PROBLEMS-REVIEW

89

1. Find the cost of gross of photographic paper (1 gross = 12 dozen) at 944 per gross; & of a gross of pencils at $1.50 per gross; doz. inkstands at $.35 each.

2. At 34 per foot what is the cost of sufficient picturemolding to go around a room 14 ft. by 17 ft.?

3. A man bought a 25-foot lot at $15 per foot. Later he bought two adjacent 20-foot lots at $24 per foot. Still later he bought a

bought all of them at a uniform price per foot and at the same total cost, how much would he have paid per foot?

4. The lots of Exercise 3 were 125 ft. deep; what is the cost of a fence surrounding the whole plot at 60¢ per running foot?

5. Find the cost of sodding a margin 15 ft. wide inside the fence along the front and side of the lot at 4¢ per square foot.

6. What will be the cost of a flagstone walk 5 ft. wide outside of the fence along the two street frontages, if the stone and labor cost 50¢ per square foot?

7. What will the owner make by selling the entire piece at $48 per foot of the main frontage?

8. In 1900 the three States producing the largest quantities of manufactured ice were Pennsylvania, 735,000 tons; New York, 455,000 tons; Missouri, 285,000 tons.

Draw on

the board straight lines representing these numbers, letting 1 inch represent 25,000 tons.

9. A line ft. long would represent similarly the product of Illinois; express this amount in tons.

POWERS AND ROOTS

Powers

1. What is the area of a square 2 inches on a side?

2. What is the area of a square 3 feet on a side? What is such a square called?

3. What is the area of a square flower-bed 4 ft. on a side? 4. What are the computations for finding the areas of the above squares?

5. How do the factors in each product compare?

2 x 2 = 4,

3 x 3 = 9,

The product of two equal factors is

either factor, or the second power of it.

4 X 4 = 16.

called the square of

Thus, in Exercise 5 above, 4 is the square (or second power) of 2,

9 is the square (or second power) of 3, and 16 is the square (or second power) of 4.

6. 25 is the square of what number? 36 is the second power of what number? 49 is the second power of what number? 81 is the second power of what number? 64 is the square of what number? 100 is the square of

7. State orally the squares of all integers from 1 to 10 inclusive.

8. Apples are packed in crates as shown in the picture; how many apples are there in a layer? How many apples are there in a box of 5 layers? Of 6 layers? When apples are 14 each, what is the cost of a box of each size?

9. Write a table of the squares of all integers from 1 to 10 inclusive.

Oral.

ROOTS

91

1. The area of a square rug is 25 sq. ft.; what is the length of a side?

2. How many squares are there in the side of a square checker-board containing 64 squares?

3. The area of a square register is 81 sq. in.; what is the length of one side?

4. The area of a square plot is 16 sq. yd.; how many yards are there in the length of one side?

5. How many 4-inch squares can be cut from 1 sq. ft. of cardboard? How many 3-inch squares?

One of the two equal factors of a number is called its square root. Since 2 X 24, 2 is the square root of 4;

5

12

12

since 3 × 3 = 9, 3 is the square root of 9. Similarly 5 is the square root of 25.

Of 81? Of 64?

6. What is the square root of 36? Of 49? Of 100? Of 144? Of 400? Of 2,500?

Written.

7. If an 8-inch square be cut from a square foot of cardboard, and as many 3-inch squares as possible from what is left, how many 3-inch squares are made? How many square inches of cardboard are wasted?

8. Draw on the blackboard a square 20 in. on a side. Rule it into square inches. With heavy lines mark off a square containing 121 sq. in. and as many squares as possible containing 49 sq. in. each. How many square inches of space are not used?

9. Test the work of Exercise 8 by adding the areas of all the parts and comparing with the area of the whole square. 10. What is the difference between the square roots of 100 and 121?

ANALYSIS OF PROBLEMS

Directions

PROBLEM: To find the altitude of a rectangle whose base is 8 in. and whose area is 152 sq. in.

1. Read carefully, noting that the base and the area of a rectangle are given, and that the altitude is required.

2. Plan: Recall that the area of a rectangle is the product of the numbers expressing its length and breadth. Hence 152 is 8 times the required number and must be divided by 8.

3. Computation.

19

8)152

4. Test. 8 x 19 = 152.

In the solution of every problem there are four main steps:

a. Read the problem; note carefully what is given and what is required.

b. Plan the work; determine how to find what is required from what is given.

c. Make the computations as planned.

d. Test the result.

1. Read and determine what is given and what is required: A train runs from New York to Cleveland, 624 mi., in 16 hr. ; find the speed in miles per hour.

2. Plan the work to be done in solving Exercise 1.

3. Make the computation. 16)624 4. What is the test? 5. Read: The area of a triangle is 176 sq. ft.; its altitude is 11 ft.; what is its base?

6. Plan the solution.

SUGGESTION.-When the calculation contains more than one process, it is better to indicate all of them in the plan. Thus, in this problem: 1. 2 x 176 the altitude times the base. times the base. The base = 352 ÷ 11=

2. 352 = 11

PLANNING PROBLEMS

Plan the solution; do not make the computation:

93

1. How many feet per minute is a train moving when traveling 42 mi. per hour?

2. A train left Cincinnati at 8:15 a. M. and arrived at St. Paul, 702 miles distant, at 2:57 A. M. the next day; find its average speed in miles per hour.

3. The base of a rectangle is 16 in., its area is 3 sq. ft.; find the altitude in inches.

4. Show how to find the perimeter of the figure.

5. Draw a figure like this. Draw a line from A to B; show how to find the area of the figure.

6. A man earns $75 a month.

17 ft.

36 ft.

A

B

8 ft.

25 ft.

12 ft.

He spends $10 per month for room rent; $3.50 per week for board, $125 per year for clothing and other expenses; in how many years will he save $1,000 at this rate?

7. 40 ft. of wire weighed 1 lb.; what was the weight of 31 mi. of this wire?

8. A, B, and C own a store; A owns of it and B owns as much as A; what part does C own?

9. The base of a rectangle is 17 in.; its perimeter is 48; what is its altitude?

10. A 100-acre farm contains 4 lots. Three of the lots contain 75 acres, 20 acres, and 75 acres respectively; how many acres does the fourth lot contain?

NOTE.-Hereafter the planning of a problem should be made a distinct and important feature of its solution. In very simple problems, the plan and the work need not be separated, but any uncertainty as to how a problem is to be worked shows clearly that a separate, correct plan is necessary. NEVER WORK AT RANDOM.