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$ 40. Numeration, Addition, Subtraction, Multiplication, and Division, are called the five ground rules of Arithmetic.
Q. How many principal rules are their in Arithmetic? What are they? Can Multiplication be performed by Addition? Can Division de performed by Subtraction ? By how many rules, then, may all the operations in Arithmetic be performed ?
§ 41. The preceding rules furnish answers to the following questions.
Ques. 1. When the cost of each one of several things is given, how do you find their entire cost?
Ans. Add the costs of the several things together, the sum will be the entire cost.
What is the entire cost of a bag of coffee at 6 dollars, a chest of tea at 4 dollars, a box of raisins at 2 dollars, and a barrel of sugar at 12 dollars ?
Ans. 24 dollars. Q. 2. When you have two unequal numbers, how do you find their difference?
A. By subtracting the less from the greater.
Q. 3. When the subtrahend and remainder are given, or known, how do you find the minuend?
A. By adding the remainder and subtrahend together. Hence the following principles.
1st. If the sum of two numbers be diminished by one of them, the remainder will be the other number.
2d. The less of two numbers added to their difference, will give the greater.
The sum of two numbers is 56, one of the numbers is 12: what is the other ?
Ans. 44. The less of two numbers is 25, and their difference 30: what is the greater ?
Ans. The less of two numbers is 35, and their difference 35 : what is the greater ?
Ans. 70. Q. 4. When you have the cost of a single thing, how will you find the entire cost of any number of things at the same rate ?
A. Multiply the cost of the single thing by the number of things.
What is the cost of 35 pears at 2 cents each ? What is the cost of 45 yards of cloth at 3 dollars per yard ?
Q. 5. When you know the number of things, and their entire cost, how do you find the cost of a single thing of the same kind ?
A. Divide the entire cost by the number of things, the quotient will be the cost of a single thing.
If 60 oranges cost 360 cents, how much do they cost apiece? If 40 yards of cloth cost 200 dollars, how much is it a yard?
APPLICATIONS IN THE PRECEDING RULES.
1. A Farmer sells a yoke of oxen for 90 dollars, 3 cows for 25 dollars each, 9 calves for 4 dollars each, and 65 sheep at 3 dollars a head. How much did he receive for them all ?
Ans. dollars. 2. The sum of two numbers is 365, one of the numbers is 221; what is the other number?
Ans. 144. 3. The difference of two numbers is 95, the less number is 327; what is the greater number? Ans.
4. A farmer sells 4 tons of hay at 12 dollars per ton, 80 bushels of wheat at 1 dollar per bushel, and takes in part payment a horse. worth 65 dollars, a wagon worth 40 dollars, and the rest in cash. How much money did he receive?
Ans. 23 dollars. 5. A farmer has 14 calves worth 4 dollars each, 40 sheep worth 3 dollars each; he gives them all for a horse worth 150 dollars: does he make or lose by the bargain ?
Ans. He loses dollars. 6. The product of two numbers is 51679680, and one of the factors is 615: what is the other factor?
Ans. 84032. 7. When the divisor is 67941, and the quotient 30620, what is the dividend?
Ans. 2080353420. 8. When the dividend is 1213193, the quotient 37, what is the divisor?
Ans. 9. A piece of cloth containing 65 yards costs 455 dollars : at does it cost per yard?
Ans. dollars. 10. A man has 6 children, all of whom are married, and each has four children; two of these grand-children are
married, and each has one child: how many children, grand-children, and great grand-children are there?
Ans. 11. The distance around the earth is computed to be about 25000 miles: how long would it take a man to travel it, supposing him to travel at the rate of 35 miles a day?
Ans. 7148 days. 12. The earth moves around the sun at the rate of 68000 miles an hour: how many miles does it travel in a day, and how many in a year?
1632000 in a day. Ans.
595680000 in a year. 13. A farmer purchased a farm for which he paid 18050 dollars. He sold 50 acres for 60 dollars an acre, and the remainder stood him in 50 dollars per acre: how much land did he purchase?
Ans. 351 acres.
OF FRACTIONS. § 42. The unit 1 represents an entire thing; as 1 apple, 1 chair, 1 pound of tea.
If we suppose 1 thing, as one apple, or one pound of tea, to be divided into two equal parts, each part is called one half of the thing.
If the unit be divided into 3 equal parts, each part is called one third.
If the unit be divided into 4 equal parts, each part is called one fourth.
If the unit be divided into 12 equal parts, each part is called one twelfth; and when it is divided into any number of equal parts, we have a similar expression for each of
The equal parts of a thing are expressed thus : } is read one half.
is read one seventh. one third.
one eighth. one fourth.
one tenth. one fifth.
one fifteenth. one sixth.
one filiieth. The }, }, 1, &c., are called fractions
Q. What does the unit 1 represent? If we divide it into two equa. parts, what is each part called? If it be divided into three equal parts, what is each part ? * Into 4, 5, 6, &c., parts ? What are such expressions called ?
§ 43. Each fraction is made up of two numbers; the number which is written above the line is called the numerator; and the one below it is called the denominator, because it gives a denomination or name to the fraction.
For example, in the fraction , 1 is the numerator, and 2 the denominator. In the fraction }, 1 is the numerator, and 3 the denominator.
The denominator in every fraction shows into how many equal parts the unit, or single thing, is divided. For example, in the fraction }, the unit is divided into 2 equal parts; in the fraction }, it is divided into 3 equal parts; in the fraction , it is divided into 4 equal parts, &c. In each of the fractions one of the equal parts is expressed. But suppose
were required to express 2 of the equal parts, as 2 halves, 2 thirds, 2 fourths, &c. We should then write, 3 they are read two halves.
two thirds. two fourths.
two fifths, &c. If it were required to express three of the equal parts, we should place 3 in the numerator; and generally, the numerator shows how many of the equal parts are expressed in the fraction. For example, three eighths are written,
and read three eighths.
nine twentieths. Q. Of how many numbers is each fraction made up ? What is the one above the line called ? The one below the line ? What does the denominator show? What does the numerator slow? In the three cighths, which is the numerator? Which the denominator ? Into how many parts is the unit divided? How many parts are expressed? In 'he fraction wine-twentieths, into how many parts is the unit divided ? llow many parts are expressed ?
6 13 9 20
4 16 5 18 6 25 7
§ 44. When the numerator and denominator are equal, the numerator expresses all the equal parts into which the unit has been divided : therefore, the value of the praction is equal to 1. But if we suppose a second unit, of the same kind, to be divided into the same number of equal parts, those parts may also be expressed in the same fraction with the parts of the first unit. Thus,
is read three halves.
twenty-five sevenths. The denominator of the first fraction, shows that a unit has been divided into 2 equal parts, and the numerator expresses that three such parts are taken. Now, two of the parts make up one unit, and the remaining comes from the 2d unit: hence, the value of the fraction is. 1}; that is, one and one half.
The denominator of the second fraction, shows that a unit has been divided into four equal parts, and the numerator expresses that 7 such parts are taken. Four of the 7 parts come from one unit, and the remaining 3 from a second unit: the value of the fraction is therefore equal to 13; that is, to one and three-fourths. In the third fraction, the unit has been divided into 5 equal parts, and 16 such parts are taken. Now, since each unit has been divided into 5 parts, 15 of the 16 parts make 3 units, and the remaining part is 1 part of a fourth unit. Therefore, the value of the fraction is 3?: that is, three and one fifth. The value of the fourth fraction is three, and of the fifth, three and four-sevenths. From what has been said, we conclude:
1st. That a fraction is the expression of one or more parts of unity.
2d. That the denominator of a fraction shows into how many equal parts the unit or single thing has been divided, and the numerator expresses how many such parts are taken in the fraction.