x- Y

x+y

\x+y x-y \x+y x-y

[HINT. Change these expressions to fractions as in Ex. 19.]

For further exercises on this topic, see the review exercises,

p. 161, and Appendix, p. 303.

EXERCISES-APPLIED PROBLEMS

The algebraic processes needed in the following exercises are simple. The principal difficulty is to read the problem carefully and intelligently.

1

1. In the figure are two weights, one weighing W1 lb. (read W sub 1 pounds) and the other weighing W2 lb. (read W sub 2 pounds). They are joined by a string which runs over a freely turning wheel at the top (the wheel having a groove cut in it for the string to run in). When the two weights thus tied together are left to themselves, the heavier one gradually descends (goes down), pulling up the lighter weight on the other side. It is shown in physics that during the motion the force T tending to break the string

FIG. 38.

(called the tension) has the value given by the formula

By means of this formula, answer each of the following questions:

(a) If W1=3 lb. and W2=14lb. what is the tension in the string during the motion? 2 lb. Ans.

(b) If W1=W2, show that the tension becomes equal to either of the weights.

In this case there is no motion.

(c) If W1 =5 lb., what must W2 be in order that the tension during the motion be 4 lb.?

(d) Show that if at any time the weight W1 be increased by q lb., the tension T will be increased by

a

2. The figure represents a beam (wood or steel or any other material) with its ends resting upon two level supports (called abutments), as is illustrated, for example, in the girder of a bridge. If a weight (usually called a load) of W lb. is suspended from the beam at the point P which is a feet from one abutment and b feet from the other, then the upward pressures (called thrusts) on the beam

W

FIG. 39.

at its ends (where it rests on the abutments) are

the first of these being the thrust at the end which is distant a feet from P, and the second being the thrust at the end which is distant b feet from P.

Whence, show the following facts about such a beam:

(a) The two thrusts are equal when P is the middle point of the beam; that is, when a=b.

(b) The sum of the thrusts is in all cases precisely equal to the load W suspended.

3. When an automobile weighing 2 tons is the way across a bridge, how much of its weight is being supported by the nearer abutment; how much by the farther one? (Answer by the formulas in Ex. 2.)

4. A man writes: "We have a set of hay-scales, and sometimes we have to weigh wagons that are too long to go on them. Can we get the correct weight by weighing one end at a time and then adding the two weights?" (Answer by means of Ex. 2.)

*103. Complex Fractions. A fraction whose numerator or denominator (or both) contains fractions is called a complex fraction.

are complex fractions. Since a fraction is an indicated division, it follows that the expression in the numerator of a complex fraction is to be divided by the expression in the denominator. A complex fraction can be reduced to a simple one by simplifying the numerator and the denominator and then performing the division indicated.

SOLUTION. This fraction may be simplified, as in Exs. 1 and 2; or we may simply multiply both terms of the fraction by 3 b (which is the least common denominator of the simple fractions). Thus, we obtain