X.

2. The length of a floor exceeds the breadth by $5$ feet. If each dimension were $1$ foot more, the area of the floor would be $42$ sq. ft. more. Find its dimensions.

3. A rectangle whose length is $6$ feet more than its breadth, would have its area $35$ sq. ft. more, if each dimension were $1$ foot more. Find its dimensions.

4. The length of a rectangle exceeds its width by $3$ feet. If the length is increased by $3$ feet and the width diminished by $2$ feet, the area will not be altered. Find its dimensions.

5. The length of a floor exceeds its width by $10$ feet If each dimension were $2$ feet more, the area would be $144$ sq. ft. more. Find its dimensions.

143. Formulas and Rules.

When the given numbers of a problem are represented by letters, the result obtained from solving the problem is a general expression which includes all problems of that class. Such an expression is called a formula, and the translation of this formula into words is called a rule.

We will illustrate by examples.

1. The sum of two numbers is $s$, and their difference $d$. Find the numbers.

lint Let x = the smaller number; lintthen x + d = the larger number. [1] lint Hence x + x + d = s, lintor 2x = s - d. x = s - d2, [1] lintand x + d = s - d2 + d = s - d + 2d2, = s + d2.

Therefore the numbers are ${\displaystyle \frac{s+d}{2}}$ and ${\displaystyle \frac{s-d}{2}}$.

As these formulas hold true whatever numbers $s$ and $d$ stand for, we have the general rule for finding two numbers when their sum and difference are given:

Add the difference to the sum and take half the result for the greater number.

Subtract the difference from the sum and take half the result for the smaller number.

2. If A can do a piece of work in $a$ days, and B can do the same work in $b$ days, in how many days can both together do it?

lint Let x = the required number of days. lint Then, 1x = the part both together can do in one day. [1] lint Now 1a = the part A can do in one day, lintand 1b = the part B can do in one day, linttherefore 1a + 1b = the part both together can do in one day

1a + 1b = 1x. lint Whence x = aba + b.

The translation of this formula gives the following rule for finding the time required by two agents together to produce a given result when the time required by each agent separately is known.

Divide the product of the numbers which express the units of time required by each to do the work by the sum of these numbers, the quotient is the time required by both together.