Hence x is greater than 9a and y is greater than 14.

From, x

If y

=

=

2, x

=

5y +2

12, one set of integral values for x and y.

Find the limits of x and y, and, if possible, one set of integral values of the unknown numbers:

17. Twice the number of pupils in a class is less than three times the number diminished by 30, and three times the number of pupils increased by 10 is greater than four times the number diminished by 25. Find the number of pupils. 18. Find the smallest whole number such that of it diminished by 3 is greater than of it increased by 3. 19. Find the limits of x, when x2+5x> 14.

Hence, the product (x+7)(x − 2) is positive and either both factors are positive or both factors are negative.

But x+7 is positive when x > −7

The second condition includes the first, therefore both factors are positive when x > 2.

The first condition includes the second, therefore both factors are negative when x < -7.

Hence x may have any value except 2 and -7 and the values between them.

Find the limits of x in each of the following:

163. Prove, if a and b are positive and unequal, that a2 + b2>2ab.

Since the square of any real number is positive,

If the letters are positive and unequal numbers, prove that:

1. a3 + b3>a2b + ab2.

2. a2 + b2 + c2>ab + ac + bc.

3. a3 + b3 + c3>3abc.

4. (a2 + b2) (a1 + b1)>(a3 + b3)2.

5. Prove that the sum of any fraction and its reciprocal is greater than 2.

6. Prove that a positive fraction is increased by adding the same positive number to each of its terms.

REVIEW EXERCISE XXI

1. If a is positive and b is negative, find the limit of x,

in

3. Given 2x-3x+5 and 11 + 2x <3x + 5, find the limits of x.

х

4. Find an integral value of x if, 4x - 11> and 20

2x > 10.

3

5. Find the limits of x when, 3(x-4)+2 >4(x-3) and 2(x + 1) <4(x − 7) + 3.

6. Prove a+b>2/ab, when a and b are positive and unequal.

7. Find the smallest integer such that of it decreased by 7 is greater than of it increased by 6.

8. Find integral values of x and y, when,

51-y<5x

21+ y>3x

CHAPTER XXII

RATIO AND PROPORTION

RATIO

164. Two numbers may be compared by dividing the first number by the second. This indicated division is called their ratio.

Thus, the ratio of 7 to 5 is indicated by the fraction, sometimes written 7:5, the colon being the ratio sign. The ratio sign is supposed to be the sign of division with the line omitted.

Two quantities of the same kind can be compared when they are measured by a common unit.

Thus, we may compare 3 inches to 1 foot if both quantities are measured by a common unit, either both being expressed as a number of inches, or both expressed as a number of feet, as 3 inches: 12 inches or ft. :1 ft. There can be no ratio between quantities of different kind, as 3 lbs. and 4 ft.

The ratio of two quantities is the ratio of their numerical measures. Thus, the ratio of 3 inches to 12 inches is the ratio of 3 to 12 or 3:12.

The first term of the ratio, which precedes the ratio sign, is called the antecedent and the second, which follows the sign, the consequent. Both terms form a couplet.

x

In the ratio xy or x is the antecedent, y the consequent and y' x and y form a couplet.

When the antecedent is greater than the consequent, the ratio is called a ratio of greater inequality.

When the antecedent is equal to the consequent, it is called a ratio of equality.

When the antecedent is less than the consequent, it is called a ratio of less inequality.

Thus, when a and b are positive numbers,

if a > b, a: b is a ratio of greater inequality. a = b, a: b is a ratio of equality.

a <b, a: b is a ratio of less inequality.

When the antecedent and consequent of a ratio are interchanged, the resulting ratio is called the inverse or reciprocal ratio of the numbers.

Thus, the inverse or reciprocal ratio of a to b is b: a or

b

α

The ratio of the squares of two numbers is called their duplicate ratio, the ratio of their cubes, their triplicate ratio. Thus, the duplicate ratio of a to b is a2: b2; the triplicate ratio is a3: b3.

165. Two quantities that can be exactly expressed in integers in terms of some common unit are said to be commensurable numbers.

The common unit is called a common measure.

Thus, 2 by using

inches and 33 inches are commensurable, since, of an inch as a common measure, 25 inches contains 37 of this unit and 3 inches contains 50 of this unit.

Hence the ratio of 25 inches to 3 inches is the ratio 37:50, a commensurable ratio.

When two quantities cannot be exactly expressed in integers in terms of some common unit of measure, they are said to be incommensurable.

Thus, the diameter of a circle and its circumference are incommensurable quantities. If the diameter can be exactly expressed in terms of some linear unit, the circumference cannot be so expressed. If the diameter is 2 inches, the circumference is 2 X 3.1416+, an approximate value only. These approximations may be found to any required accuracy by using the necessary number of decimal places but will never be absolutely accurate. The ratio of the diameter of a circle to its circumference is therefore an incommensurable ratio.