would naturally think that the purchaser of this first Folio in 1623 made a fine investment. What would an original investment of $ 5 amount to in 1912 at 6% compound interest? 7. If at the beginning of the year 1, one cent had been invested at 4% compound interest, what would the amount be in 1915? What would be the radius of a sphere of gold that would represent the value of the investment in 1916, if a cubic foot of gold is worth $ 362,900 ? 8. If log 2.3010 find the value of x in the equation 2-10. 9. Compute the value of 32 by means of logarithms. (Harvard.) 10. About 300 years ago the Indians sold Manhattan Island to Peter Minuit for $24. Suppose this money had been put out at compound interest at 6%, how much would it have amounted to at the present time? 11. According to the will of Benjamin Franklin, the cities of Boston and Philadelphia each received £ 1000 in July 1791 to be invested at 5 % compound interest for 100 years. In July 1891 the total amount of the fund in Boston was $391,168.68 and in Philadelphia $100,000. How much should have been realized by the terms of the will? (£1000 $5000.) 12. A chain of letters is started for the purpose of aiding an old railroad man who is ill. Number 1 sends a letter to each of 5 friends, each of them in turn sends a letter to 5 friends, and so on. If the chain ends with letter number 50 and each person who receives a letter sends 10 cents, how much does the man receive? XXIX. GENERAL REVIEW 646. 1. If a 3, b = 2, c = 1, find the value of each of the = following expressions: (1) 2 a2b2; 2(a2 - b2); (2 a2 - b2)2; 2(a2 - b2)2. (4) 1+3+5+... +(2n-1)= n2. 3. Solve the following problems by translating the verbal language of the problem into an equation with one unknown: (1) In five years a boy wil be double the age he was five years ago. How old is he? (2) I have as many brothers as sisters said a boy. And I, said one of his sisters, have twice as many brothers as sisters. How many brothers and sisters were there? (3) Can there be three consecutive integers such that their sum is three times the smallest ? (4) The sum of three consecutive numbers is three times the middle number. What are the three numbers? Does this problem lead to an equation or to an identity? 4. Express in algebraic language the following theorems: (1) The product of two numbers is equal to the difference of the squares of their half sum and their half difference. (2) Every integer that is a perfect square diminished by unity is equal to the product of the number that is one less than its square root by the number that is one more than its square root. (3) The difference between the squares of two consecutive integers is an odd number, obtained by increasing by unity. twice the smaller of the two numbers. b 5. The lengths of the sides of a triangle are a = 5 inches, = 4 inches, c = 3 inches. Indicate the semi-perimeter by and find the area, A, of the triangle, if A =√s(8 − a) (s — b)(s — c). 6. If 2 s represents the perimeter of a triangle and a, b, c its sides, verify the following: s+t, 7. Having given the trinomials r + s + t, r + s — t, r · — r+s+t, from the sum of the first three subtract the sum of the last three increased by the sum of the second and third. 8. If A=(p+q) + (r + s), B = (p + q) − (r + s), C=(p−q) +(r− s), D=(p − q) − (1 − s), find the value of A+B+ C + D and of A x D by type multiplication. 9. Prove that [m-(p+q) + r] — {m − [(p + q)−r]} + {m −(p + r) + q } = m − p + q − r. 11. Show that (1+x+x2 + 203 + 204 + 205)(1 − x)= 1 − 26. Show also that (1 x + x2 - 203 +24 — 205)(1 + x)= 1 − x6. 12. Divide 3 x1o + 14 x32 + 9 x2 + 2 by 3x2o — x2 + 2. 13, Divide x3m x31 by x2m + xm+n + x2n. 14. What value must the coefficient k have in order that 5 x3 + 9 x2 + kx + 2 may be exactly divisible by x2 - 3x +2? (6) [(p − q)2 — (r − s)2] ÷ (p − q−r + s). 16. Divide a2(b + c) − b2(c + a) + c2 (a + b) + abc by a-b + c. 17. Square and cube each of the following: 2a+1; x-7; 3x-5y; ax2 + by2; x+≤ p ; { m − 1 n. 18. Square each of the following: a3 — a2 + a 1; x−y+z+1; ax2 + bx + c. 19. Verify the following identities: (1) (a2 + b2)2 = (a2 — b2)2 + (2 ab)2. (2) (a2 + b2 + c2)2 = (a2 + b2 — c2)2 + (2 ac)2 + (2 bc)2. (3) [(n + 2)2 − (n + 1)2] − [(n + 1)2 — n2] = 2. (4) Show that the identity (n + 1)2 - n2 = 2n + 1 expresses that the difference between the squares of two consecutive integers is always an odd number. 20. Evaluate each of the following expressions: (1) 2 × 5+12÷4-7+6-4×7. (2) 6 × 7-32 x 5+8x5-7. 6 21. Extract the square roots of the following polynomials: (1) 3a2 - 2a+1-2 a3 + a1. (2) 16x+9y8 - 30 x2y + 49 x1y* — 40 x1y2. 22. Factor into prime factors: (1) 7 a1 + 7 a2b2 14 a3b. (2) m2(a − b) + n2(b − a). (3) 7 pqx2-42 pqx+63 pq - 7 prx2 + 42 prx- 63 pr. (4) xy-x+y-1. (5) xx' + xx'' + x2 + x'x''. (7) a2b + b2c + c2a + a2c + b2a + c2b+3 abc. (8) 7x2 - 28x4. (9) - 2 uv - u2 — v2 2 uw - 2 vw — w2. (10) x3 (x2 — y2) — y3 (x2 — y2) — xy (x − y)2 (x + y). (11) (a2 + b2)2 — (a2 — b2)2. (12) x(x-1)-(x − 1)2 + x2 - 1. (15) 28 +10x+21. 23. Solve the following equations by factoring: (1) x2 - (a+b)x + ab = 0. (2) p2 + 3p+2=0. (6) x3-2x2-4x+8=0. (9) s4 + s2 – 12 = 0. (10) 3+ k2= 0. (11) Find a number such that if 3 and 5 are subtracted from it in turn, the product of the two remainders is 120. (12) Find two numbers such that their difference is 2 and the sum of their squares is 130. 24. Find the H. C. F. of 2-3x+2, x2 - 2x + 1, x2 + x −2. 25. Find the H. C. F. of x2+2x+1, x 10 x2+9, x+2x2-5x-6. |