rule first given? What is to be the denomination of the third term? How are the other terms to be arranged? What is the rule for canceling? What term is to be placed first above the line? How are the other terms to be arranged? What is the given note?

SOLUTION OF ARITHMETICAL PROBLEMS BY ANALYSIS.

An arithmetical question is solved analytically, when the operation is guided entirely by the conditions embraced in the question itself.

Take the following illustration:

Ex. 1. If 3 men perform a piece of work in 6 days, in what time will 9 men perform the same labor?

It is obvious that if it take 3 men 6 days to perform the proposed labor, it will take one man 3 times 6, or 18 days, to perform the same. But 9 men operating together, will perform 9 days work in one day; consequently, they will do the whole in 2 days; for 18-9-2, Ans.

2. If 21 men earn 63 dollars in a given time, how much will 42 men earn in the same time?

63÷213, the number of dollars one man will earn in the given time. Therefore, $42×3= $126, the answer required. Or the ratio of 21 men to 42 is 2; and, $63×2=$126, the same as before.

Solutions of this kind may, therefore, be effected by the following general principle:-Find the ratio of the two given terms which are of the same kind, and by this ratio multiply the term corresponding in kind with the one required.

3. If 42 men can make 3 rods of wall in a given time, how much can 8 men make in the same time?

The ratio of 42 men to 8 men is 42:8=1; therefore, 34 of one rod, which is the distance required. 4. If a staff 4 feet long, cast a shadow 6 feet, how high is that steeple, whose shadow measures 75 feet?

The ratio of 6:75=12, and 12×4=50 feet, Ans. Or, the shadow is 13 as long as the staff; hence, 75÷3= 25, and 25x2=50 feet, the same answer as before.

5. An express traveling at the rate of 60 miles per day, had been absent 5 days, when a second express was dispatched on the same rout, traveling 75 miles per day. How many miles must the second travel to overtake the first?

60×5=300, the whole number of miles traveled by the first express before the second started, and consequently, the number of miles the second had to gain. But the first travels 60, and the second 75 miles per day; hence, 75—60—15, the number of miles gained daily, by the second express. 15 miles are, therefore, gained in traveling 75 miles, consequently, one mile is gained in traveling 5 miles; and since 300 miles are to be gained, 300 × 5=1500 miles, answer.

6. If 6 men in 14 days earn 84 dollars, how much will 9 men earn in 11 days?

$84-6 $14, the money one man will earn in 14 days, and $14÷14=$1, the wages of one man for one day; therefore, $1×9=$9, the money 9 men will earn in one day, and $9× 11-$99, the money 9 men will earn in 11 days.

7. If 6 persons spend $300 in 8 months, how much will be sufficient for a family of 15 persons 20 months?

300÷6=50, and 50÷8=$64, the money spent by one person in one month; then, $64×15×20=$1875, answer.

8. If 12 men build 36 feet of wall in 9 days, how many men would build 108 feet in 16 days?

36÷12=3, and 3÷9, the distance built by one man in one day; and×16-16=5, the distance one man would build in 16 days; therefore, 108-5-204, the number of men required.

9. A merchant owning of a vessel, sold of his share for $1456. What was the value of the whole vessel ?

of. If then,

the

cost $1456, 1456÷8=$182, value of of the vessel; hence, 182×15=$2730, answer. 10. If of a yard cost of a dollar, what will 40 yards cost? If of a yard cost $7, of a yard will cost of that sum, or } of $7=$27, and one yard or will cost 5 times that sum, or $; therefore, $35 x 40-$1400-$58.333,+ Ans.

11. If 240 men perform a piece of work in 8 months, how many men must be employed to finish the same work in 2 months?

The ratio of 2: 8=4, and 240×4=960 men, answer.

APPLICATION OF CANCELING TO ANALYTICAL SOLUTIONS.

12. If 8 pounds of tea cost $12, what will 32 pounds cost?

The two terms of the same name here given are 8 and 32, and their ratio is 4, and is obtained by

Therefore, $12 × 4= $48, answer.

The third term is here multiplied by the ratio of the first and second, as required for analytical solution. The terms are also canceled and multiplied as directed by the rule for canceling.

13. If 16 horses consume 84 bushels of grain in 24 days, how many bushels will suffice 32 horses 48 days?

In the preceding sum, it is evident that the given quantity of grain is to be increased by the ratios of 16 horses to 32 horses, and of 24 days to 48 days. Hence,

We therefore see by the above example, that the effect of the operation is to increase the quantity of the same name as the required quantity, by all the given ratios. The same is true in all cases, that is, every statement for canceling is a complete analysis of the question under consideration.

14. If 8 men build 9 feet of wall in 12 days, how many men must be employed to build 36 feet in 4 days?

The number of men required will obviously depend on the the ratios 9: 36, and 4: 12, the former of which is 4, and the latter, 3. Therefore, 8 men × 4×3=96 men, the number required. The above statement canceled, gives the same result, thus:

The same number would have been obtained, had the numbers been canceled in any other order; thus,

8. 36. 12
and 8 x 12-96.
9. 4'

Hence we perceive, that in a correct solution of any sum by canceling, a complete analysis of that sum is given.

15. If 10 men make 300 pairs of boots in 20 days, how many men must be employed to make 450 pairs in 30 days? Ans. 10 men.

Statement,

10. 450. 20
300. 30'

If 10 men make 300 pairs in 20 days, they would make 15 pairs in one day; and if 10 men make 15 pairs in one day, one man would make one and a half pairs per day; and in 30 days, he would make 45 pairs; therefore, 450÷45=10, Ans.

16. If of a yard cost of a pound sterling, what will § of a yard of the same cloth cost?

40

If yard cost of a pound, the whole yard would cost of a pound, and of the same would cost of of a pound= 6 of a pound; consequently, would cost 5 times that sum, or of a pound, or 15 s. Ans.

30

40

17. If of a house cost 49 pounds, what will be the value of of the same? Ans. 10 £. 10 s.

18. A merchant bought a number of bales of velvet, each containing 12917 yards, at the rate of $7 for 5 yards, and sold the same at the rate of $11 for 7 yards, and gained $200 by the transaction. How many bales were there?

6

He paid of a dollar per yard, and received of a dollar for the same. Hence, -=55 49 Hence, -55-33-35 of one dollar, the amount gained on one yard. Therefore, $200÷3 7000 = 6 " the whole number of yards; and 700012917 3500-31500-3500-9 bales, Ans.

27

27

27

6

=

7000

6

19. If 7 horses consume 24 tons of hay in 6 weeks, how many tons will 12 horses consume in 8 weeks? Ans. 62 tons. 20. If 14 men finish a piece of work in 42 days, how long will it take 21 men to do it? Ans. 28 days.

21. If of a farm be valued at $895, what is the whole farm worth? Ans. $1611.

22. If 7 horses consume 29 bushels of oats in 5 weeks, how many will 12 horses consume in 6 weeks? Ans. 592 bushels.

23. A merchant owning of a vessel, sold of his share

for $1200; what was the value of the whole vessel, at the same rate? Ans. $1645.714.+

24. There is a pole in the mud, in the water, and 8 feet out of the water. What is its length? Ans. 53 feet.

25. In a certain orchardof the trees bear apples, pears, plums, 30 of them peaches, and 20 cherries. How many trees does the orchard contain? Ans. 600.

26. A certain school is classified as follows: study grammar, study geography, arithmetic, write, and 9 learn to read. How many are there in all, and how many in each study? Ans. Whole number 80. In grammar 5, geography 30, arithmetic 24, writing 12, and 9 read.

SIMPLE AND COMPOUND PROPORTION IN FRACTIONS.

In stating such sums in Simple or Compound Proportion as consist of fractions, it is only necessary to compare terms as already directed, and then, if they are solved without canceling, having inverted the divisor, to divide the product of the numerators by the product of the denominators. If, however, they are to be solved by canceling, arrange the numerators of the several fractions as directed to arrange whole numbers, when whole numbers only are given, and place each denominator opposite its own numerator.

Note. Before stating the sum, mixed numbers, if any are given, must be reduced to improper fractions.

Ex 1. If of a yard cost of a pound, what will of a yard cost?

The

Statement,

7. 3. 5 15. 14. 3'

is inverted, that its numerator may stand below the line' as the same term would stand if it were a whole number.