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Numbers and Their Application to Math and Science

Odd Solutions for HW Numbers Lesson 7

  1. Use Pascal's Triangle to expand (2x + 3)6 by examining (2x + 3)2, (2x + 3)3, ...

    A.
    (2x + 3)2 = 1×(2x)2×30 + 2×(2x)1×31 + 1×(2x)0×32
    = 4x2 + 12x + 9
    (2x + 3)3 = 1×(2x)3×30 + 3× (2x)2×31 + 3×(2x)1×32 + 1×(2x)0×33
    =8x3 + 36x2 + 54x + 27
    (2x + 3)6 = 1×(2x)6×30 + 6×(2x)5×31 + 15×(2x)4×32 + 20×(2x)3×33 + 15×(2x)2×34 + 6×(2x)×35 + 1×(2x)0×36
    =64x6 + 576x5 + 2160x4 + 4320x3 + 4860x2 + 2916x + 729

    Simplify completely problems 2-6 using a common denominator.  SHOW WORK!

  2.  
  3. 1+1= 13 + 7=20
    7 13 91 91

  4.  
  5. 1+1+1 =143 + 91 +77= 311
    7 11 131001 1001

  6.  
  7. Find 25% of 16.
    A. Either .25×16 or 1/4 × 16 = 4.

  8.  
  9. The owner of a $50,000 (state equalize value, which should be about half the market value) house must calculate how much a proposed 2 mill road improvement tax will cost him. Help him!
    A. $50000•2/1000 =$100

  10.  
  11. Divide 50 by 1/2 and add 3.
    A. 103=50 × 2 + 3.
  12.  
  13. Simplify completely:
    2/3 + 1/2 =  4/6 + 3/6 = 7/6  = 7/6 × 6/1  = 7  = 7
    5/12 - 1/4 5/12 - 3/12 2/12 1/6 × 6/11
  14.  
  15. Simplify completely:
     35   ÷ 15  ×  6   =  35 ×  34 ×  6 =  1735  ×  234  ×  26   =  4
    1734717157 117 151517

    For problems 16-18:
    Egyption fraction is another name for unit fraction. In ancient Egypt, these were the only fractions allowed. Other fractions between zero and one were always expressed as a sum of distinct Egyption fractions. The greedy algorithm was commonly used to render fractions, such as 3/5, into unit fractions. The algorithm begins by finding two consecutive unit fractions that the given fraction is between ( 1/2 < 3/5 < 1/1). Using the smallest fraction, subtract it from the given fraction. This new number plus the smaller fraction is the result. The greedy Egyption number for 3/5 is 1/2 + 1/10 (3/5 - 1/2 = 6/10 - 5/10 = 1/10). Of course, there is no guaranty the result is a unit fraction, so more than 2 fractions may well be required. (See MMPC 1996, part II, problem 1.)

  16.  
  17. Find the greedy Egyptian fraction for 2/13.
    2/13 = 1/7 + 1/91 this comes from the fact that 1/7 < 2/13 < 1/6 and 2/13-1/7=(14-13)/91

  18.  
  19. Using your corrected list of the first 15 Fibonacci Numbers from homework 2 problem 3, find the approximate ratio of each consecutive pair. Bonus: what is the exact limiting value this approaches?
    A. -, 1, 2, 1.5, 1.666..., 1.6, 1.625, 1.619, 1.6176, 1.61818, 1.617977, 1.6180555, 1.61802575, ....
    Note: your results may be one less or 0.618...=(sqrt(5)-1)/2 if you put the big number on the bottom.
  20.  
  21. Read section 11.2 of your geometry textbooks for further examples for lesson 6. You will be held responsible for the reading on tests. Do problems 11.2: 15-20.

    1. Make a conclusion if given: 1) If x=3, then y=4. 2) y=5.
      A.x is not 3.

    2.  
    3. Joanne had a date, but her mother, whose word was law, told her, "If you don't apologize to your brother for the way you treated him, then you're not going out tonight." Joanne went on her date. Is it true that she apologized to her brother?
      A. Contrapositive: If you go out tonight, then you apologized to your brother. Yes, it's true.

    4.  
    5. All equilateral triangles have three 60° angles. m (angle ABC) = 59°.
      A. Triangle ABC is not equilateral.

    6.  

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