15. How many square feet in a yard squared ? 16. How many square inches are there in 1 square foot? 17. How many square feet in 1 square yard? 18. How many square yards in 1 square rod? 19. How many square feet in 1 square rod? 40 square rods make 1 rood; 4 roods make 1 acre. 20. How many rods make 1 acre? 21. If a piece of board is 6 inches wide, how long must it be to contain a square foot? 22. If a piece of board is 3 inches wide, how long must it be to contain a square foot? 23. How long must it be to contain a square foot, if it is 2 inches wide? If 1 inch wide? If 4 inches wide? 24. If cloth is a yard wide, how much in length will make a square yard? 25. How much lining of a yard wide will line one yard of cloth one yard wide? 26. If cloth is two thirds of a yard wide, how much in length will it take for a square yard? 27. How much cloth a yard wide will it take to line 7 yards of cloth of a yard wide? How much wide will line 14 yards wide? 28. How long must a strip of land one rod wide be to contain an acre? How long, if 2 rods wide? If 3 rods wide? If 4 rods wide? If 8 rods wide? If 10 rods wide? 29. How long must a piece of land be to contain acre, if it is 4 rods wide? of an 30. If a piece of land is 10 rods in length how long must it be to contain an acre? 31. A man has an acre of land 16 rods in length; how wide is it? 32. How many steps must the owner take to walk round it if he take 5 steps to a rod? 33. A man has an acre of land 8 rods wide; how long is it? How many rods of fence will it take to fence it? 34. If a road 4 rods wide is laid out through my land, how much of the road will it take in length to make an acre? How many acres will there be in one mile of the road? 35. If it passes through my land for half a mile, and I am paid at the rate of 30 dollars an acre for the land occupied by the road, what will be the amount of damages due me? 36. If land in the city is worth 45 cents a square foot, what will be the cost of a building-lot 30 feet front and 60 feet from front to rear? 37. There are two pieces of land; one of them 12 rods square, the other 13. Which is nearest an acre? 38. There is a piece of land 12 rods square; how much does it fall short of an acre? 39. A painter tells me it will cost 20 cents a square yard to paint the floor of a room in my house; supposing the room is 5 yards wide and 64 yards long, what will the painting of it come to? 40. What will the painting of an entry cost, at the same rate; that is 14 yards wide and 7 yards in length? 41. A stone-cutter agrees to lay a hammered stone door-step for 50 cents for every square foot of hammered surface. The stone is 5 feet long, 34 feet wide, and 9 inches thick; what is the surface of the top, the two ends, and the front edge, added together? What will be the cost of the stone? 42. How many men could stand on of a mile square, allowing each man 1 square yard to stand upon? There are various ways of finding the answer to the above questions. To encourage the student's invention, some of them will be here suggested. First Method. As there are 30+ square yards in one square rod, multiply 80 (the number of rods in one fourth of a mile,) by itself; and this product by 304. 80×80=6400; 6400× 301-18,000+12,000+1600-193,600, answer. Second Method. Multiply 80 by 54, which will give the number of men in one line one fourth of a mile long; multiply this product by itself. 80×5=440; 4402—16,000+32,000 +1600-193,600, answer. Third Method. As there are 301 square yards in one square rod, in 10 rods square there will be 100 times this number, or 3025; multiply this by 8, which will give the square yards in a rectangle 80 rods long and 10 rods wide; multiply this product by 8, which will give the square yards in a square 80 rods on a side. 30X100 3025; 3025X8 =24,200; 24,200×8=192,000+1600—193,600, answer. There are still other ways of solving the question, which the student may discover for himself. SECTION XVI. CONSTRUCTION OF THE SQUARE. If I place three dots in a row, and place three such rows side by side, this will represent to the eye the square of the number 3. Thus, In the same way you may represent the square of 4, 5, or any number whatever. I will now ask your attention to the square of 4. We may make it by making a row of 4 dots, and placing 4 such rows side by side. But there is another way of coming at the square of 4. We will take the square of 3, as shown above, and see what additions we must make to it, in order to make it the square of 4. You observe that it must be wider by one row and longer by one row than it is now. We will then add a row above the others, and also a row on the right hand. Thus, Thus, * * * I have made the additions by stars to distinguish them from the dots. You now see there is something wanting to complete the square, single star in the corner. - a You observe, therefore, that you obtain the square of 4 by adding to the square of 3, twice 3 plus 1. We will now take the square of 4, and by additions to it obtain the square of 5. Adding a row of 4 at the top, and a row of 4 at the right hand, there will be one wanting at the corner to complete the square. Adding this, which makes twice 4+1, we have the square, complete. If therefore we have the square of any number, we can find the square of a number one greater by adding twice the first number plus 1. The square of 5 is 25; what must you add to this square to make the square of 6? What must you add to the square of 6 to make the square of 7? What must you add to the square of 7 to make the square of 8? What must you add to the square of 9 to make the square of 10? The square of 15 is 225; above method? The square of 20 is 400; what is the square of 16, by the what is the square of 21? The square of 40 is 1600; what is the square of 41? The square of 50 is 2500; what is the square of 51? What is the square of 60? of 61? of 70? of 71? of 80 ? of 81? of 90? of 91? We will now return to the square of 3, and I ask your close attention once more. Supposing we have the square of 3 before us, and we wish to make such additions to it as shall make the square of 5. As 5 is two greater than 3, we must one. If we add 2 rows of 3 at the top, and 2 rows of 3 at the right hand, the figure will stand thus, add two rows instead of * Here you see there are four stars wanting to complete the square. I have marked their places by the circle (O). If you suppose these four to be added the square will be complete, and will be the square of 5. The question is now, what has been added to the square of 3 in order to make the square of 5? You observe there are added 6 stars or two rows of three at the top, 6 on the right hand, and 4 in the corner, to make the square of 5. But we can express this in a different way. We may consider 5 as consisting of two parts, 3 and 2 added together. We will call 3 the first part, and 2 the second part of 5. Now by the figure you perceive that the square of 5 is made up, first, of the square of the first part, that is, the nine dots; then the stars at the top are the product of the first part multiplied by the second, and adding to these the stars on the right hand, we have twice the product of the first part into the second; and, last, we have, in the corner, the square of the second part. To state it briefly once more: regarding 5 as made up of the two parts, 3 and 2, the square of 5 we find is equal to the square of the first part+twice the product of the two parts+ the square of the second part. This is called expressing the amount of a square in the terms of its parts. Examine and answer the following questions: 1. If we regard the number 6 as made up of two parts, 4 and 2, how will you express the square of 6 in the terms of its parts? 2. Regard the number 7 as consisting of two parts, 5 and 2; what is the square of 7 in the terms of its parts? You can draw the figure for yourself and see the application of the principle in the above cases. It is of no consequence in what way the number is divided; the operation will bring out the exact square of the whole number in all cases. To show this we will take the number 10, the square of which is 100. We will first divide 10 into the parts, 7 and 3; then, by the formula given above, the sq. of 7-+-twice the product of 7 into 3+ sq. of 3, will be the sq. of the whole number. The sq. of 7=49, twice 7×3=42, the sq. of 3-9; 49+42+9=100. We will now divide 10 into the parts 6 and 4, proceeding as above; we find 36+48+16=100. Again, we will divide 10 into the equal parts, 5 and 5, 25+50+25=100. Finally divide 10 into the parts 8 and 2. 64+32+4=100. We will now apply the above method to the purpose of finding some squares of larger numbers. 3. What is the sq. of 25? dividing into 20+5; Ans. 400+ 200+25625? 4. What is the sq. of 35, or 30+5? Ans. 900+300+ 25=1225. 5. What is the sq. of 46 or 40+6? Ans. 1600+480+ 36=2116. 6. What is the sq. of 55? of 64? of 75? of 83? of 92? 7. What is the sq. of 125? divide into 100+25. 100 sq. =10000, twice 100X25-5000, 25 sq.-625. Ans. 15,625. 8. What is the square of 150? of 230? of 510. The same formula will embrace the examples mentioned in the first part of this section, when the second part of the number is 1, for example, 9. What is the sq. of 5 or 4+1? Here twice the product of the two parts is merely twice the first part, inasmuch as multiplying by 1 does not increase the number; and the sq. of 1 is only 1. The answer, therefore, by the formula, is 16 +8+1=25. |