12. ab

=

=

10; what is b if a 1? 2? 3? 4? 10? 20?

13. By what principle may we write

c(x + y + z) = cx + cy + cz,

x, y, and z denoting the bases of 3 rectangles whose altitudes are each equal to c? (Answer by sketching the proper figure and pointing out the rectangles whose areas represent each of the products in the equation.)

=

14. If a 8 and b = 5, find the values of the following expressions and tell what ones are equal:

15. If 9 and y = 4, tell which of the following

x =

express true relations and write the correct relation sign in other cases: (See p. 255.)

§133. Statements in Words and in Symbols.

1. Write the symbolic statements for these verbal phrases and statements. Let x stand for the number when there is but one number to be symbolized in the problem: (1) A certain number increased by 15.

(2) Twice a number diminished by 8.

(3) Seven times a number increased by three times the number.

(4) The square of a number, divided by 8.

(5) The sum of the square and the first power of a number.

(6) Eight times a number, divided by three.

(7) One-third of 10 times a number.

(8) Three times a certain number, diminished by one, equals 20.

(9) Eighteen times the square of a number equals 72. (10) Twenty-five times a number, increased by 5, equals 30 times the number, diminished by 15.

(11) One-eighth of the sum of a certain number and 18. (12) Six times the difference between a certain number and 3 equals 18.

(13) The product of the sum and the difference of x and 3 equals 18 (x being greater than 3).

(14) The difference between x and 18 is greater than 18; is less than 25; is equal to 20 (x > 18 in each case).

2. State in words what these expressions mean. For example, (1) means "double a certain number, diminished by 9":

(10) (x−1)(x−1)

(11) 7x-2+16

(12) 5x+7=42

(13) (x−4)(x+4)=20

(14) (a+b)(a−b) = a2 — b2 (15) (a+b)2=a2+2ab+b2

3. Translate into symbols these verbal phrases and statements, using a and b, or x and y, for the two numbers: (1) The sum of two numbers equals 25.

(2) The difference of two numbers equals 15.

(3) The sum of the squares of two numbers is less than 27.

(4) The square of the sum of two numbers equals 100. (5) The difference of the squares of two numbers equals 9.

(6) The sum of the squares of two numbers equals seven times the difference of the numbers.

(7) The product of two numbers equals their sum. (8) The quotient of two numbers equals their difference. (9) A certain number increased by 1 equals another number diminished by 3.

4. Translate into words these symbolic expressions. For example, (1) means "one-ninth of the difference between 6 times a certain number and its square":

5. Find the number which may be put in place of the

letter in each of these equations to furnish true equations:

NOTE.—First multiply both sides of (7) by 21.

6. Find the numbers of problem 1 (8), (9), and (10).

$134. Problems. FOR EITHER ARITHMETIC OR ALGEBRA.

1. The mercury column in a thermometer rose a certain number of degrees one day, and 3 times as many degrees the next day. It rose 12° during the 2 days. How many degrees did it rise each day?

ARITHMETICAL SOLUTIONS:

A certain number denotes the rise the first day.

3 times this number denotes the rise the second day.
Hence, 4 times a certain number denotes the rise in two days.
4 times a certain number equals 12° (by the given problem).
Once the number equals 3o, the rise the first day (Principle IV).
3 times the number equals 9°, the rise the second day (Principle III).
Check: 3° + 9° 12°, the rise in two days.

ALGEBRAIC SOLUTION:

=

Let x denote the first day's rise.

Then, 3x denotes the second day's rise.
x + 3x denotes the rise in 2 days.

4x

3x

X= 3o, the first day's rise (Principle IV).
9°, the second day's rise (Principle III).
Check: 3° + 9° 12°.

=

2. A man bought 4 times as many hogs as cows, and after selling 5 hogs he had 23 hogs left. How many cows did he buy?

ARITHMETICAL SOLUTION:

A certain number represents the number of cows bought.
4 times this number represents the number of hogs bought.
4 times this number minus 5 denotes the number of hogs left.
Then 4 times this number, minus 5, equals 23 (by the problem).
4 times this number equals 23 plus 5 (Principle I).

4 times this number

This number

=

7, the number of cows (Principle IV). Check: 4 X 7 5

ALGEBRAIC SOLUTION:

=

23.

Let x denote the number of cows bought.
Then, 4x denotes the number of hogs bought.
5 denotes the number of hogs left.

4x

3. Two masses were placed on one scale pan of a balance and found to weigh 18 lb. One of the masses was then placed in each pan, and it required 4 lb. additional on the light pan to balance the scales. What was the weight of each mass?

ARITHMETICAL SOLUTION⚫

A certain number of pounds denotes the weight of the heavier mass.
Another number of pounds denotes the weight of the lighter mass.
The first number plus the second number denotes the combined
weight (18 pounds).

The first number minus the second denotes the difference of the
weights, or the additional weight, which equals 4 pounds.
2 times the first number equals 18 plus 4 equals 22.

The first number equals 11.

11 plus the second number
The second number

=

18. (Principle IV).
7. (Principle II).
The weights are, then, 7 lb. and 11 lb.

=

Check: 11 lb. + 7 lb.

=

18 lb., and 11 lb. - 7 lb.

=

4 lb.