Need help creating an algebraic expression

Question

There are 1700 bottles of water arranged into 4 groups. The second group has 20 more bottles than the first group. The third group has twice as many bottles as the second group. The fourth group has 10 more bottles than the third group. How many bottles are in each group? Solve this problem in two ways: with the aid of a diagram and with algebraic equations. Explain both solution methods, and discuss how they are related.

Answer 1

Method 1: Diagram

   Group 1:     xxxxxxxxxxxxxxx

   Group 2:     xxxxxxxxxxxxxxx + 20

   Group 3:     xxxxxxxxxxxxxxx + 20

                       xxxxxxxxxxxxxxx + 20

    Group 4:    xxxxxxxxxxxxxxx + 20

                       xxxxxxxxxxxxxxx + 20 + 10

     Total = 6 * xxxxxxxxxxxxxxx + 110  = 1700

                      xxxxxxxxxxxxxx = 1590 / 6 = 265

            

       Group 1:   265

       Group 2:   285 

Group 3: 570

Group 4: 580

Total: 1700

 Method 2: Algebraic Equations

    Total – 1700

    a = count in Group 1

    b = count in Group 2

    c = count in Group 3

    d = count in Group 4

  Note that we may write equations in terms of a:

      b = a + 20

      c = 2 * b = 2 * (a + 20)

      d = c + 10 = 2 * (a + 20) + 10

    total = a + b + c + d = 1700

              a + (a + 20) + (2 * (a + 20)) + (2 * (a + 20) + 10 = 1700

              6 * a + 110 = 1700

               a = 1590 / 6

                a = 265

Now that a is known, it is easy to find b, c, and d,

The two methods use similar use of the total and the relationship of each quantity on the quantity in the first group (a),

Answer 2

The question asks how many bottles are in each group, so we have to see how many unknown variables we have to solve for. In this case from the question we can identify four unknown variables one for each group:

1. a = how many bottles in group 1
2. b = how many bottles in group 2
3. c = how many bottles in group 3
4. d = how many bottles in group 4

Ok, since we have 4 unknown variables, we will need to make four equations using the information provided.

First, we see there are a total of 1700 bottles, logically the sum of all the groups will have to equal this:

1) 1700 = a + b + c + d

Next, we see that group 2 has 20 more bottles than group 1. So, group 1 total plus 20 equals group 2 total

2) b = a + 20

Then, the third group has twice as many bottles as the second group. This means two times the amount of the second group will equal the amount of the third group

3) 2*b = c

Now we need one more equation. Here we see that the fourth group has 10 more bottles than the third group. So the total amount of the third group plus 10 equals the amount in the fourth group.

4) c + 10 = d

That's how you create the algebraic expressions. Now you use substitution or elimination to solve the problem.