11. Find the duty on 120 doz. linen cuffs at 7s. 6d. a dozen, the duty being 40 a dozen and 20% ad valorem.

12. Find the duty on a fur rug valued at $50, duty 35% ad valorem. At what price should the rug be sold so as to make a profit of 20% ?

13. Calculate the duty levied on 5 doz. straw hats valued at $30 a dozen, the duty being $7 a dozen and 20% ad valorem.

14. What is the duty on 800 yd. of silk valued at $1.15 a yard at 60% ad valorem? At what price per yard should the silk be sold to make a profit of 20% ?

15. Calculate the duty on 12 horses valued at $250 each at 25% ad valorem.

16. An Axminster carpet, 18 ft. by 12 ft., valued at $1.50 per square yard, is imported. Find the duty at 60¢ per square yard and 40% ad valorem.

17. What is the duty on 5 doz. opera glasses valued at $3.50 each at 45% ad valorem?

18. What is the duty on 500 lb. of glue valued at 50¢ per pound, the rate of duty being 15¢ per pound and 20% ad valorem?

19. A suit of clothes imported from England costs a merchant $38.80. The duty is 60% ad valorem. Taking £1= $4.85, find the invoice price in pounds sterling.

20. Find the duty on 1200 yd. of linen invoiced at 71⁄2d. per yard at 45% ad valorem. At what price per yard should it be marked so that the dealer may give a discount of 10% and make a profit of 20% ?

CHAPTER IV

INVOLUTION

IN the language of mathematics, 3 x means three times the number x; i.e. 3x may stand for three times any number whatever.

α

a+b stands for the sum of any two numbers, a and b.

b stands for the difference of two numbers, a and b. In other words, ab is the number which when added to b gives a for sum.

ab stands for the product of two numbers, a and b. stands for the quotient obtained by dividing a by b. In other words, is that number which when multiplied by b b gives a for product.

a2 stands for the square of a. a3 stands for the cube of a, (a+b)2 stands for the square of the sum of two numbers, a and b.

(a - b)2 stands for the square of the difference of two numbers, a and b.

7 ab stands for seven times the number ab, which is itself the product of two numbers, a and b.

bers plus one half their difference.

(a+b)x stands for the product of a number, x, by the sum of two numbers, a and b.

(a - b)x stands for the product of a number, x, by the difference of two numbers, a and b.

=

SOLUTION.

20 x2.

?

4 x x 5x = 4 × x × 5 × x = 4 x 5 x x x x

= × ×

Example 2. Multiply 7 x2y by 3 xy2.

SOLUTION. 7 x2y × 3 xy2

=

(7 × x x x × y) × (3 × x ×

y × y) = 7 x x x x x x × 3 × y × y × y = 21 x3y3,

Example 3. What is the square of a+b?

Multiply a+b by a, and write the result, a2+ab. Next multiply a+b by b and write the result, ab+b2. Then add. The following is a graphical representation of the result:

Let ABCD be the square on AB, ALHG be the square on AL. Then HYCK is the square on b, and GHKD = LBYH, because their dimensions are equal to each other. Now, ABCD = ALHG + GHKD + HYCK + LBYH = ALHG + HYCK+2GHKD.

=

··· (a+b)2=a2 + b2+2 ab. Hence, we have the following important conclusion:

The square of the sum of two numbers equals the square of the first number plus the square of the second number plus twice the product of the two numbers.

The process of finding the power of a number is called.

involution.

Example 1. Square x + 9.

SOLUTION. (+9)2 = x2 + 92 + 2 × 9 × x = x2 +81 + 18x, or x2+18x+81.

Example 2.

SOLUTION.

What is the square of 43?

432 = (40 + 3)2

432 = (40 + 3)2 = 402 + 32 + 2 × 40 × 3=

Find by Statement (3) the square of:

1. Find the square of .1, .2, .3, .4, .5, .6, .7, .8, .9.

2. Find the square of the reciprocals of the numbers. from 1 to 20 inclusive.

9 2

9 10

3. Find the squares of 3, 4, 8, 7, 10, 11, 18, 16.

4. Find the squares of §, 3, 4, 11, 13, 17, 23, 61.

5. How does the square of a fraction compare in value with the fraction itself?

6. Find the cubes of the reciprocals of the numbers from 1 to 20 inclusive.

7. Find the value of 13 +23+ 33 + 43 + 53 + 63 + 73 + 88 +93 +103.

8. Square each of the following numbers and give the product correct to four decimal figures: (a) 1.732, (b) 9.256, (c) 5.401, (d) 8.129, (e) .6834, (ƒ) .7609, (g) 9.482, (h) .7071, (i) .7746, () .9487.