Problems for Arithmetic or Algebra 1. A farmer has three times as many acres of corn as of potatoes, and five times as many acres of grass as of corn. In all he has 95 acres. How many acres has he of each? 2. A has seven business as B has. times as much money invested in A has $15,000 more invested than B has. How much has each invested? 3. A certain number plus of itself plus of itself equals 148. Find the number. 4. Find two numbers whose sum is 52 and one of which is three times the other. 5. A man bought a horse and sold him so as to gain as much as the cost. The selling price was $126. was the cost? What 6. There are baskets of peaches of two sizes at a fruit store. In each of the four smaller baskets there is a certain number, and in each of the six larger baskets there are three times as many. In all there are 440 peaches. How many are there in the different baskets? 7. A boy had a certain number of papers. He bought as many more, and then sold of what he had. He had 10 papers left. left. How How many had he at first? 8. A man and his two sons worked 12 days. The older son received the same wages as the father, and the younger son as much as the father. In all they received $48. How much did each receive per day? 9. The width of a certain field is distance around the field is 56 rods. of its length. The Find its area. The length 10. A triangular field contains 25 acres. of one side is 52 rods. What is the perpendicular distance from this side to the opposite angle? Square Root A perfect square is the product of two equal factors or of two equal groups of factors. The square root of any number which is a perfect square may be found by first finding its factors. To find the approximate square root of a number which is not a perfect square, another process is necessary. Note that the square of a number consisting of one figure is composed of not more than two figures; the square of a number consisting of two figures is composed of not less than three and not more than four figures; the square of a number consisting of three figures is composed of not less than five and not more than six figures, etc. Hence when a number is squared the square of the units' figure is contained within the first group of two figures, reckoning from the right, the square of the tens' figure within the second group of two figures, and so on. The square of 234 is 5'47'56. The square of 4 is contained within the first group, or section, 56. The square of the 3, or 30, is contained within the second section, 47, or 4700. The square of the 2, or 200, is contained within the third section, 5, or 50,000. The excess in each of these sections over the squares of the separate parts of 234 is due to the fact that the square of the whole combined is more than the sum of the squares of the separate parts. = = Square Root- Algebraic Explanation 1. Find the square of a + b. a + b a+b a2 + ab ab + b2 a2 + 2 ab + b2 a2 + b (2 a + b) 2. Square 43. 40 +3 40 +3 402 + 3 x 40 8 x 40 + 32 Since a and b may represent any numbers, we see from the multiplication that the square of the sum of two numbers is always equal to the square of the first number, plus twice the product of the first and the second, plus the square of the second; or the square of the first, plus the second, times the sum of twice the first and the second. 402 + 2 (3 × 40) + 32 402+3 (2 × 40 + 3) In the number 43, since the 4 is tens, its real value is 40, and the number may be written in the form 40 +3. The square of the number in this form gives a result in the same form as in the case of the square of a + b. 3. Find the square root of 1849. 402 = 1849/40 +3 2 x 40 80 249 2 x 40+ 3 = 83 249 By dividing the figures of the number into sections we know that the square root of the number will consist of two figures. The greatest number of tens whose square is contained in 1849 is 4 tens, or 40, and the square is 1600. After subtracting 1600 from the number, the 249 remaining must contain twice the product of the tens and the units, and the square of the units. Since the square of the units must be a comparatively small number, we may regard 249 as consisting approximately of twice the product of the tens and the units. We know that twice the product of the tens is 80; hence the units must be approximately the number of times that 80 is contained in 249, or 3. Then 3 (2 × 40 + 3) or 3 × 83, or 249, will complete the parts of the square of 40+ 3, leaving no remainder. Find the square root of the following numbers and explain the process: Percentage 1. If goods are bought for $1250 and sold for $1500, what per cent is gained? 2. I sold a car load of corn for $242 and gained 10%. What per cent should I have gained if I had received $253 ? 3. If a tax of $60 is paid on a building worth $4000, what is the value of a building upon which a tax of $13.50 is paid, at the same rate? 4. Find the amount of difference between buying goods at discounts of 25% and 10% and at discounts of 10% and 25%, if the list price in each case is $300. 5. I bought corn for $860 and wheat for $735. On the corn I gained 18% and on the wheat I lost 14%. How much did I gain by the whole transaction? 6. Which is the better investment, 5% stock at 80 or 7% stock at 120 ? 7. I bought flour at $4.20 a barrel, on 6 months' credit. I sold it immediately for cash, at an advance of 10%. I put the proceeds at interest at 6%. When I paid for the flour, at the end of 6 months, how much had I gained per barrel? 8. Find the bank discount of a note of $525 for 60 days, at 51 per cent. 9. $400. PITTSBURG, PA., Jan. 5, 1901. Three months after date, for value received, I promise to pay C. M. Henry, or order, Four Hundred Dollars, with interest at 5%. This note was discounted at a bank at 6%, Feb. 24, 1901. Find the proceeds. |
