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Let's take a closer look at how independent and dependent variables work by working through some examples:

Sample Problem

Jim gets paid $10 per hour. The amount Jim gets paid in a week depends on the number of hours he works that week. If all he did was put in two hours shelving books at the library, he will barely be able to afford to buy a book. Good thing he works at the library. On the other hand, if he works 12-hour days in the assembly line of an automobile factory, he can afford to buy all the books he wants. Same rate (input), different pay (output). 

Sample Problem

Jim gets paid $10 per hour. The amount Jim gets paid in a week depends on the number of hours he works that week. Let p represent the amount Jim gets paid in a week. Let h represent the number of hours Jim works that week. Then p = 10 · h.

The letters p and h are called variables because they are not fixed numbers. This fact reminds us: Be sure to have your numbers spayed or neutered. The quantity h varies because Jim may work a different number of hours each week. The quantity p varies because p depends on h.

Example 1

Tara throws a party every month. She's a little desperate for attention. The number of cupcakes she bakes for her guests depends on how many kids will be at the party. Tara likes to have two cupcakes per kid. She would also like to have some adult friends, but good luck with that, Tara. How should we express this situation algebraically?


Example 2

Tara throws a party every month. The number of cupcakes Tara bakes depends on how many kids will be at the party. Tara likes to have two cupcakes per kid. Let c be a variable representing the number of cupcakes Tara bakes, and let k be a variable representing the number of kids who will be at the party. You can also let l represent Tara's sense of inner loneliness, but that one won't play a part in this particular equation: c = 2 · k.Which variables vary in this situation?


Example 3

Fred and John are brothers. John, who is the older of the two and was largely deprived of attention as a young boy, is constantly trying to one - up Fred. Both brothers enjoy clothes shopping. However, because of John's competitiveness, every time Fred buys a new pair of jeans, John will go out and buy a pair that is $10 more expensive. Both pairs still look exactly the same, and all John is actually doing is demonstrating a lack of fiscal responsibility. Express in symbols the relationship between the amount of money Fred spends on a pair of jeans and the amount of money John spends on a pair of jeans. Let F be the amount of money Fred spends, and J the amount of money John spends.


Exercise 1

Ruth gets $2 every time she helps her dad gather sticks in the yard. Apparently, there is a major stick epidemic in their neighborhood. Let t be the number of times Ruth helps her dad gather sticks in the yard, and let R be the amount of money Ruth gets for picking up the sticks. What is the relationship between R and t?

Exercise 2

Ruth gets $2 every time she helps her dad gather sticks in the yard. Apparently, there is a major stick epidemic in their neighborhood. Let t be the number of times Ruth helps her dad gather sticks in the yard, and let R be the amount of money Ruth gets for picking up the sticks. Does R depend on t or does t depend on R?

Exercise 3

Ruth gets $2 every time she helps her dad gather sticks in the yard. Apparently, there is a major stick epidemic in their neighborhood. Let t be the number of times Ruth helps her dad gather sticks in the yard, and let R be the amount of money Ruth gets for picking up the sticks. Why are R and t good letters to use as variables for this problem?

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