Answer/Discussion on Further Problem Solving

(Back to the tutorial on further problem solving)

Beginning Algebra
Answer/Discussion to Practice Problems
on Further Problem Solving

Answer/Discussion to 1a

A rectangular garden has a width that is 8 feet less than twice the length. Find the dimensions if the perimeter is 20 feet.

Step 1: Understand the problem.

Make sure that you read the question carefully several times.

We are looking for the length and width of the rectangle. Since width can be written in terms of length, we will let

L = length

Width is 8 feet less than twice the length:

2L - 8 = width

Step 2: Devise a plan (translate).

Step 3: Carry out the plan (solve).

*Mult. ( ) by 2
*Combine like terms

*Inv. of sub. 16 is add 16

*Inv. of mult. by 6 is div. by 6

Step 4: Look back (check and interpret).

If length is 6, then width, which is 8 feet less than twice the length, would have to be 4. The perimeter of a rectangle with width of 4 feet and length of 6 feet is 20 feet.

FINAL ANSWER:

Width is 4 feet.

Length is 6 feet.

(return to problem 1a)

Answer/Discussion to 1b
In Nebraska on I-80, the speed limit is 75 mph. How long would it take you to travel 525 miles in Nebraska on I-80 if you went the speed limit the whole time?

Step 1: Understand the problem.

Make sure that you read the question carefully several times.

Since we are looking for how long it would take you to travel, we can use the distance/rate formula:

d = rt

The variables in this formula represent the following:

d = distance
r = rate
t = time

Step 2: Devise a plan (translate).

Plugging the values into the formula we get:

Step 3: Carry out the plan (solve).

*Inverse of mult. by 75 is div. by 75

Step 4: Look back (check and interpret).

If you go at a rate of 75 miles per hour for 7 hours, you would travel 525 miles.

FINAL ANSWER:

It would take 7 hours.

(return to problem 1b)

Answer/Discussion to 1c
How much 25% antifreeze and 50% antifreeze should be combined to give 40 liters of 30% antifreeze?

Step 1: Understand the problem.

Make sure that you read the question carefully several times.

We are looking for the amount of 25% antifreeze and 50% antifreeze needed to get 40 liters of 30% antifreeze.

x = number of liters of the 25%.

Since the two mixtures together need to be 40 liters, then we can take the total (40) and subtract from it the "given" number of liters (x):

40 - x = number of liters of the 50%.

Step 2: Devise a plan (translate).

Step 3: Carry out the plan (solve).

*Remove ( ) by using dist. prop.

*Combine like terms

*Inv. of add. 2000 is sub. 2000

*Inv. of mult. by -25 is div. by -25

Step 4: Look back (check and interpret).

If there are 32 liters of the 25%, then there would have to be 40 - 32 = 8 liters of the 50% solution.

If you have 32 liters of 20% solution and 8 liters of 50% solution you do get 40 liters of 30% alcohol solution.

FINAL ANSWER:

32 liters of the 25% antifreeze.
8 liters of the 50% antifreeze.

(return to problem 1c)

Answer/Discussion to 1d
You recently came into $20,000 (lucky you!) and you want to place part of your money in a savings account paying 7% per year and part in a certificate of deposit paying 9% per year. If you wish to obtain an overall return of $1700 per year, how much would you place in each investment?

Step 1: Understand the problem.

Make sure that you read the question carefully several times.

We are looking for how much she invested in EACH account.

x = amount invested in 7%

Since the two accounts together need to be $20000, then we can take the total (20000) and subtract from it the "given" number in the 7% account (x):

20,000 - x = amount invested in 9%

Note that you could have reverse those, the problem would still work out the same.

Step 2: Devise a plan (translate).