 5 times 7 dollars, which is 35 dollars. What is the value of 8 barrels? 8 times 7 dollars, which is 59 dollars. What is the value of of a barrel? of 7 dollars, which is 5 dollars. Hence, the following rule.

RULE. Multiply the price of a unit of the given quantity by the number expressing the quantity, the product will be the value of the quantity required.

1. If the price of a bushel of potatoes is 45 cents, what is the value of 40 bushels?

2. What is the value of a piece of cloth measuring 12 yards, at 20 cents a yard?

3. If an acre of land is worth 32 dollars, what is the value of 325 acres?

Ans. 10660 dollars.

4. If the price of an acre of land is 32 dollars, what is the value of of an acre? Ans. 28 dollars.

PROBLEM VI. THE VALUE OF ANY QUANTITY BEING GIVEN,

TO FIND THE PRICE OF A UNIT OF THAT QUANTITY.

Suppose the value of 5 barrels of flour to be 35 dollars, what is the price of 1 barrel? The price of 1 barrel must be of 35 dollars, which is 7 dollars. If 8 barrels be worth 59 dollars, what is 1 barrel worth? 1 barrel must be worth of 594 dollars, which is 7 dollars. If of a barrel be worth 5 dollars, what is a barrel worth? A