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. 12×9=108, the dividend required.

PROOF. 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. 13×10=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. 5×3=15; and 60÷15=4, the number required.

PROOF. 5×3×4=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?

the solution of a great variety of examples, both in arithmetic and practical life.

In the preceding sections the pupil has become acquainted with the method of analyzing particular principles, and thence deducing general truths or rules.

We will now illustrate the application of analysis to the solution of examples or problems.

OBS. Business men have a method of solving practical questions, which is frequently shorter and more expeditious than that of arithmeticians fresh from the schools. If asked, by what rule they perform them, their reply is, "they do them in their head," or by the 66 no rule method."

Their method consists in analysis. It might with propriety be called the common sense rule. A little practice will give the learner great facility as well as satisfaction in its application.

MENTAL EXERCISES.

Ex. 1. If 8 barrels of flour cost 40, how much will 5 barrels cost?

Analysis.-1 is 1 eighth of 8: therefore 1 barrel will cost 1 eighth as much as 8 barrels; and 1 eighth of $40 is $5. Now it is obvious that 5 barrels will cost 5 times as much as 1 barrel; and 5 times $5 are $25, the answer required.

Or, we may reason thus: 5 barrels are of 8 barrels; 5 barrels will therefore cost as much as 8 barrels. Now 1 eighth of $40 is $5, and 5 eighths is 5 times $5, which is $25.

2. If 7 lbs. of tea cost 42 shillings, what will 10 lbs. cost?

3. If 9 sheep are worth $27, how much are 15 sheep worth?

4. If 10 barrels of flour cost $60, what will 12 barrels cost?

5. Suppose 30 gallons of molasses cost $15, how many dollars will 7 gallons cost?

QUEST.-Obs. How do business men solve practical questions? What do they call it? In what does it consist? What might it with propriety be called?

6. If a man earns 54 shillings in 6 days, how much can he earn in 15 days?

7. If 12 men can build 48 rods of wall in a day, how many rods can 20 men build in the same time?

8. A gentleman divided 90 shillings equally among 15 beggars: how many shillings did 7 of them receive?

9. Suppose 75 pounds of butter lasts a family of boarders 25 days, how many pounds will supply them for 12 days?

10. If 7 yards of cloth cost $30, how much will 9 yards cost?

11. If 10 barrels of beef cost $72, how much will 8 barrels cost?

12. If 7 acres of land cost $50, what will 12 acres cost?

13. A farmer buying an ox cart, paid $15 down, which was of the price of it: what was the price of the cart; and how much does he owe for it?

Analysis. The question to be solved is simply this: 15 is of what number? If 15 is, is of 15, which is 5. Now if 5 is 1 tenth, 10 tenths is 10 times 5, which is 50.

Ans.

3

S $50 is the price of the cart, and $50-$15-35, the sum unpaid.

Note. In solving examples of this kind, the learner is often perplexed in finding the value of 1, &c. This difficulty arises from supposing that if 10 of a certain number is 15, 10 of it must be of 15. This mistake will be easily avoided by substituting in his mind the word parts for the given denominator.

Thus, if 3 parts cost $15, 1 part will cost of $15, which is $5. But this part is a tenth. Now if 1 tenth cost $5, then 10 tenths will cost 10 times as much.

14. A man bought a yoke of oxen, and paid $56 cash down, which was of the price of them: what did they cost?

15. A merchant bought a quantity of wood and paid $45 in goods, which was 5 of the whole cost: how much did he pay for the wood?

16. A whale ship having been out 24 months, the Cap

tain found that his crew had consumed of his provisions how many month's provision had he when he embarked, and how much longer would his provisions last? 17. How many times 7 in of 35?

Analysis of 35 is 7, and is 4 times 7, which is 28. Now 28 is 4 times 7. Ans.

18. How many times 6 in 7 of 45?

24. of 48 are how many times 4 ?

25. of 64 are how many times 7?

26. of 100 are how many times 12?

10

27. of 110 are how many times 8?

I

28. of 180 are how many times 10?

29.

5

12 of 84 are how many times 9 ?

30. How many yards of cloth, at $7 per yard, can be bought for of $54?

31. How many barrels of flour, at $5 per barrel, can be bought for of $60?

32. A man had $64 in his pocket, and paid of it for 10 barrels of flour: how much was that per barrel? 33. 40 is of how many times 6?

Analysis. Since 40 is, is of 40, or 8; and is 9 times 8, or 72. Now 6 is contained in 72, 12 times. Ans. 12 times.