24. Find two numbers in the ratio 3 : 4 (suggestion, 3 x and 4 x) of which their sum is to the sum of their squares as 7: 50.

25. Find two numbers in the ratio of 5 : 4, such that their sum has to the difference of their squares the ratio of 1: 18.

26. Find two numbers such that if 7 is added to each they will be in the ratio of 4: 3; and if 11 is added to the greater and subtracted from the smaller the results will be in the ratio of 5: 2.

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27. If 7 x 4 z: 8x - 3 % = 4 y is a mean proportional between x and y.

28. 3Vy+a: 3Vy a = m : n;

7z: 3y-8z, prove that z

find x.

29. If mx + ny : px + qy = my + nz : py + qz, show that

n: q = m: p.

30. If 2a+3b: 3 a

abc: d.

4b2c3d: 3c- 4d, prove that

31. If 2 men working 9 hours a day can do a piece of work in 32 days, in how many days can x men working y hours a day do the work?

If a b c d, prove that:

32. a: a + c = a + b : a+b+c+d.

33. a cac b3d + bd2 = (a + c)3 : (b + d)3.

c3
c + ď

35. (a+b+c+d) (a-b-c+d) = (a−b+c-d) (a+b―c-d). 36. Show that, when four quantities of the same kind are proportional, the sum of the greatest and the least is greater than the sum of the other two.

38. Each of two vessels contains a mixture of wine and water; a mixture consisting of equal measures from the two vessels contains as much wine as water, and another mixture consisting of four measures from the first vessel and one from the second is composed of wine and water in the ratio of 2: 3. Find the proportion of wine and water in each of the two vessels.

Ans. In the first the wine is, in the second . 39. If the increase in the number of male and female criminals is 1.8%, while the decrease in the number of males alone is 4.6% and the increase in the number of females is 9.8%, compare the number of male and female criminals respectively.

Ans. Number of female criminals four-fifths the number of male criminals.

APPLICATION OF QUADRATIC EQUATIONS AND RATIO AND PROPORTION TO GEOMETRY

509. EXAMPLE 1. If the sides of a triangle are divided by a line drawn parallel to the base, so that the upper segment on one side is equal to the lower segment on the other side, how large is this segment, if the other two are respectively 15.125, and 8 feet? Solution.-Let EC= x.

AD

15.125

B

FIGURE 2

c By 2482, x=151×8=121

x = ±11.

The negative value has no meaning in this problem.

EXAMPLE 2. A line DE drawn parallel to the base BC of a triangle ABC meets the side AB in D and the side AC in E. The upper segment of one side is 5 feet, and the lower segment of the same side is as much greater than the upper segment of this side as the upper segment of the second side is less than the upper segment of the first side. If the second side is 2.8 feet, how long is the first side?

Solution 1.-Let x be the first side; then the upper segment is AD 5, and the lower DB = x — 5.

=

B

9

A

.. 5 AE

=x

E ..

AE = 15

and

10,

-X,

EC 2.8
=

C

FIGURE 3

But, by Geometry, it is known that AD AE = DB : EC.

Hence, by substituting the values of AD, AE, DB, EC,

5 15-xx-5: 2.8 15+ x,

(15x) (x-5)=5(x-12.2).

REMARK. The second value, x = 1, is not a solution, because the entire side AB would be less than a part of it, AD = 5.

Solution 2.-Let x be the amount in feet by which the lower segment DB is greater than the upper segment AD; then the first lower segment DB will be 5 + x, and the second upper segment AE will be 5-x, and the second lower segment will be 2.8—(5―x)=x-2.2, and the following proportion is obtained.

Therefore the first lower segment is x + 5 = 9 feet, and the side AB=9+ 5 = 14.

BQx (Fig. 4). By hypothesis, AB = }; AP = |,

Let

By removing fractions,

(a + 2 x √3) (a √3-2x)

= 8 ax.

By multiplying the second parenthesis and the term on the right by

EXAMPLE 4. Place a rectangle whose sides are in the ratio a: b in a circular sector whose central angle is 90°, so that two of the corners of the rectangle lie on the arc and the other two on the radii of the sector. How long are the sides of the rectangle?

Solution.-Let the POQ (Fig. 5) be 90° and the radius of the circle be r; then the length of the chord PQ will be found by the theorem of Pythagoras to be r1 2. Hence,

Since the ORD is isosceles, it is possible to form the equation,

5. From the right angle of a right triangle a perpendicular is drawn to the hypotenuse, dividing it into two segments in the ratio 34. How long is the hypotenuse if the perpendicular is three feet?

6. From the right angle of a right-angled triangle draw a perpendicular to the hypotenuse; the perpendicular divides the hypotenuse into two segments, one of which is six inches longer than the other. How long is the hypotenuse if the perpendicular is four inches?

7. Two circles are tangent externally, and two tangents, common to both circles, are drawn. The distance of the intersection of the tangents from the point of contact of one of the tangents with the larger circle is 2.4 times as large as the radius of the smaller circle; the radius of the larger circle is 5 inches longer than the radius of the smaller circle. What is the distance of the intersection of the tangents from the center of the larger circle?

8. A line of the length a is to be divided harmonically, so that one of the external segments is the fourth part of the other external segment. What are the lengths of the segments?