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Solving Averages, Investments, Piggy Bank Problems, and Geometry Using Different Methods, Exams of Algebra

University of Houston (UH)Algebra

Solutions to various problems involving averages, investments, piggy bank problems, and geometry using different methods such as setting goals, factoring, completing the square, and the quadratic formula.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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koofers-user-zh8 🇺🇸

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1310, 2.2 Applications and Word Problems
We’ll study four kinds: Averages, Investments, Piggy Bank problems, and Geometry.
Averages
You know that the average of a collection of data is the sum divided by the number of
data points. You may not know that you can do many different kinds of problems
involving averages.
1. Suppose you want an A in a course that has 5 tests, all equally weighted, and you
know 4 of your scores…you can then set a goal and see what you have to score on the
last test.
Suppose you have the following grades: 87, 88, 93, 89 and you need a 90% average
to get an A…what do you need to score on the last test?
2. Suppose you know that 3 consecutive natural numbers average to 9. What are the
numbers?
What is the product of the numbers?
What is the smallest number?
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1310, 2.2 Applications and Word Problems We’ll study four kinds: Averages, Investments, Piggy Bank problems, and Geometry. Averages You know that the average of a collection of data is the sum divided by the number of data points. You may not know that you can do many different kinds of problems involving averages.

  1. Suppose you want an A in a course that has 5 tests, all equally weighted, and you know 4 of your scores…you can then set a goal and see what you have to score on the last test. Suppose you have the following grades: 87, 88, 93, 89 and you need a 90% average to get an A…what do you need to score on the last test?
  2. Suppose you know that 3 consecutive natural numbers average to 9. What are the numbers? What is the product of the numbers? What is the smallest number?
  1. Let’s talk about a situation in which Janet is paid twice as much per hour as Mark is and Susan earns two dollars an hour less than Mark. If their average earnings per hour is $6.00 how much does Mark earn?
  2. The sum of four consecutive even integers is 636. Find the average of the integers.
  1. Michael invested $10,000 for 18 months at 10% simple interest. How much did he earn at the end of 18 months?

And there’s the ever popular piggy bank problems :

  1. A piggy bank has 25 coins – all nickels and quarters – that add up to $5.65. How many nickels are there in the bank? Here, we’ve got two things to track: the number of coins and the value of the coins. We’ll work the problem in dollars. How many quarters? What is the product of the number of nickels and quarters? What is the average number of each type of coin?

And there’s geometry problems

  1. A rectangle’s length is 7 inches greater than it’s width. If the perimeter of the rectangle is 110 inches, find its length and width.
  2. A rectangular garden is three times as long as it is wide. If the total area of the garden is 432 square feet, find the length and width.

2.3 Quadratic equations – mainly Completing the Square! Recall factoring! (x  2 ) (x  3) = (^) x^2  5 x  6 Now we’ll look at going further: If x^2  5 x 6  0 , what is x? Note we are finding the x axis intercepts of the parabola here. Well, this is a question that we need to answer. And we need to answer it with the trinomial in factored form because if a + b = 0, then all you know is that a =  b. BUT if you know that ab = 0, then you know the identity of a or b or both! Amazing but true. So, if x^2  5 x 6  0 , you know that (x  2) (x  3) = x 2  5 x 6  0 , thus x  2 = 0 or x  3 = 0 so x = 2 or x = 3. You may need to brush up on factoring in order to work these problems by factoring. Recall the quadratic formula! Another way to solve for x. x = 2 a  b  b^2  4 ac for ax^2 bxc 0

Another one: x 2  16  0 Factoring QF Algebraically Quickly: x 2  81  0

Now for today’s material: Solving for x using Complete the Square Given (^) x 2 x 12  0 Here’s the steps:

  1. If a is a number other than one, divide both sides by a. Regard this as a new quadratic a = 1 b = _________ c = _________
  2. Take b and divide it by 2. Square the result and add it to both sides. 2 2 b       = Now we have: _____________________________________________________
  3. Rewrite the first 3 terms as )^2 2 b ( x 
  4. Subtract c from both sides:
  5. Take the square root of both sides:

Let’s do another one: 2 x 2  x 6  0 Solve for x using Complete the Square.

  1. If a is a number other than one, divide both sides by a.
  2. Take b and divide it by 2. Square the result and add it to both sides.
  3. Rewrite the first 3 terms as )^2 2 b ( x 
  4. Subtract c from both sides:
  1. Take the square root of both sides:
  2. Subtract 2 b from both sides
  3. Report both values for x.
  1. Take the square root of both sides:
  2. Subtract 2 b from both sides
  3. Report both values for x. The steps:
  1. If a is a number other that one, divide both sides by a.
  2. Take b and divide it by 2. Square the result and add it to both sides.
  3. Rewrite the first 3 terms as )^2 2 b ( x 
  4. Subtract c from both sides:
  5. Take the square root of both sides:
  6. Subtract 2 b from both sides
  7. Report both values for x. More practice x 2  3 x 5  0

The steps:

  1. If a is a number other that one, divide both sides by a.
  2. Take b and divide it by 2. Square the result and add it to both sides.
  3. Rewrite the first 3 terms as )^2 2 b ( x 
  4. Subtract c from both sides:
  5. Take the square root of both sides:
  6. Subtract 2 b from both sides
  7. Report both values for x. x 2  6x  5  0

The steps:

  1. If a is a number other that one, divide both sides by a.
  2. Take b and divide it by 2. Square the result and add it to both sides.
  3. Rewrite the first 3 terms as )^2 2 b ( x 
  4. Subtract c from both sides:
  5. Take the square root of both sides:
  6. Subtract 2 b from both sides
  7. Report both values for x.  2x 2  6x 3.