__Appendix 2__

1. An unknown number added to ten more than fifteen times itself gives one hundred six. Find the number.

The relationship is (the unknown number) + 10 more than 15 times (the unknown number) gives 106. Tag the unknown number by *x*. Translated to algebraic symbols, the problem becomes *x* + 10 + 15*x* = 106. This equation simplifies to **16 x + 10 = 106**.

2. A 106-foot length of rope is cut into three pieces. The second piece is 10 feet longer than seven times the length of the first, and the third piece is eight times the length of the first piece; find the lengths of all three pieces.

The three pieces are related to each other in the following way:

• The second piece is 10 feet more than seven times the length of the first piece.

• The third piece is eight times as long as the first piece.

Let the length of the first piece be represented by *x*. If we do this, we obtain the following algebraic relationships:

• First piece length = *x*.

• Second piece length = 10 + 7*x*.

• Third piece length = 8*x*.

We know that first piece length + second piece length + third piece length = 106. Translating to algebra gives the equation *x* + 10 + 7*x* + 8*x* = 106. Simplifying the left-hand side gives **16 x + 10 = 106**.

3. Given a rectangle of perimeter twice 53 meters and whose length is five more than seven times its width, find its length and width:

The perimeter is given by 2(length) + 2(width). We have the following from the diagram:

• Length = 7*x* + 5.

• Width = *x*.

In the language of *x*, the perimeter becomes 2(7*x* + 5) + 2*x* = 14*x* + 10 + 2*x* = 16*x* + 10.

The perimeter’s quantitative value in this problem measures twice 53 meters, which is 2(53 meters) = 106 meters. This gives the following relationship: Perimeter in *x* language = perimeter’s quantitative value. Expressing this relationship in algebraic equation language yields **16 x + 10 = 106**.

4. If a power tool rents for $16 a day plus a one-time $10 processing fee, how many days can you rent the tool if you have four $20 bills, two tens, a five, and a dollar bill to spend?

We want to find the number of days that we can rent using the available money. Let *x* = number of days. The money we spend for the rental is 16 times (the number of days) + $10 processing fee, or equivalently in *x* language, 16*x* + 10.

The available amount of money is 4(20) + 2(10) + 5 + 1 = 80 + 20 + 5 + 1 = 106. This gives the following relationship: Money spent in *x* language = available amount of money. Expressing this in algebraic language yields the equation **16 x + 10 = 106**.

5. Next Saturday Barbara will be doing a job for a client that pays her $36 an hour. The job requires specialized computer services that cost $20 an hour to use in addition to a $40 setup fee. When she arrives on location for the job, she notices a $50 bill that the client left to say thanks for coming in on the weekend. How many hours does she need to work so that her total profit (including her tip) for the day is $106?

We want to find the number of hours Barbara needs to work to yield a profit of $106. Let *x* = number of hours. Profit is calculated as Barbara’s earnings minus her costs. Here are those values in *x* language:

• Barbara’s earnings = (hourly wage)*x* + gift = 36*x* + 50.

• Barbara’s costs = (hourly charge)*x* + setup fee = 20*x* + 40.

Barbara’s quantitative profit in dollars equals $106.

This gives the following relationship: Barbara’s profit in *x* language = Barbara’s quantitative profit. Expressing this relationship in algebraic symbols yields

Simplification gives 36*x* + 50 – 20*x* – 40 = 106 or **16 x + 10 = 106**.