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21. How many yards of carpet 1 yd. wide must be purchased for a floor 20 ft. 3 in. long, 16 ft. wide, when the strips run lengthwise?

When the strips run lengthwise, 5 strips are required, but 6 strips must be purchased. The length of the room being 6 yd., the number of yards is 6 x 6, etc.

22. How many yards of carpet 27 in. wide will be needed for the floor of a room 20 ft. 6 in. long, 16 ft. wide, when the strips run crosswise?

Number of strips: 20 ft. ÷ 27 in. = 246 in. ÷ 27 in. = 9; 10 strips must be used. Length of each strip, 16 ft., or 5 yd. Quantity of carpet to be bought: 53 yd. x 10, etc.

23. A floor is 20 ft. 3 in. long, 16 ft. wide. How many yards of carpet are needed (a) when the carpet is 1 yard wide and the strips run crosswise? (b) Find the number of yards of carpet 27 inches wide that must be bought when the strips run lengthwise. (c) When they run crosswise.

Matching Patterns. — In "making" a carpet for a given floor, the strips are sewed together in such a way as to "match the patterns." This frequently requires that a portion of each strip except the first be cut off, the amount varying according to the pattern.

24. How many yards of carpet, 27 inches wide, are required for the floor of a room 20 ft. 3 in. long, 16 ft. wide, when 4 inches are wasted on each strip except the first, and the strips run lengthwise?

25. How many inches must be cut off every strip except the first to match the pattern when the first strip is 20 ft. 3 in. long, and the pattern is repeated (a) every 9 inches? (b) Every 12 inches? (c) Every 8 inches?

Deductions for Openings

In building walls by the cubic yard; in painting, plastering, etc., by the square yard, contractors do not make a full allowance for openings. In some cases no deduction is made for openings below a certain size; in other cases one half of the area of the openings is deducted.

In ascertaining the quantity of material required, the actual surface or volume is used, due consideration, however, being given to material necessarily wasted.

26. (a) At 30¢ per square yard, find the cost of plastering the walls and the ceiling of a room 15 ft. long, 12 ft. wide, and 10 ft. high, making one half allowance for 2 doors, each 9 ft. by 3 feet., and 2 windows each 6 ft. by 3 ft. (b) Find the cost of tinting the walls and the ceiling at 8 per square yard, making the same allowance. (c) At 35 cents per square yard, find the cost of painting the baseboard, which is 1 ft. wide, the doors and the windows, making no allowance in the last for the space occupied by the glass.

27. The outside dimensions of the walls of a cellar are 36 ft. by 24 ft. (a) How many square feet remain for the floor of the cellar, if the walls are 2 ft. thick? (b) How many square feet are occupied by the walls? (c) If the walls are 9 ft. high, how many cubic yards of material do they contain, if there is one opening 9 ft. by 4 ft. and three openings each 3 ft. square? (d) How many tons of stone are required, if one ton is sufficient for a perch of 161 cu. ft.? (e) Find the cost of building the walls at $3 per perch of 22 cu. ft., if the walls are measured on the outside and one half allowance is made for the openings.

The walls are considered by the contractor as equivalent to a wall (36 ft.+ 24 ft. + 36 ft. + 24 ft.) long, 9 ft. high, 2 ft. thick.

28. A finished road consists of 3 inches of fine broken stone laid on 12 inches of coarser material. (a) How many cubic yards of each must be spread before rolling for a mile of road 27 feet wide, assuming that the steam roller will compress it into three fourths of the space it occupied when loose? (b) How many cart loads will be required at cu. yd. to the load?

29. A printer has an order for 1500 cards 3" by 21′′. He has 25 sheets of pasteboard measuring 24" by 18". What is the largest number of cards he can get out of this material?

30. Two rectangular cisterns, with lids, are to be made of sheet iron. One measures 12' x 8' x 6', the other 16' x 9' x 4'. (a) Find the number of square feet of material required for each, making no allowance for seams. (b) What is the capacity of each in cubic feet?

31. Find the difference in the number of cubic inches between a solid 12" x 8" x 6" and one 14" x 10" x 8".

32. How many board feet of inch boards will be required for a covered box having inside dimensions 12" × 8" x 6'?

33. A window sash whose outside dimensions are 5' x 3' 6" contains 4 panes of glass of the same size. The frame is 2" wide and the panes are separated from each other by strips 1" wide. Make a drawing showing the dimensions of each pane.

34. A stair carpet covers 18 steps 10", tread and 7"'rise, with 18" extra at both the bottom and the top of the staircase. How many yards of carpet are required?

SECTION XVI

EQUATIONS IN BUSINESS

Formulas

A builder is asked what load a certain beam will bear. Turning to his handbook he finds the following formula for the safe load of a rectangular beam supported at both ends and uniformly loaded over the entire span.

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An explanatory note states that w represents the width of the beam in inches, d its depth in inches, and its length in feet between the points of support. The value of C is given in a table which shows the equivalent in pounds for different woods; yellow pine, oak, spruce, white pine, hemlock, etc.

If the beams are of spruce 3 inches wide, 6 inches deep, and 12 feet long, and the table gives 70 pounds as the safe unit for spruce, these figures are substituted in the foregoing formula, thus:

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2. What is the safe load when yellow pine is used in a similar case, its unit being 100 pounds?

Equations

The foregoing formula constitutes an equation which consists of two members connected by a sign of equality, one of the members containing an unknown number whose value is to be determined.

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The expression 6 + 5 = 11 is not an equation; it is called an

In these two examples the result is determined by substituting the given numbers and performing the indicated operations. In the following, intermediate steps are required to obtain the result.

3. A builder wishes to ascertain the depth of a yellow pine beam 12 feet long between supports and 3 inches wide that will sustain a load of 1800 pounds uniformly distributed.

He substitutes in formula (a) the given numbers, producing the following equation :

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He first simplifies the second member by cancellation, making it 50 d2, which gives the following:

1800 = 50 d2,

or, as it is customary to make the first member the one containing the unknown number:

50 d2: =

1800.

Dividing both members by 50, the equation becomes

d2 = 36.

Extracting the square root of both members,

d = 6. Ans. 6 inches deep.

Test by substituting 36 for d2 in the formula, which should give 1800 lb. for the safe load.

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