Algebra Period 4

Word Problems.. Continuted...

Generally, you see these type of word problems for geometry, age problems, and finding two integers that have some sort of relationship to each other.

GEOMETRY:
The length of a rectangle is twice its width. The perimeter is 48 inches. What is the length and width? Let w = width and l = length of the rectangle Using P = 2l + 2w
2l + 2w = 48 and What else do we know? l = 2w
We can't solve either of these 2 equations because they each have 2 different variables. But...we can substitute in for one of the variables and "get rid of it"! :)
2l + 2w = 48 Substitute 2w in for length: ( because we know l = 2w)
2(2w) + 2w = 48
6w = 48
w = 8 inches
l = 2w so l = 2(8) = 16 inches

AGE PROBLEMS:
Mary and Sam's ages sum to 50

Mary's age is 10 less than twice Sam's age. Find their ages
Let M = Mary's age and S = Sam's age
M = 2S - 10
M + S = 50
Substitute (2S - 10) for Mary's age so you'll only have 1 variable:
(2S - 10) + S = 50
3S - 10 = 50
3S = 60
Sam (S) = 20 years old
Mary = 2S - 10 = 2(20) - 10 = 30 years old.


Finding 2 integers:
The sum of 2 integers is 26.
One integer is 10 more than 3 times the other. Find the 2 integers.

x = one integer and y = other integer x + y = 26

x = 3y + 10 Substitute in for x:
(3y + 10) + y = 26 4y + 10 = 26
4y = 16 y = 4
x = 3(4) + 10 = 22

Algebraic Inequalities:

TRANSLATING WORDS:

Some key words to know:

AT LEAST means greater than or equal

AT MOST means less than or equal

I need at least $200 to go to the mall means I must have $200, but I'd like to have even more!

I want at most 15 minutes of homework means that I can have 15 minutes, 
but I'm hoping for even less! 
Because the answers in an inequality are infinite, the word problems usually are worded to either ask for the least or the greatest answers in the solution set.
 
For example if the problem asks for 2 consecutive odd integers that sum to at least 50, it will say: "give the smallest" in the solution set.

n + (n + 2) > 50
n > 24

The smallest odd integers in the set are 25 and 27

For example if the problem asks for 2 consecutive odd integers that sum to at most 50, it will say: "give the greatest" in the solution set.

n + (n + 2) < 50
n < 24

The greatest odd integer in the set is 23, so the integers are 23 and 25 (n + 2)