1.16: Using Algebraic Models
Bailey is planting an outdoor garden. She wants the length of the planting area to be twice as large as the width. If she has 128 square feet of soil to work with, what dimensions must the garden be?
Guidance
Word problems are some of the hardest types of problems for students to grasp. There are a few steps to solving any word problem:
- Read the problem at least twice.
- Cross out any unnecessary words, circle any numbers or words that represent mathematical operators, or translate words into mathematical expressions.
- Write an equation and solve.
To help you with steps 2 and 3, generate a list of words that represent: add, subtract, multiply, divide, equal, etc. Here are a few to get you started.
add | sum plus and | increase more (than) | multiply | times of product | double (x2) triple (x3) |
---|---|---|---|---|---|
subtract | difference minus | decrease less (than) | equal | is total to | made/make spend/spent |
divide | quotient half (÷2) | third (÷3) | variable |
how many __ how much __ what amount (of) __ |
See if you can add anything to these lists. Then, use this chart to help you with decoding word problems.
Example A
Two consecutive numbers add up to 55. What are the two numbers?
Solution: First, translate the statement. “Consecutive” means numbers that are one after the other. So if the first number is \begin{align*}x\end{align*}, then the second number will be \begin{align*}x + 1\end{align*}. And, they add up to 55. The equation is: \begin{align*}x+(x+1)=55\end{align*}
We put \begin{align*}x + 1\end{align*} in parenthesis to show that it is a separate number. Solve the equation.
\begin{align*}x+x+1 &=55\\ 2x+1 &= 55\\ 2x &= 54\\ x &= 27\end{align*}
The smaller number is 27, and the larger number will be 28. \begin{align*}27 + 28 = 55 \end{align*}
Sometime you may encounter problems with “consecutive even numbers” or “consecutive odd numbers.” All even numbers are divisible by 2, so the smallest should be \begin{align*}2x\end{align*}, then the next even number would be \begin{align*}2x + 2\end{align*}. For consecutive odd numbers, they will always be an even number plus 1, 3, 5, etc. So, the smaller odd number would be \begin{align*}2x + 1\end{align*} and the larger odd number would be \begin{align*}2x + 3\end{align*}.
Example B
Over the Winter Break, you worked at a clothing store and made $9.00 an hour. For the two weeks you worked 65 hours of regular pay and 10 hours of overtime (time and a half). How much money did you make?
Solution: First, we need to figure out how much you make for overtime. Time and a half would be \begin{align*}\$ 9.00 + \$ 4.50 = \$ 13.50\end{align*} an hour. So, you made:
\begin{align*}\$ 9.00(65)+ \$ 13.50(10) = \$ 585.00+ \$ 135.00 = \$ 720.00\end{align*}
Example C
Elise is taking piano lessons. The first lesson is twice as expensive as each additional lesson. Her mom spends $270 for 8 lessons. How much was the first lesson?
Solution:Translate each statement.
Call the regularly priced lessons \begin{align*}l\end{align*}. Then, the first lesson will be \begin{align*}2l\end{align*}.
"Mom spends $270 for 8 lessons" \begin{align*} \rightarrow 2l + 7l = \$ 270\end{align*}
Solve:
\begin{align*}2l+7l &= 270\\ 9l &= 270\\ l &= 30\end{align*}
The regularly priced lessons are $30. The first lesson will be $60.
Intro Problem Revisit Bailey wants the length to be twice as long as the width. If the width is w, then the length will be 2w. The area of the soil is 128 square feet and the formula for area is \begin{align*}A=l \cdot w\end{align*}.
\begin{align*} 128&=2w \cdot w \\ 128 &= 2w^2 \\ 64 &= w^2 \\ 8 &= w \end{align*}
The answer could also be \begin{align*}-8\end{align*}, but because we are talking about length, w cannot be negative. Therefore, the width is 8 feet and the length is 2w or 16 feet.
Guided Practice
1. Bob is twice as old as his daughter. In that same year, his granddaughter is one-tenth his daughter's age. His granddaugther is 3. How old is Bob?
2. Javier needs to get a tank of gas. Gas costs $3.79 per gallon. How much money does Javier need to fill up his 16 gallon tank?
Answers
1. Rewrite each statement as a mathematical one.
\begin{align*}b &= 2d\\ 10g&=d\\\end{align*}
Now, plug in the granddaughter's age and work through each equation.
\begin{align*}10\cdot 3&=30\\ b&=2\cdot 30\end{align*}
Bob is 60 years old.
2. This problem wants to know how much money Javier needs to fill up his gas tank. Gas costs $3.79 per gallon and he needs 16 gallons of gas. It will cost \begin{align*}\$ 3.79 \cdot 16= \$ 60.64\end{align*} to fill up his tank.
Practice
Answer each question to the best of your ability.
- The average speed on highway 101 is 65 miles per hour (mph). Assuming you drive the speed limit, how long will it take you to drive 350 miles? Use the formula \begin{align*}distance = rate \cdot time\end{align*}. Round your answer to two decimal places.
- Using the information in #1, how many miles did you drive on highway 101 if you drove for 2.5 hours?
- The sum of two consecutive numbers is 79. Find the two numbers.
- The sum of two consecutive odd numbers is 44. Find the two odd numbers.
- You borrowed $350 from your parents for a new Wii and games. They are not going to charge you interest, but you need to pay them back as quickly as possible. If you pay them $15 per week, how long will it take you to pay them back?
- George is building a rectangular, fenced-in dog run. He has 120 feet of fencing and wants the length to be 20 feet greater than the width. If you use all the fencing, find the length and width of the dog run.
- Cynthia is selling chocolate bars for a fundraiser for school. Each bar costs $1.50. If she needs to raise $225, how many chocolate bars does she need to sell?
- Harriet bakes and sells cookies to local stores. Her cost for one dozen cookies is $2.75 and she sells them to stores for $7.00 (per dozen). How many dozen cookies does she need to make to earn $500? Round to the nearest dozen.
- A football field is a rectangle where the length is 100 yards. If the total perimeter is 1040 feet, what is the width of a football field? Leave your answer in feet.
- Challenge The sum of three consecutive even numbers is 138. What are the three numbers?
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Image Attributions
Here you'll learn how to write and solve an algebraic equation from a word problem.