Algebra I Unit 1 Problem Solving Skills Practice

1. Using the four integers 2, 3, 6 and 8 once each -- in any order -- and three arithmetic operations selected from among addition, subtraction, multiplication, and division, write expressions whose values are the target numbers given below.  You are allowed to use grouping symbols.

        (a) 3                        (b) 24                        (c) 36                        (d) 30

2. Each person in your group will pick any number.  Add 4 to it and then double your answer.  Now subtract 6 from that result and divide your new answer by 2.  Write down your answer.  

• Now, check the answers in your group.
• Compare your final answers with your original number.
• Do you notice a pattern?

3. Each person in your group will choose any number.  Double it.  Subtract six and add the original number.  Now divide by three.  Now check the numbers in your group.  

• Do you notice a pattern?  
• By using a variable such as x in place of your number, what is the pattern in terms of x?

4. Jess takes a board that is 50 inches long and cuts it into two pieces, one of which is 16 inches longer than the other.  How long is each piece?

5. When describing the growth of a population, the passage of time is sometimes described in generations, a generation being about 30 years.  One generation ago, you had two ancestors (your parents).  Two generations ago, you had four ancestors (your grandparents).  Ninety years ago, you had eight ancestors (your great-grandparents).  How many ancestors did you have 300 years ago?  900 years ago?  Do your answers make sense?

6. Before you are able to take a bite of your new chocolate bar, a friend comes along and takes ¼ of the bar.  Then another friend comes along and you give this person 1/3 of what you have left.  Make a diagram that shows the part of the bar left for you to eat.  

7. Later you have another chocolate bar.  This time, after you give away 1/3 of the bar, a friend breaks off ¾ of the remaining piece.  What part of the original chocolate bar do you have left?  Draw a diagram to show your answer.

8. Consider the sequence of numbers 2, 5, 8, 11, 14,…, in which each number is three more than its predecessor.

• Find the next three numbers in the sequence.
• Find the 20th number in the sequence.
• Using the variable n to represent the position of a number in the sequence, write an expression that allows you to calculate the nth number.  The 200th number in the sequence is 599.  Verify this with your expression.  

9. Another number puzzle:  Pick a number, add 5 and multiply the result by 4.  Add another 5 and multiply the result by 4 again.  Subtract 100 from your result and divide your answer by 8.  How does your answer compare to the original number?  Try a couple more numbers to see the pattern.  Use a variable x to show the pattern.

10. A group of ten people were planning to contribute equal amounts of money to buy some pizza.  After the pizza was ordered, one person left.  Each of the other nine people had to pay 60 cents extra as a result.  How much was the total bill?

11. To buy a ticket for a weekly state lottery, a person selects 6 integers from 1 to 36, the order not being important.  There are 1947792 such combinations of six digits.  Alex and nine friends want to win the lottery by buying every possible ticket (all 1947792 combinations), and plan to spend 16 hours a day doing it.  Assume that each person buys 1 ticket every five seconds.  What do you think of this plan?  Can the project be completed in a week?

12. Find whole numbers m and n that fit the equations .  Is it possible to find whole numbers m and n that fit the equation ?  If so, find an example.  If not, explain why not.  

13. Here’s an ad in the local newspaper:  

Amazing Investment opportunity at Algebank!  Double your money instantly!  Invest any amount!  No         amount is too small.  our bank will double the amount of money in your account every month.  Watch         your money grow!

        A service charge of $100 will be deducted from your account at the end of every month.

        Explore:  Do you think this is a good deal?  Why or why not?  

        a).  Rey was interested in this investment.  After calling to make sure that the $100 would be deducted         after his money was doubled, he decided to join.  However, after his service charge was deducted at the         end of the fourth month, he discovered that his bank balance was exactly $0!!  How much money did he         start out with?  

        b).  Three students invested their money.  Gable started with $45, Max with $60, and Lara with $200.          The figure shows a way to keep track of what happened to Lara’s investment:

• Use arrows in this way to show that happened to Lara’s, Max’s and Gabe’s investment for the first five months.

• Give advice to each of these students.

14.  Which one is larger?  How did you know?

        4^3 or 2^7?                                4^15 or 2^29?

15.  Look for the pattern and find the values of x and y.

                a + 45 = 101

                b + 445 = 1001

                c + 4445 = 10001

                d + 44445 = 100001

                ...

                x + y = 100000001

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18.  A specialty coffee store wants to create a special mix using two coffees, one priced at $6.40 per pound and the other priced at $7.28 per pound.  How many pounds of the $7.28 coffee should be mixed with 9 pounds of the $6.40 coffee to sell the mixture for $6.95 per pound?

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