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5. If the same or equal quantities are subtracted from unequals, the remainders will be unequal.

6. If equal quantities are multiplied by the same or equal quantities, the products will be equal.

7. If equal quantities are divided by the saine or equal quantities, the quotients will be equal.

8. If a quantity is both multiplied and divided by the same or an equal quantity, its value will not be altered.

9. The whole of a quantity is greater than a part. 10. The whole of a quantity is equal to the sum of all

its parts.

Obs. The term quantity signifies any thing which can be multiplied, divided, or measured. Thus, numbers, yards, bushels, weight, time, &c., are called quantities.

285. The following principles will at once be recognized by the pupil as deductions from the four Fundamental Rules of Arithmetic, viz: Addition, Subtraction, Multiplication, and Division.

286. When the sum of two numbers and one of the numbers are given, to find the other number.

From the given sum subtract the given number, and the remainder will be the other number.

Ex. 1. The sum of two numbers is 25, one of which is 10; what is the other number?

Solution. 25-10=15, the other number. (Art. 40.) Proof. 15+10=25, the given sum. (Art. 184. Ax. 10.)

2. A and B together own 36 cows, 9 of which belong to A: how many does B own?

3. Two farmers bought 300 acres of land together; one of them took 115 acres : how many acres did the other have ?

QUEST.-284. What is an axiom? What is the first axiom? The second? Third ? Fourth? Fifth? Sixth? Seventh? Eighth ? Ninth ? Tenth? Obs. What is meant by quantity? 286. When the sum of two numbers and one of them are given, how is the other found ?

287. When the difference and the greater of two numbers are given, to find the less.

Subtract the difference from the greater, and the remainder will be the less number.

4. The greater of two numbers is 37, and the difference between them is 10: what is the less number?

Solution. 37–10=27, the less number. (Art. 40.)

Proof. 27+10=37, the greater number. (Art. 39. Obs.)

5. A had 48 dollars in his pocket, which was 12 dollars more than B had : how many dollars had B? 6. D had 450

sheep, which was 63 more than E had : had E?

how many

1

289. When the difference and the less of two numbers are given, to find the greater.

Add the difference and the less number together, and the sum will be the greater number. (Art. 39.)

7. The difference between two numbers is 5, and the less number is 15: what is the greater number?

Solution. 15+5=20, the greater number.
Proof. 20–15=5, the given difference. (Art. 40.)

8. A is 16 years old, and B is 8 years older: how old is B ?

9. The number of male inhabitants in a certain town, is 935; and the number of females exceeds the number of males by 115: how many females does the town contain ?

Quest.–287. When the difference and the greater of two numbers are given, how is the less found ? 289. When the difference and the less of two numbers are given, how is the greater found ?

290. When the sum and difference of two numbers are given, to find the two numbers.

From the sum subtract the difference, and half the remainder will be the smaller number.

To the smaller number thus found, add the given difference, and the sum will be the larger number.

10. The sum of two numbers is 35, and their difference is 11 : what are the numbers?

Solution. 35 -11=24; and } of 24=12, the smaller number. And 12+11=23, the greater number.

Proof. 23+12=35, the given sum. (Art. 284. Ax. 10.)

11. The sum of the ages of: 2 boys is 25 years, and the difference between them is 5

years :

what are their ages?

12. A man bought a chest of tea and a hogshead of molasses for $63; the tea cost $9 more than the molasses: what was the price of each ?

291. When the product of two numbers and one of the numbers are given, to find the other number.

Divide the given product by the given number, and the quotient will be the number required. (Art. 74.)

13. The product of two numbers is 84, and one of the numbers is 7 : what is the other number?

Solution. 84-7=12, the required number. (Art. 72.) Proof. 12x7=84, the given product. (Art. 54.)

14. The product of A and B's ages is 120 years, and A's age

is 12 years : how old is B? 15. A certain field contains 160 square rods, and the length of the field is 20 rods : what is its breadth ?

Quest.–290. When the sum and difference of two numbers are given, how are the numbers found? 291. When the product of two numbers and one of them are given, how is the other found ?

Note.-The area of a field is found by multiplying its length and breadth together. (Art. 163.) Hence the area of a field may be considered as a product.

292. When the divisor and quotient are given, to find the dividend.

Multiply the given divisor and quotient together, and the product will be the dividend. (Art. 73.)

16. If a certain divisor is 9, and the quotient is 12, what is the dividend ?

Solution. 12 x 9=108, the dividend required.
Proor. 108--9=12, the given quotient. (Art. 72.)

17. A man having 11 children, gave them $75 apiece : how

many dollars did he give them all ? 18. A farmer divided a quantity of apples among 90 boys, giving each boy 15 apples : how many did he give them all ?

293. When the dividend and quotient are given, to find the divisor.

Divide the given dividend by the given quotient, and the quotient thus obtained will be the number required. (Art. 73. Obs. 2.)

19. A certain dividend is 130, and the quotient is 10; what is the divisor?

Solution. 130--10=13, the divisor required. (Art. 72.) PROOF. 13x10=130, the given dividend. (Art. 73.)

20. A gentleman divided $120 equally among a company of sailors, giving them $10 apiece : how many sailors were there in the company ?

QUEST.-292. When the divisor and quotient are given, how is the dividend found ? 293. When the dividend and quotient are given, how is the divisor found ?

21. A farmer having 600 sheep, divided them into flocks of 75 each : how many flocks had he?

294. When the product of three numbers and two of the numbers are given, to find the other number.

Divide the given product by the product of the two given numbers, and the quotient will be the other number.

22. There are three numbers whose product is 60 ; one of them is 3, and another 5: it is required to find the other?

Solution. 5X3=15; and 60-15=4, the number required.

Proof. 5X3 X4=60, the given product.

23. The product of A, B, and C's ages is 210 years : the

age of A is 5 years, and that of B, is 6 years : what is the age of C?

24. Three boys together had 1728 marbles ; two of them had a dozen apiece : how many had the other ?

SECTION XI.

ANALYSIS.

295. The term analysis in physical science denotes the separation of a compound body into its elements or constituent parts.

In arithmetic it signifies the resolving or separating numbers into the parts or the factors of which they are composed. (Arts. 26, 57. Obs.)

Analysis may be applied with advantage not only to the development of mathematical principles, but also to

Quest.–294. When the product of three numbers and two of them are given, how is the other found ? 295. What is meant by analysis in physical science? What in arithmetic? To what may analysis be advantageously applied ?

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