337. We may extract any root of both sides of a proportion, or we may raise both sides to any power, without destroying the proportion.

To illustrate this in the case of extracting a root, consider the proportion, 1949: 441. Taking the square root of each term we have 1: 3 7: 21, or 21, and this equality being true shows that the proportion is not destroyed.

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In the case of raising to a power, take the proportion, 1 : 2 24. By cubing each term we obtain 1 8 8 64, or : = 8: 1280 , and since this equality is true the proportion has not been destroyed

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SIMPLE PROPORTION.

The above are illustrations of examples in Simple Proportion, or proportions in which only four quantities are used.

338. To solve examples by simple proportion:

Rules. 1. Make the number that is of the same denomination as the required answer, the third term. Make the fourth term.

2. Determine by the nature of the example whether the answer is to be greater or less than the third term; if it is to be greater, place the greater of the other two numbers for the second term; but if it is to be less, place the smaller for the second term and the other for the first.

3. Multiply the third term by the second term and divide the product by the first term. The result will be the value of x.

339. To illustrate the use of the rule take the following examples:

How many feet of planking will be required for a bridge 530 feet long, if 18,800 feet of planking are required for one 300 feet long?

The number that is of the same denomination as the required answer is 18,800, which we place for the third term; a is made the fourth term. It is evident that more feet of planking will be required for a bridge 530 feet long than for one 300 feet long, therefore, the answer will be greater than the third term. Hence, 530 is the second term and 300 the first. Following the rule the above result is obtained.

If a wheel that is 8 feet in circumference turns 4,250 times on a journey, how many times will a wheel that is 10 feet in circumference turn on the same journey?

The number which is of the same denomination as the required answer is 4,250, which is made the third term; x is the fourth term.

A wheel 10 feet in circumference will not turn as many times on the journey as one which measures 83 feet. It is apparent that the answer will be less than the third term, therefore 8 is used as the second term and 10 as the first.

340. A Direct Proportion is one in which both couplets are direct ratios.

341. An Inverse Proportion is one in which one of the couplets is an inverse ratio. If the proportion is inverse, it is formed in the same manner as if direct.

342. Let us now take a case of inverse proportion:

A train running 27 miles per hour covers a certain distance (189 miles) in 7 hours. How long does it take a train running at 32 miles per hour to cover the same distance?

Now if we use a direct proportion we have 27 : 32 :: 7 : x; but in this case x equals 8 hours, which implies that the train running 32 miles per hour takes longer to cover the distance than one running only 27 miles per hour. Obviously this is not so. We must then use not a direct but an inverse proportion. Inverting one couplet we have 27: 32 x 7, or, 32: 277: x. Either of these (or any other inverse proportion formed by them) gives

343. In using proportion great care must be taken to perform the exact operation required by the conditions of the problem. For example: If a joint 16 feet long requires 80 rivets, what number will a joint 11 feet long require?

Here we wish to find the number of rivets required:

16 ft. 11 ft. 80 rivets number of rivets required.

Find the value of x in each of the following:

5. If 4 tons of coal cost $24, what will 18 tons cost at the same rate?

Ans. $108.

6. How many men will be required to build 32 rods of wall in the same time that 5 men can build 10 rods? Ans. 16 men.

7. If a railroad train runs 30 miles in 50 minutes, in what time will it run 260 miles? Ans. 7 hrs. 13 min.

8. If 15 tons of lead cost $105, what will 765 tons cost?

Ans. $5,355

COMPOUND PROPORTION.

344. A Compound Proportion is an expression of the equality of two ratios, one or both of them being compound ratios.

345. In solving a compound proportion the following rules must be observed:

1. Make the number to which the answer is the consequent the third term.

2. Of each remaining couplet consisting of antecedent and consequent of the same kind, form a ratio as if the answer depended on that ratio alone.

3. Multiply all the means, and divide the product by the product of all the given extremes; the quotient, or the answer, is the consequent of the third term.

346. Let us make a practical application of the use of the rule.

If 5 men earn $275 in 18 days, how much can 12 men earn in 15 days? The answer will be in dollars, and so we place $275 as the third term. The other terms take their places forming couplets as first and second terms, and the complete proportion is expressed as follows:

Because 12 men can earn a greater amount than 5 men if other conditions are left out of consideration, we arrange our couplet with the greater number appearing after the other, thus, 5 12. And similarly, because total earnings for 15 days are smaller than for 18 days if other conditions are left out of consideration, we arrange the corresponding couplet with the smaller number appearing after the other, thus, 18: 15.

NOTE. One of the simple ratios (5: 12) entering into this proportion would seem to imply that x, the answer sought, must be greater than 275; while the other (18: 15) would seem to imply that it must be less. This, however, involves no inconsistency, since in the determination of a both ratios are combined.

Now, multiplying the means and dividing the product by the product of the extremes:

As in simple proportion, we use cancellation with the following result:

1. How many men working 10 hours a day for 15 days are required to build a wall 160 ft. long, 40 ft. high, and 3 ft. thick, if 75 men in 12 days of 9 hours each build a wall 120 ft. long, 30 ft. high, and 2 ft. thick? Ans. 144 men.

2. If 15 masons lay 38.75 perches of foundation in 83 days, how many perches can 24 masons lay in 17 days?

Ans. 124 perches.

3. How many pounds of wool will make 150 yds. of cloth 36 inches wide, if 12 ounces make 23 yds. 54 inches wide?

Ans. 30 lbs.

4. A bar of iron 3 ft. long, 24 in. wide, and 12 in. thick weighs 45 lbs How much will a bar weigh that is 7 ft. long, 3 in. wide, and 2 in. thick? Ans. 180 lbs.

GENERAL REVIEW.

Among the processes frequently used in Arithmetic are those involving denominate numbers. A familiarity with the operations in which they occur requires thorough knowledge of all the principles previously studied and necessitates careful work on the part of the student. In changing from lower to higher denominations or vice versa an error, though small in itself, may make the final answer entirely wrong. Thus we see that absolute accuracy is the prime requisite in this work.

Some students may at first find it hard to master the study of roots, but with thorough and consistent application this should prove to be as easy as any other part of Arithmetic. The individual problems are longer but therein lies their chief difficulty; they are simply applications of the fundamental processes studied in Part I. Solving the following problems will thoroughly prepare you for the Examination.