§81. By pursuing this investigation, we shall find that whenever we multiply both the numerator and the denominator by the same number, no matter what that number may be, the fraction made by that multiplication, will be equal to the first fraction. Hence, there is an equality between the following fractions, 1, 2, 3, 4, 1, 12, &c.; and also between the following,,,, 15, 18, &c.

4

5

18

6

6

16

§82. The principle just explained, leads to another which is of much importance. Suppose we multiply by 8, both terms of the fraction; and obtain 12. It is plain that both terms of the fraction 1 can be divided by 8, to bring the fraction back to. So also, if both terms in & be multiplied by 6, the fraction will be 1, which means just as much as; and, of course, if both terms in 14 be divided by 6, the fraction will be brought back to 2, which is equal to. So, in general, if we divide both the numerator and the denominator of a fraction by the same number, we have a new fraction which will be equal to the first. Thus, may be changed to ; to; 1 to 4.

§83. It is evident, that of several fractions of equal value, that which has the least denominator is the most easily understood. Thus, & of an apple is much better known at first sight, than 5 of an apple. And when a fraction is brought to as small a denominator as it can be changed to, we say it is reduced to its lowest terms.

§84. In order to reduce a fraction to its lowest terms, divide both the numerator and the denominator by any number that will divide each without a remainder. Thus, in the fraction, both terms may be divided by 5, by which we obtain ; and both terms of this last fraction may be divided by 3, by which we obtain §.

EQUATIONS.-SECTION 6.

Problems.

1. In an orchard, of the trees bear apples, of them bear pears, bear plums, and 81 bear cherries. How many trees are there in the orchard; and how many of each sort?

Now, we know that if we multiply by 4 we obtain x alone; that is, we destroy the fraction, and make it a whole number. And we know, that if we multiply the first member by 4; and also the last member by 4, we shall not destroy the equation. See §77. We will therefore multiply both members by 4, for the purpose of destroying the fraction in the first term. It will then become

Next, we will multiply both members by 5, to destroy the fraction in the second term. This will make

Then we will multiply by 11, to destroy the remaining fraction, which will make

55x+44x+40x+17820-220x.

2. In a certain school, of the boys learn mathematics, of them study Latin and Greek, and 6 study English grammar. What is the whole number of scholars?

come

After the question is stated, the equation will be

Multiplying by 5,

Multiplying by 4,

Transposing and uniting,

3. A gentleman has an estate, of which is woodland, of it pasture, and 105 acres embrace the pleasure grounds, gardens, and orchards. How many acres does it contain? Ans. 630 acres.

4. After paying away dollars yet in my purse.

and of my money, I find 22 How much had I at first?

Ans. $40.

5. A man bought a lot of ground, for which he agreed to pay as follows: of the money on taking possession, of it in 6 months, and $250 at the end of the year. How much did he pay in all? Ans. $600.

6. A post is one-fourth of its length in the mud, one-third in the water, and 10 feet above the water.

whole length?

What is its

Ans. 24 feet.

7. In a Christmas pudding, is flour, milk, eggs, suet and fruit, and of a pound of spices and other ingredients. What is the weight of the pudding?

The equation will be +*+*+*+3=x.

Ans. 15 pounds.

8. A lady being asked what her age was; replied, if you add }, 1, and † of my age together, the sum will be 18. How old was she?

After the question has been stated, the equation will be,号+砉+2=18. Ans. 24 years.

9. What sum of money is that, whose part, part, and part, added together, amount to 94 dollars?

Ans. $120.

10. A person found upon beginning the study of his profession, that he had passed of his life before he commenced his education, of it under a private teacher, the same time at a public school, and four years at the university. What was his age? Ans. 21 years.

11. How much money have I in my pocket, when the fourth and fifth part of it together, amount to $2.25 ?

Ans. $5.

12. The 3d part of my income, said a person, I expend in board and lodging, the 8th part of it in clothes and washing, the 10th part of it in incidental expenses, and yet I save $318 a year. What was his yearly income?

Ans. $720.

13. A gentleman bequeaths, in his will, the half of his property to his wife, one-sixth part to each of his two sons, the twelfth part to his sister, and the remaining $600 to his What was the amount of his property?

servant.

15. A farmer mixes a quantity of grain, so that 20 bushels less than of it is barley, and 36 bushels more

than of it is oats. How many bushels are there in the whole; and how many of each sort?

x
2

In stating the question; -20- barley, and +

36= oats.

Ans. 96 bushels in all; 28 of barley, and 68 of oats. 16. A teacher being asked how many scholars he had, replied, If I had as many more, half as quarter as many more, I should have 88. he?

many more, and

How

many had

In stating the question, he has x; and as many more is another x; &c.

Ans. 32.

17. In a mixture of copper, tin, and lead; 16 pounds less than was copper, 12 pounds less than was tin, and 4 pounds more than was lead. What was the weight of the whole mixture; and also of each kind?

Ans. 288lb.; and also 128lb., 84lb., and 76lb. 18. What is that number whose part exceeds its

part,

by 12?

The statement is the same as, of it equals

of it Ans. 144.

+12.
19. What number is that whose part exceeds its
by 72 ?

part Ans. 540.

20. A certain sum of money is to be divided amongst three persons, A, B, and C, as follows. A is to receive $3000 less than half of it, B $1000 less than the third of it, and C $800 more than the fourth of it. What is the sum to be divided; and what does each receive?

Ans. $38400; and also, $16200, $11800, $10400. 21. A man driving his geese to market, was met by another, who said, Good morrow, master, with your hundred geese. He replied, I have not a hundred; but if I had as