Estimation

• Estimation is the process of arriving at an approximate answer to a question.
• Back of the envelope (napkin) computation.
• We do this for a number of reasons.
  • We often just want a quick answer to check to see if we are right.
  • We sometimes can't do an actual count, think about trying to find crowd size, where people are constantly moving, coming and going.
  • We don't have our calculator with us.
• In general, we round to something that makes it easy
• Sometimes it is good to estimate high
  • Going on a date, estimate the bill
  • Traveling, estimate the amount needed to get home.
• Examples:
  • Problem 1 page 14

    Estimate to the nearest 10
          533 + 47.8 + 821.6 + 654.93 + 321 + 0.98
    
             Round each
    	     533-> 530, 47.8 -> 50, 821.6 -> 820, 
    	     654.93 -> 650, 321 -> 320, 0.98 -> 0
    
    	     530
    	      50
    	     820
    	     650 
    	     320
    	       0
                ----
    	    2,370 
    	 

  • Problem 3

    297,700 x 4087 -> 300,000 x 4,000 = 1,200,000,000
         

  • Problem 13

         Estimate the answer
         The income earned for 27 hours at $8.25
             30 x 8 = 240
         

  • Problem 15

         Estimate the answer
         The cost of 6 grocery items if the cost is 
             $4.32, $2.79, $0.79, $7.62, $12.38, $4.99
    	  4      3      1      8      12      5  ≈ 32
          
         

  • Problem 17

         Estimate the answer
         The distance traveled when driving 57 mph for 3.2 hours
    
         Note: d =rt   distance = rate × time
    
         d = 60 × 3 = 180
         

  • Problem 18

         Estimate the answer
         One fourth of an annual salary of 47,600
            1/4 of 48,000  is 12,000
    
         

Problem Solving

• George Polya developed a method for solving problems which he published in his book How To Solve It
• The steps include (page 22)
  1. Understand the problem.
     • Read it carefully, twice, and then once again
       • What are you being asked to find?
       • What information are you given?
     • Make a sketch: this helps you understand and organize your information
     • Make a list
     • Do you have enough information to solve the problem?
  2. Devise a plan to solve the problem
     • Have you seen a similar problem before?
     • Can you break the problem into smaller problems?
     • Can you use an formula?
     • Can you simplify the problem?
  3. Carry out your plan.
  4. Check your answer
     • Does it make sense?
     • Is it reasonable?
     • Can you somehow check against the original problem?
     • Does it match an estimated solution?
• Keep this in mind as you attempt the homework problems.
• Examples:
  • Problem 1 page 29

    	 Reading a map.
    	 The scale on a map is 1 inch = 18 miles.
    	 How long is a distance is a route on the map 
    	 if it measures 4.25 inches.
    
    	 This is a ratio problem 
    	  1 inch     4.25 inches
    	 --------  = -----------
    	 18 miles     x   miles
    
    	 x*1 = 4.25 * 18
    	   x = 18*4.25 = 76.5
    	 

  • Problem 3

    	     At a given time of day, the ratio of the 
    	     height of an object to the length of its shadow is the same
    	     for all objects.  If a 3 ft stick casts a 1.2 ft shadow,
    	     find the length of the shadow of a 48.4 ft tree.
    
    	     This again is a ratio
    
    	     3:1.2 as 48.4:x
    
    	      3     48.4
    	     --- =  ----
    	     1.2      x
    
    	     3x = 1.2 × 48.4
    	     x = 19.4
    	 

  • Number 10

    	     Luann has two packages for which to choose to purchase 
    	     school pictures.  Package 1 costs $27 and includes 23 pictures
    	     Package 2 costs $17 and contains 10 pictures.  If all the
    	     pictures are the same size, which package offers the best
    	     price per picture
    
    	     Calculate the cost per picture
    	        Package    Price Pictures  $/picture 
    		   1       $27     23       27/23 =  1.17
    		   2       $17     10       17/10 = 1.70
    
    	      Package 1 is the best.
    	 

  • Problem 13

    	 $840.62 per month for 15 years
    	 $620.28 per month for 30 years
    
    	 Monthly Payment  Time   Payments   Total Cost
    	   840.62          15   15x12= 180 180x840.62 = 151,311.60
    	   620.28          30   30x12= 360 360x620.28 = 223,300.80
    
    	   223,300.80-151,311.60=$71,989.2
    	 

  • Problem 14

    	    A student had grades of 79, 93, 91 and 68.  What do they need on 
    	    a last test to have an 80% average.
    
    	    79+93+91+68+x/500 =.8
    
    	    331+x = .8 * 500 
    
    	    x = 400-331 = 69