Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

EXPLANATION.—7 days is written for the 3d term, because it is the same kind as is required in the result.

1. If 11 men require 7 days, 5 men will require more than 7 days; therefore, the greater number (11) is written for the ad term, and the lesser (5) for the 1st.

2. If 147 cords require 7 days, 150 cords will require more than 7 days; therefore, the greater number (150) is written for the 2d term, and the lesser (147) for the 1st.

3. If 14 hours a day require 7 days, 10 hours a day will require more than 7 days; therefore, the greater number (14) is written for the 2d term, and the lesser (10) for the 1st.

Multiplying the 3d term by the continued product of the 2d, and dividing by the continued product of the 1st, gives 22 days.

NOTE.—Employ cancellation wherever possible.

SOLUTION BY ANALYSIS:

14 16

[ocr errors]
[ocr errors]

1. Since time for 11 men working 14 hr. a da. to cut 147 c.

= 7 da., 2. the 1 man

to cut 147 c.

= 11 x 17 da. 3. 1

1 66
6666 to cut 147 c.

14 x 11 x 7 da. 4. 1 "

1 " "" to cut 1 c.

14 x 11 x 17 da.

147 5. 1 "

1 " 66 66

to cut 150 c. 150 x 14 x 11 x 17 da.

147 6.

10 a da. to cut 150 c.

150 x 14 x 11 x 1 da.

10 x 147 7.

5 men

10 " a da. to cut 150 c. 150 x 14 x 11 x 17 da.

22 da.

5 x 10 x 147 NOTE.—After the pupil becomes familiar with each step, he may pass from step 1 to step 4, and then to step 7. By waiting until the last step, numerous cancellations simplify the work.

1 "

[ocr errors]

=

2. If four men build a wall 72 ft. long, 5 ft. wide, and 2 ft. high in 12 days, how many men can build a wall 120 ft. long, 6 ft. wide, and 11 ft. high in 9 days? 1. Known: 4 men, 72 ft. I., 5 ft. w., and 2 ft. h., 12 da. 2. Unknown: X 120 6“

11

OPERATION:

72 : 120
5 : 6

:: 4 men : x men.
2 : 1.5

9: 12 Or, 72 x 5 x 2 x 9 : 120 x 6 x 1.5 x 12 :: 4 men : x men.

120 x 6 x 1.5 x 12 x 4 men .. X

8 men. 72 x 5 x 2 x 9.

men =

Study the example above until you can give the reason for the arrangement of each couplet.

Exercise XX

1. If a man travels 145 miles in 5 days, traveling 12 hours a day, how many days will be required for him to travel 435 miles, traveling 6 hours a day?

2. If 18 men, working 10 hours a day, finish a task in one day, how many days will it require 10 men to do the same work, by working 9 hours a day?

3. A contracted to build a wall 500 ft. long in 20 days; 20 men built 300 ft. in 15 days: how many additional men must be employed to complete the wall in the required time?

4. If a cistern 171 ft. long, 101 ft. wide, and 13 ft. deep holds 546 barrels, how many barrels will a cistern hold that is 16 ft. long, 7 ft. wide, and 15 ft. deep?

5. If 27 men can perform a piece of work in 19 days,

how many men can do another piece of work | as great in of the time?

6. How many hours a day ought 30 men to labor to do in 10 days a piece of work which is as great as a similar job which 25 men, working 12 hours a day, did in 12 days?

7. If a bin 8 ft. long, 6 ft. wide, and 4.5 ft. deep, will hold 180 bushels of wheat, how deep must another bin be, that is 12 ft. long and 9 ft. wide, to hold 405 bushels?

8. A contractor engaged to pave 15 miles of a road in 12 months, and for that purpose employed 100 men. At the end of seven months he had completed only 6 miles. How many more men did he need to finish the work in the time prescribed ?

9. In how many days will six persons consume 5 bushels of potatoes, if 3 bushels and 3 pecks last 9 persons 22 days?

10. If 18 men build 24 rods of fence in 12 days, working 7 hours a day, how many men will it take to build 80 rods in 14 days, working 6 hours a day?

187. Cause and Effect. Problems in compound proportion may easily be solved by cause and effect, but this method reduces the problem to simple proportion and becomes rather mechanical. It does not develop the power to reason so well as does the method by compound proportion.

Problem number 10 will be solved as an illustration. The statement may be arranged thus:

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Considering all terms abstract, we have

X x 14 x 6 x 24 18 x 12 x 7 x 80.

18 x 12 x 7 x 80 ..x =

60. 14 x 6 x 24

Ans. 60 men.

PARTITIVE PROPORTION

188. Partitive proportion is the process of dividing a number into parts bearing a given relation to each other.

EXAMPLES

1. Divide 21 into parts proportional to 3 and 4. EXPLANATION.—Since the parts bear the ratio of 3 to 4, for every 7 (sum of 3 and 4) there is a 3 and a 4. Therefore, one part is 1 of 21, or 9, and the other is # of 21, or 12.

2. Divide $39 into parts proportional to ], }, and 1. EXPLANATION.—Reducing the fractions to a common denominator, we have 2, 12, and iż. Now the parts are proportional to 6, 4, and 3.

Proceed as in example 1. The parts are $18, $12, and $9.

Exercise XXI

1. Divide 105 into parts proportional to 5, 7, and 9.

2. Three persons gain $2640, of which A is to receive $6 as often as B receives $4, and C $2. What is each one's share?

3. Divide $1596 into parts proportional to }, å, and .

4. If common salt is composed of 71 parts of chlorine to 46 parts of sodium, how many pounds of each element are there in 468 pounds of common salt?

5. A, B, and C bought a house for $9000, A furnishing $4500, B $2500, and C $2000. The house rents for $810; find each one's share of the rent.

CHAPTER VII

SERIES

189. A series (Latin series, a row) is a succession of numbers formed according to some common law.

190. The terms of a series are the numbers composing it. 191. The extremes are the first and last terms. 192. The means are the intermediate terms.

193. An ascending series is one in which each term is greater than the one preceding it.

194. A descending series is one in which each term is less than the one preceding it.

195. There are many kinds of series, but it is customary to consider only two kinds in arithmetic.

These two are called progressions.

I. ARITHMETICAL PROGRESSION

196. An arithmetical progression is a series of numbers increasing or decreasing by a common difference.

197. The common difference is the difference between any two adjacent terms.

198. In an arithmetical progression there are five parts, any three of which being given, the other two can be found. The five parts with which we are concerned are:

1. The first term, represented by a.
2. The last term,

l.

« ΠροηγούμενηΣυνέχεια »