2. Required the interest on $247.50, for 5 years, at 6 per cent.

3. The equation trpi contains four quantities, any three of which being known, the fourth may be found. For example, to find p when the other quantities are given. Take

trp=i; divide both members by tr,

This formula gives the following rule.

To find the principal, when the interest, time, and rate are known, divide the interest by the product of the time and rate.

4. The interest being $90.10, the time 4 years, and the rate 5 per cent., required the principal, according to the rule.

Let the learner find, from trpi, formulæ for the time and the rate, translate these formulæ into rules, and perform by the rules the particular examples subjoined.

5. The interest, rate, and principal given, find the time.

6. The interest being $54, rate 6 per cent., principal $150, what is the time?

7. The interest, rate, and principal given, find the late. 8. The interest being $33.705, time 5 years, and principal $96-30, what is the rate?

9. Required the amount of p dollars, for t years, at the rate r, simple interest.

Since the amount is the sum of principal and interest, trp being the interest, as before, we have

a=p+trp; or, a=p(1+tr).

Hence the following rule.

To find the amount, when the principal, time, and rate

are known, multiply the time and rate together, add 1 to the product, and multiply this sum by the principal.

10. The principal being $630-20, the rate 4 per cent., and the time 7 years and 6 months, what is the amount?

11. Let us find the value of p from the equation, p+trpa. Separate the first member into factors, (1+tr)pa; divide by 1+tr,

To find the principal, when the amount, time, and rate are given, multiply the time and rate together, add 1 to the product, and divide the amount by the sum.

12. The amount being $124, the rate 6 per cent., and the time 4 years, what is the principal ?

Remark. The principal, in this case, is called the present worth of the amount.

13. Required the present worth of $400, due in 3 years and 3 months, at 5 per cent.

14. From the same equation, let us find the formula for r.

To find the rate, when the amount, time, and principal are given, divide the difference of the amount and principal by the product of the time and principal.

Remark. This rule is virtually the same as that given by problem 7th, since the difference of the amount and principal is the interest.

15. The amount being $368.875, time 3 years, and principal $325, required the rate.

16. In a similar manner, and from the same equation, let the learner find the formula for t, convert it into a rule, and perform by the rule, the subjoined particular example.

17. The amount being $1012-50, principal $750, and rate 5 per cent., required the time.

ART. 80. 1. Separate a into two parts, such that one shall be m times the other.

2. A and B have together a dollars, of which B has m times as much as A. How many dollars has each? 3. Separate a into two parts, such that the second shall

4. Separate a into three parts, so that the second shall be m times, and the third n times the first.

5. What number is that whose mth part, added to its nth part, makes the number a?

6. What number is that whose mth part exceeds its nth part by a ?

7. Separate a into two parts, so that the mth part of one, added to the nth part of the other, shall make the number a.

Let x one part; the other will be a -X.

and of my money, I had

8. After paying away and

a

dollars left. How many dollars had I at first?

9. A man and his boy together could do a piece of work in a days, and the man could do it alone in b days. Required the number of days in which the boy could do it alone.

10. The heirs to an estate received a dollars each;

but if there had been b less heirs, they would have received c dollars each. Required the number of heirs.

11. A man has 4 sons, each of whom is a years older than his next younger brother, and the eldest is m times as old as the youngest. Required their ages.

12. A father gave his children m oranges apiece, and had a oranges left; but in order to give them n oranges apiece, he wanted b oranges more. How many children had he?

13. The sum of two numbers is a, and of the

m

greater added to of the less makes b. Required the

numbers.

n

Remark. This and the two following questions may be performed by means of two unknown quantities. 14. One of and m cows cost a dollars; but one cow and n oxen cost b dollars. Required the price of a cow, and that of an ox.

1

m

15. There are two numbers, such that the first with

of the second makes a; and the second with of the

n

first also makes a. Required the numbers.

SECTION XXVII.

EXTRACTION OF THE SECOND ROOTS OF NUMBERS.

. ART. 81. 1. Some market women, counting their eggs, found that each had 12 times as many eggs as there were women, and that they all together had 300. Required the number of women.

We see that x must be a number which, multiplied by itself, shall produce 25; and we know that 5.5=25 Hence,

x=5 women, Ans.

The equation 12 x2300 is called an equation of the second degree, or a quadratic equation, because it contains the second power or square of the unknown quantity.

In general, an equation of the second degree is such as, when reduced to its simplest form, contains at least one term in which there are two, but no term in which there are more than two unknown factors.

When an equation with one unknown quantity contains the second power only of that quantity, it is called a pure equation of the second degree, or a pure quadratic equation. The equation given above is of this kind.

The first power of a quantity, with reference to the second, is called the second root or square root, and finding the first power when the second is given, is called extracting the second or square root. The second root of a quantity, then, is such as, being multiplied by itself, will produce that quantity.

The second powers of the first nine figures are as follows.

(1, 2, 3, 4, 5, 6, 7, 8, 9. Roots.

1, 4, 9, 16, 25, 36, 49, 64, 81. Powers.

Hence, when a number consists of only one figure, its second power cannot contain more than two figures. The least number consisting of two figures or places is