5.6.7.8

5 · 6 · 7 · 8 · 2 · 3 · 4 · 5 25

Required equation, 6.7.8.9.···9·1·2·3·4

27. Answer.

9

13. What is the direct ratio of 4 to 7? (See Example 1, page 71.).

(Ex. 2, p. 72.)

14. What is the direct ratio of 6 to 9? 15. Required the inverse ratio of 8 to 12. 16. Give the inverse ratio of 12 to 20. 17. What is the direct ratio of § to (Ex. 3, page 73.) 18. Required the inverse ratio of to 3. (Ex. 4, p. 73.) 19. What is the direct ratio of 7x5 to 14x 10. (Ex. 5, pages 73 and 74.)

20. Give the direct ratio of 4× 21. Give the direct ratio of ×

to × 7. (Ex. 6, p. 74.) tox.

22. Required the inverse ratio of 7×9 to 21 × 27. (Ex

ample 7, pages 74 and 75.)

23. What is the inverse ratio of 6 x 8 to 3x 4? 24. Find the inverse ratio of 11 x 12 to 22 x 36. 25. What is the inverse ratio of 1

8, page 75.)

26. Give the inverse ratio of

to

3 (Example

1 to 3.

27. Give the ratio of 5 x 8' to 15 x 16', when 8 and 16 are inverse.* (Example 9, page 75.)

28. What is the ratio of 7 x 9 to 14 x 15'?

29. Required the ratio of x' to 3x. (See Example 10, page 76.)

30. Find the ratio of x to 8'x1.

4 23 13

15

31. Give the ratio of xx to XX. (See Example 11, page 76.)

32. Required the ratio of 'XX to '× 72 × 1.

1

11

33. What is the ratio of xxx 33 to 1 ×3? (Example 12, pages 76 and 77.)

* The inverse terms of a ratio will hereafter be marked by a dash, (as has been shown;) thus, 8', 16', as in this example-the terms not marked, will be considered direct.

34. What is the ratio of xxx to ×××? 35. Required the ratio of 43x 9x 7 to 9x18x11.

NOTE.-Mixed numbers should always be reduced to improper fractions, and whole numbers to the fractional form before the equation is formed. The above example is the same as if the ratio of 2×× 23 to 39x18x had been required.

In the same manner the pupil may find the ratio in the examples which follow, taking care to notice particularly those terms in each ratio which are DASHED, and to remember that they are inverse terms.

36. Required the ratio of 8 to 9; 5 to 63; 7×5 to 8×91.

37. What is the ratio of 23 x 27' to 69× 81'; 7× 32 to 284× 102' ?

38. Give the ratio of 37× 84' x to 72 × 37' x 1; 3× 5 to

ARTICLE XIV.

RATIONAL EQUIVALENTS.

REMARKS ON RATIONAL EQUATIONS.

We are now about to show how equations may be used for the solution of problems in which the RATIONAL EQUIVALENT of a required quantity is demanded. This will form the SECOND, and by FAR the most important use of RATIONAL

EQUATIONS.

The Rules of SINGLE and DOUBLE Proportion, or as they are generally termed, the "SINGLE and DOUBLE RULE OF THREE," have been generally adopted by arithmeticians for finding a FOURTH proportional to three specified numbers, i. e., to find a FOURTH number, which should bear the same ratio to a THIRD, that a SECOND number bears to the FIRST, and by this means it was proposed to ascertain the cost of any specified quantity of a certain commodity, when the value of another quantity of the same commodity, was known, The SIMPLEST, and consequently the BEST mode of accom

plishing this object, is therefore, a desideratum in the science of numbers.

We will suppose, e. g., that 6 peaches are worth 7 apples, and that we wish to know, from this statement, how many apples 15 peaches would be worth. In this case, it is plain, that the number of apples which must be given for 15 peaches, should bear the SAME direct ratio to 7 apples, that 15 peaches bears to 6 peaches. Now, the direct ratio of 15 to 6 is 15, (fifteen-sixths,) and hence, if 6 peaches are worth 7 apples, it follows, that 15 peaches are worth 15 of 7 apples, equal to 15 of 7=17=2=&=171⁄2, apples which is the quantity sought, or answer.

15

5

It is evident from this statement, that 6 peaches constitute a given quantity, (Def. 11, page 71,) equal in value to 7 apples, and that 15 peaches constitute a required quantity, (Def. 12, page 71,) whose equivalent value is demanded or required to be found.

We will now form an equation, in which the given quan. tity (6 peaches,) will constitute the left hand member, and its equivalent value, (7 apples,) the right hand member, as before, and if the solution, according to the principles already explained, shall lead to the same result as that just found, we will then have established the principle, that equations may be used as a substitute for proportion; Thus,

Given equation,

Simple equation,

Peaches. Apples.
6 7

Required equation,-6-1-2-1-2=174. Answer.

We will now demonstrate the same problem according to the principles of proportion, viz. DIVIDE the PRODUCT of the SECOND and THIRD terms, by the FIRST, and the QUOTIENT will be the FOURTH term, or ANSWER. In stating the proportion, we must realize that 6 peaches have the same ratio to 15 peaches, that 7 apples have to 15 times 7 apples, divided by 6, and we shall then have the proportion 6 to 7 as 15 to

15x7=105=35=173, and to write this proportion, we set down the number thus, (Def. 8, page 71,) 6:7::15:171⁄2, in which 17 represents apples, and is the answer required. Now, in comparing the solution by equations to that by proportion, we are, at first sight, struck with what appears to be a circumstance greatly in favor of the latter, viz. its APPARENT brevity; but, however desirable brevity may be, it is true in this, as in many other cases, that the plain road is the best guide, while the byways, though shorter, are less plain, and therefore, not a safe guide for any but those who have frequently travelled them, and become familiar with all the difficulties in their path. If, therefore, the student of arithmetic is not willing to follow the BROAD ROAD of science, he will, in all probability, lose his course and fail to accomplish his object.

The reason of this is obvious.-The pupil is taught to add, subtract, multiply, and divide numbers, and then told, that in some cases he must multiply, and in others divide, to obtain the answer, without any previous knowledge of the principles upon which we arrive at such a conclusion. To him, therefore, it is a leap in the dark. He is told how to produce an effect, while the cause is buried in obscurity. After toiling long and hard, in attempting to do what none but a proficient in science is capable of performing, he is told that the difficulties will vanish, if the numbers are stated in the form of a proportion, and this must be done without having any previous mental training, which would instruct him into the nature of proportion. It has, therefore, happened as a natural consequence, that the pupil could very easily furnish the answer, if some wiser head would state the proportion. Under these circumstances, it is not strange that the pupil should become dissatisfied, and make but little progress in this branch of study.

To avoid this, and furnish what is believed to be a more intelligent mode of conducting arithmetical calculations, we propose the use of RATIONAL EQUATIONS, as a substitute for PROPORTION. The advantages of the equating system over any other heretofore recommended will be more clearly seen by furnishing numerous examples than by any attempt to explain it abstractly. We have shown that when a given and required quantity are known, (i. e., when these quantities

have been distinguished from each other,) there can be no difficulty in forming a rational equation; for the given quantity is placed on the left, and its equivalent value, or cost, on the right of the sign of equality, and this expression constitutes the first, or given equation. The second, or simple equation, is found by simply transferring the number or numbers, (greater than a unit,) which form the given quantity, from the LEFT hand member of the given equation, to the RIGHT, placing a unit in their former situation. Lastly, the required equation, is formed by placing the required quantity, in both members of the simple equation. This procedure is universal in the solution of rational equations, and only requires that we should notice whether the given and required quantities (as a whole, or if they consist of factors, whether any or all these factors,) are direct or inverse. If they are direct, the signs of the given quantity must be changed when they are transferred; but if they are inverse, they must be transferred without changing their signs, and, lastly, the direct terms of the required quantity, must be placed directly (without changing their signs,) in both members of the simple equation; but the inverse terms must be placed DIRECTLY in the left hand, but INVERSELY (signs changed,) in the right hand member of the simple equation, which (right hand member,) reduced to its simplest form, will be the equivalent of the required quantity, or the ANSWER Sought.

The process of forming and solving RATIONAL EQUATIONS will, therefore, be best explained by giving a careful attention to the following

EXAMPLES IN MENTAL ARITHMETIC.

1. If one pound of sugar costs 10 cents, what will 7 pounds cost?

Proceed thus: One pound GIVEN, and seven pounds RE

QUIRED.

If one pound of sugar equals (cost,) ten cents, then seven pounds (of sugar,) will equal seven times ten cents, equal to seventy cents.

Hence, seven pounds of sugar at ten cents a pound, will cost seventy cents.

2. What will be the cost of 5 yards of cloth at 9 dollars a yard? Here one yard is GIVEN, and five yards REQUIRED.