Intermediate Arithmetic

count of 15% was given on the remainder. How much more than the cost of material did the school receive?

24. A coiled basket made from buri raffia was sold for P2.45. If this price allowed 40% of the cost of material for labor, for how much should the basket have been sold in order to get 50 % of the cost of material for labor?

25. It took 2 bundles of buntal for a basket and cost P.05 for nito used, 50% of the cost of material was allowed for labor, and the basket sold for P1.50. What was the cost of the buntal per bundle?

26. If maguey costs P15 a picul and a picul is sufficient to make 10 baskets and nito for them costs P1, for how much each must the baskets be sold in order to allow 25 % for labor?

27. It took a boy 3 days to make a basket which he sold for P1.50. The cost of the dumayaka petioles, nito, and air roots used was 40% of the selling price of the basket. How much per day did the boy get for his work?

28. Juan and Pedro each made a basket from banban stems. They sold the two baskets for P5, Juan receiving 871% as much as Pedro. How much did each receive?

29. Balangot costs P.40 a kilo and 4 pairs of slippers can be made from a kilo. Twisted abacá costs P 1.20 a skein and a skein will make two pairs of slippers. If the balangot slippers sell for P.40 a pair and a boy can make 2 pairs a day, and abacá slippers sell for P2 and it takes the same boy 5 days to make a pair, which would give the boy the higher daily wages, and how much?

The maAn agent

30. Mr. Santos made 10 round mats in 100 days. terial used was 20 kilos of tikug worth P.30 a kilo. sold the mats on 15% commission. If the agent's commission was P9, how much did Mr. Santos earn per day?

18. Oral. 3x3=9

POWERS AND ROOTS

POWERS

3 x 3 x 3 = 27

3 x 3 x 3 x 3 = 81

9, 27, and 81 are powers of 3.

A power is a product of equal factors.

1. What is the product of 4 taken twice as a factor, or what is the second power of 4?

2. What is the product of 3 taken three times as a factor, or what is the third power of 3?

3. What is the product of 2 taken 4 times as a factor, or what is the fourth power of 2?

A power is named according to the number of its equal factors. The product of two equal factors is the second power, or square, of the number used as a factor. The product of three equal factors is the third power, or cube, of the number used as a factor. The powers above the third are called simply fourth, fifth, etc.

The second power is called a square because the area of a square is the product of two equal factors, length and breadth; and the third power is called a cube because the volume of a cube is the product of three equal factors, length, breadth, and thickness.

An exponent is a small figure placed at the right and a little above a number to show how many times the number is to be used as a factor, or what power of the number is to be found. 75 is read, the fifth power of 7. (1)2 is read, the square of §. the cube of 4. 5, 2, and 3 are exponents.

4. Learn the squares of the numbers from 1 to 25.

5. Learn the cubes of the numbers from 1 to 10.

48 is read,

Of 36?

Of 81?

Of 64?

Of 100 ?

Of 64?

Of 125?

1. What are the two equal factors of 9? 2. What are the two equal factors of 16? 3. What are the three equal factors of 8? 4. What are the four equal factors of 16? Of 81? Of 10,000?

A root of a number is one of its equal factors.

The square root of a number is one of its two equal factors; the cube root, one of its three equal factors; the fourth root, one of its four equal factors; etc.

The radical sign (√) placed before a number indicates that its root is to be found.

The index of the root is a small figure placed in the opening of the radical sign to show what root is to be found.

When the index is omitted, the square root is indicated.

√25 = 5 is read, "the square root of 25 is 5"; V64 = 4 is read, “the cube root of 64 is 4"; √625 = 5 is read, "the fourth root of 625 is 5."

21. Oral.

12=1

9281

Square Root

102=100

992 = 9801

100210,000 9992998,001

From the above illustrations we see that the square of a number of one figure contains either one or two figures; the square of a number of two figures contains either three or four figures; etc. Hence, if the figures of a number be separated into groups of two figures each, beginning at the decimal point, the number of groups will show the number of figures in the square root of the number.

If the number is a whole number, the left-hand group may contain either one or two figures.

How many figures are there in the square root of:

In finding the square of a number containing tens and units,

we may separate the tens and units.

Thus 262 (20+6)2.

Hence, the square of a number composed of tens and units is equal to the square of the tens + twice the tens times the units + the square of the units.

f2 + 2 × (t × u) + u2 may be written, t2 + (2 × t + u) × u.

13. Separate the following into tens and units and find their squares, using the formula, t2 + 2 × (t × u) + u2: 12, 15, 22, 24, 31, 33, 45, 52, 64, 71, 85, 91.

22. Oral.

The area of a square is the product of two equal factors, length and breadth.

Finding the square root of a number, therefore, is equivalent to finding the length of one side of a square whose area is the given number.

Find the length of one side of a square whose area is 1156 sq. m.

Since the number contains four figures, its square root will contain two tens and units. In the diagram:

figures

A tens2, D =

units2,

tens x units, C = tens x units.

A + B + C + D = t2 + 2 × (t × u) + u2.

900, the square of 30, is less than 1156, and 1600, the square of 40, is greater than 1156. Therefore, the length of one side of the square whose area is 1156 sq. m lies between 30 m and 40 m.

The greatest square of tens in 1156 sq. m is 900 sq. m (A). The

length of one side of A

is 30 m.

1156 sq. m 900 sq. m =

256 sq.m.

The area of B + C + D = 256 sq. m.