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Algebra.

282. ALGEBRA APPLIED TO SOME PROBLEMS IN RATIO.

EXAMPLE I.

The consequent is c; the ratio is r. What is the antecedent?

Let x= the antecedent.

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From the above learn that the antecedent is always equal to the product of the consequent and the ratio.

1. Consequent 75; ratio 11. Antecedent?
2. Consequent 92; ratio 4. Antecedent?
3. Consequent .56; ratio } Antecedent?

EXAMPLE II.

The antecedent is a; the ratio is r. What is the consequent ?

Let x= the consequent.
Then

a
-=r
x

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From the above learn that the consequent is always equal to the quotient of the antecedent divided by the ratio.

1. Antecedent 75; ratio 5. Consequent?
2. Antecedent 96; ratio 1. Consequent?
3. Antecedent; ratio : Consequent?
4. Antecedent .25; ratio Consequent ?

283. TO FIND Two NUMBERS WHEN THEIR SUM AND

RATIO ARE GIVEN.

EXAMPLE

The sum of two numbers is 36 and their ratio is 3. What are the numbers?

Let Then and

x= the smaller number.

3x= the larger number, x + 3x = 36

4x= 36

x = 9, the smaller number. 3 x=27, the larger number.

PROBLEMS.

1. The sum of two numbers is 196, and their ratio is 3. What are the numbers ?

2. The sum of two numbers is 294, and their ratio is 24. What are the numbers ?

3. The sum of two decimals is .42, and their ratio is 2). What are the decimals ?

4. The sum of two numbers is s, and the ratio of the larger to the smaller is r. What are the numbers ?

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* Observe that any number you please may be put in the place of s, and any number greater than 1 in the place of r; so that when the sum of two numbers and the ratio of the larger to the smaller are given, the smaller number may be found by dividing the sum by the ratio plus 1.

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1. One side of a triangle may be regarded as its base. The perpendicular distance from its base (or from its base extended) to the opposite angle, is its altitude.

2. What is the altitude of the first of the above triangles ? Of the second? Of the third ?

3. Convince yourself by measurement and by paper cutting that every triangle is one half of a parallelogram having the same base and the same altitude as the triangle.

4. To find the area of a triangle, Find the area of the parallelogram having the same base and altitude, and take one half of the result. Or, as the rule is given in the older books,Take one half of the product of the base and altitude."

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PROBLEM. — If the above figure represents a piece of land and is drawn on a scale of } inch to the rod, what part of an acre of land does it represent?

285. MISCELLANEOUS REVIEW. 1. The specific gravity of granite is 2.7.* How much does a cubic foot of granite weigh?

2. A certain vessel is exactly large enough to contain 1000 grains of water. It will contain only 700 grains of petroleum. What is the specific gravity of the petroleum ? †

3. The specific gravity of gold is 19.3. How much does a cubic foot of gold weigh?

4. A cubic foot of sulphur weighs 125 lbs. What is the specific gravity of sulphur?

5. A cubic foot of steel weighs 487.5 lb. What is the specific gravity of steel?

6. What is the ratio of .01 to .001 ? Of .1 to .01 ? Of 1 to .1?

7. What is the ratio of 1 bu. to 1 pk.? Of 1 pk. to 1 qt. ? 8. What is the ratio of $371 to $15? Of $15 to $371?

9. What is the area of a rhomboid whose base is 16 inches and whose altitude is 16 inches ?

10. Is the rhomboid described in problem 9 equilateral ?

11. The ratio of the perimeter of one square to the perimeter of another square is 4. What is the ratio of the areas of the two squares ?

12. Draw three triangles, the base of each being 4 inches and the altitude of each being 2 inches. Make one of them a right-triangle; another, an isosceles triangle, and the third having angles unlike either of the other two. What can you say of the area of the right-triangle as compared with each of the others ?

(P. 359.)

* See page 185, Exercise 10. + This means what is the ratio of the weight of the petroleum to the weight of the same bulk of water?

PROPORTION.

286. The terms of a ratio are together called a couplet. Two couplets whose ratios are equal are called a proportion.

The two couplets of a proportion are often written thus : 6:18 10:30, and should be read, the ratio of 6 to 18 equals the ratio of 10 to 30.

Couplets are sometimes written thus : 20:4::50:10, and read, 20 is to 4 as 50 is to 10.*

287. TO FIND A Missing TERM IN A PROPORTION.

EXAMPLE I.

36:12::x: 25.

The ratio of the first couplet is 3 ; that is, the antecedent is 3 times the consequent. Since the ratios of the couplets are equal, the ratio of the second couplet must be 3, and its antecedent must be 3 times its consequent. Three times 25 = 75, the missing term

PROBLEMS.

= x: 225

Find the missing term.

1. 90 : 15 ::x:180
2. 48:12::x : 150
3. 75: 30 ::x: 140

4. 20: 60
5. 30 : 50 = x: 175
6. 90: 20

= x:140

* The ratio sign (:) may be regarded as the sign of division (;-) with the horizontal line omitted, and the proportion sign (::) the sign of equality (=) with an erasure through its center, thus: (= =).

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