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tract twice the width of the walk to get E F, the side of the inner square.

Ex. 1. What is the area of a frame formed by squares whose sides A B and E F are 10 in. and 8 in. respectively? Here, area of outer square A B C D 10 × 10=100 sq. in. inner square EFGH=8×8=64 sq. in.

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square frame 100-64-36 sq. in.

Ex. 2. A square plot of ground, E F G H, whose side E Fis 50 yds. long, has a gravel walk 2 yds. wide running round the outside of it. What is the area of the walk?

Here, length of inner square=50 yds.

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= 54 yds.

Hence, area of outer square=54 × 54-2916 sq. yds.

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inner

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= 50 × 50=2500 sq. yds. gravel walk 2916-2500-416 sq. yds.

How many

Ex. 3. A path 2 yds. wide runs round the inside of a square court, A B C D, whose side AB is 24 yds. long. flagstones 1 yd. square will be required to pave the path? Here, length of outer square=24 yds.

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inner

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Hence, area of outer square=24 × 24=576 sq. yds.

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inner
path

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20 yds.

=20×20=400 sq. yds.
=576-400=176 sq. yds.

And as each flagstone will cover a space of 1 sq. yd., the number required will be 176.

Ex. II.

(1) The side of a square grass-plot, A B C D, measures 15 yds., but in its centre is a square flower-bed, E F G H, 9 yds. long. What is the extent of the grass surface?

[First, find the area of the larger square, A B CD, including both the flower-bed and the grass surface, and then of the smaller square, E F G H, representing the flower-bed. The area of the grass surface will be the difference between the areas of the two squares.]

(2) A square court whose side is 500 ft. long, has a square garden within it 400 ft. long; required the area of the part outside the garden.

(3) A room is 25 ft. square; in the central part is a Turkey carpet which measures 18 ft. square; find how many sq. ft. of oilcloth will be required to cover the rest of the floor.

(4) There is a plantation in the form of a hollow square, length externally 156 yds. and depth 15 yds. Find the area of the plantation and of the inner square.

[To find the side of the inner square, twice the depth of the plantation must be taken from the side of the outer square.}

(5) From a square lawn 32 ft. long, a flower-border 3 ft. wide is taken off all round; find the area of the border.

(6) A square court is 60 ft. long, and a path 6 ft. wide goes round the court inside it; find how many tiles a foot square will be required to pave the path.

(7) The side of a square is 84 yds., and a path 8 yds. wide goes round the square outside it; how many stones 1 yd. square will be required to pave the path?

[In this case, twice the width of the path must be added to the side of the inner square.]

(8) The canvas of a painting is 27 in. square; the frame is 2 inches wide. Find the area of the whole picture when framed, and of the surface to be gilded.

19. To find the value of any square surface when the side is given.

RULE. Find the area of the square, and multiply the number of square units in it by the value of a square unit, or proceed by Practice.

Ex. Find the value of a square garden whose side is 20 yds. long, at £2 13s. 4d. per sq. yd.

Area of garden=20×20=400 sq. yds.

Value of garden-£2 13s. 4d. × 400-£1066 13s. 4d.

Or thus, by Practice,

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£ s. d.

400 0 0 value of 400 sq. yds. at £1 per sq. yd.

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(1) A piece of ground is 25 yards square.

at 2s. 3d. per square yard.

Find its value,

(2) A square plot of land 125 yds. long is sold for half-acrown a square yard. Find its value.

(3) A square lawn 90 ft. long is to be laid down with turfs a foot square, at 1s. 3d. per 100.

Find the cost.

[Area of lawn=90 × 90=8100 sq. ft.
No. of turfs required=8100, or 81 hundred.
Cost of turfs=1s. 3d. x 81.]

(4) What will be the expense of carpeting a floor 8 yds. square, at 2s. 9d. per square yard?

(5) A square court whose side is 9 yds. is paved with tiles, at a cost of £10 16s. How much is that per square yard?

[Area of court=81 sq. yds. Cost of 81 sq. yds. = £10 16s. Cost per sq. yd.£10 168.81.]

(6) The cost of trenching a square garden 71 yds. long is £57 15s. 2 d. How much is that per square yard?

(7) Find the cost of carpeting a square room 23 ft. long, with carpet at 4d. a square foot.

(8) Drugget is worth 2d. a square foot. of the quantity required to cover the floor

square.

Find the value of a room 20 ft.

20. When the side is given in several different denominations, such as yards, feet, and inches, it must be expressed in a single denomination before it can be multiplied by itself.

This may be done by reducing the side to its lowest denomination, or by expressing the lower denominations as a fraction or decimal of the highest.

Another method sometimes adopted is called Duodecimals, or Cross Multiplication. We proceed to exemplify each method separately.

21. To find, by reduction, the area of a square when its side is given in different denominations.

RULE. Reduce the side to its lowest denomination, and multiply the result by itself; the product will be the area expressed in the same denomination.

Thus, if the side is reduced to feet only, the arca will be expressed in square feet, which may be reduced to square yards by dividing by 9.

If the side is reduced to inches, the area will be expressed in square inches, which may be reduced to square feet by dividing by 144, and then to squarc yards (if required) by further dividing by 9.

[The lengths of sides are expressed in Linear Measure, areas in Square Measure.]

Ex. 1. Find the area of a square whose side is 7 yds. 2 ft. Here, 7 yds. 2 ft. =23 ft.

Area of square = 23 × 23=529 sq. ft.=58 sq. yds. 7 sq. ft. Ex. 2. Required the value of a square whose side is 4 it. 3 in., at 3s. Od. per square foot.

Here, 4 ft. 3 in. =51 in.

Area of square=2601 sq. in.

£ S.

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Find, in square yards, feet, and inches, the areas of the

squares whose sides are respectively :—

(10) 3 yds. 2 ft. 6 in.

(11) 12 yds. 1 ft. 3 in.

(12) 30 yds. 2 ft. 9 in.

(13) 16 yds. 1 ft. 7 in.

(14) 1 yd. 2 ft. 3 in.

(15) 8 yds. 1 ft. 9 in.

(16) What is the area, in yards, feet, and inches, of a square table whose side measures 3 ft. 9 in.?

(17) Required the area of a square grass-plot whose perimeter is 25 yds. 1 ft. 8 in.

(18) A square sheep-fold is 14 ft. 6 in. long. Find its area, and the cost of the fence which encloses it, at 8d. a foot.

(19) A square playground is 17 yds. 2 ft. long. What will be the expense of gravelling it, at 9d. per square yard?

22. To find, by fractions, the area of a square whose side is given in different denominations.

RULE. Express the length of the side as a fraction of the highest denomination, and multiply the rc

sult by itself; the product will be the area expressed in the same denomination.

When the side is given in yards and feet, express the feet as a fractional part of a yard, thus:

1 ft.=yd. 2 ft. 3 yd. Also 1 ft. 6 in. = yd.

When the side is given in feet and inches, express the inches as a fractional part of a foot, thus:

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Ex. 1. If the side of a square is 5 yds. 2 ft., find its area. Here, 5 yds. 2 ft.=53 yds.

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Area 53×534×H=282=321 sq. yds.

Ex. 2. The side of a square is 3 ft. 9 in.; required its area. Here, 3 ft. 9 in. =3 ft.

Area 32x32-15-225-14 sq. ft.

Ex. V.

Express in square yards, and the fraction of a square yard, the area of the squares whose sides are respectively:

(1) 14 yds. 1 ft.

(2) 3 yds. 2 ft. (3) 7 yds. 1 ft. 6 in.

Express in square feet, and the fraction of a square foot, the

area of squares having the following lengths of sides:

(4) 7 ft. 6 in.

(8) 10 ft. 4 in.

(5) 3 ft. 3 in.

(9) 7 ft. 9 in.

(6) 8 ft. 1 in.

(10) 11 ft. 5 in.

(7) 19 ft. 2 in.

(11) 2 ft. 11 in.

(12) 6 ft. 8 in.
(13) 13 ft. 10 in.

(14) 14 ft. 7 in.

(15) 10 ft. 1 in.

(16) Find the value of a square plot of

long, at 6s. 9d. per square yard.

ground 27 yds. 2 ft.

(17) Find the expense of paving a square courtyard, whose boundary wall measures 57 yds. 1 ft., at 3s. 9d. per square yard.

23. To find, by decimals, the area of a square when its side is given in different denominations.

RULE. Express the length of the side as a decimal of the highest denomination, and multiply the result by itself; the product will be the area expressed in the same denomination.

Express the inches as a decimal part of a foot,

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