Word Problems in Two Variables



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Algebra II Name _____________­­­­­­­­__________

Word Problems in Two Variables

For the following word problems there needs to be…

1. two variables involved

2. two equations involved

*** Each problem will have two variables that need to be defined, and two equations that need to be created! ***
You can solve by any of the following four methods:
1. 2.

3. 4.
Part 1 – Quantity/Worth Problems in Two Variables

After you define your variables:

1. Make one equation dealing strictly with the quantity or amount of things (# of

coins/questions/objects/pounds/etc).

2. Then, make a second equation that assigns the value or worth to the “things”

(25 cents/ 5 lbs/ 5pts/ etc)

OR


1. If you are being asked to find the value or worth, create two equations by translating

sentences into math equations. (most likely will have quantity times a variable equaling a total value.)


EX 1. A history test has 20 questions. Mr. Johnson used multiple choice and short essay questions. The multiple choices are worth 4 points each. The essay questions are worth 8 points each. The test has a total of 100 points. How many of each type of problem are on the test?


EX 2. Meg has $3.75 in dimes and quarters. The number of quarters is 2 less than 3 times the number of dimes. Find the number of coins of each type.


EX 3. An M&M storage warehouse has a total of 30 bins. Some bins hold 20 tons of m&ms each and some hold 15 tons of m&ms each. How many of each type of bin are there if the capacity of the warehouse is 510 tons of m&ms.

EX 4. The Funsten’s are planning a 3 month vacation across Europe this summer. Mrs. Funsten goes to the store and buys 2 memory cards and 1 battery charger for $49.75. Without realizing his wife went to the store yesterday, Mr. Funsten goes to the store and buys 3 memory cards and 2 battery chargers and his bill was $86.00. How much is a memory card? A battery charger?


TRY:

1. At a senior citizen center in Ambler, 35 tables were set up for games of checkers (2 per table) and bridge (4 per table). There were 92 players occupying the 35 tables. Find the number of checker players and the number of bridge players.


2. The receipts for the Upper Dublin Homecoming football game were $154,500, paid by 30,000 spectators! Bleacher seats sold for $4.75 and reserved seats sold for $5.95 each. How many seats of each type were sold?

3. Mrs. Decktor has $21.40 in dimes and quarters, for a total of 100 coins in her purse! How many of each kind of coin does Mrs. Decktor have?

4. The bill for five glasses of iced tea and four salads is $39.50, but the bill for four cups of iced tea and five salads is $43.75. What would the bill be for a glass of iced tea and one salad?

5. Regal Warrington charges $10.50 for an adult ticket and $9.00 a student ticket. On the Saturday that the latest Twilight movie came out, the Regal sold 1385 total tickets making $12,874.50 in sales. How many student tickets were sold for the new Twilight movie that Saturday?


Part 2 – Numbers and Perimeter/Area Word Problems in Two Variables

Remember!

Area of a rectangle:

Area of a square:

Perimeter of a rectangle:

Sum/together/added/plus/more than/increased by/wider/longer/etc means 

Difference/less than/subtract/decreased by/shorter/more narrow/etc means 

Times/product/multiplied by/twice a number/etc means 

Divided by/quotient/half a number/etc means 


Solve the following using a system of equations.
EX 1. The sum of two numbers is 25 and their difference is 7. Find the numbers.

EX 2. One number is 4 less than eleven times another. The sum of two numbers is 92. Find the numbers.


EX 3. The difference between the length and width of a rectangle is 7cm. The perimeter of the rectangle is 50 cm. Find the length and width.

EX 4. The length of a rectangle is 5 more than twice the width. The perimeter is 130 inches. What is the area?

TRY:

1. The larger of two numbers is 3 more than twice the smaller. If twice the larger is decreased by 5 times the smaller, the result is 0. Find the two numbers.


2. Twice one number is 15 less than a second number. When 13 is added to the second number, the result is 7 less than 9 times the first number. Find the two numbers.

3. The perimeter of a rectangle is 22m. The length of the rectangle is 1 m less than 3 times the width. Find the length and width.


4. Ms. Hurlock has $24 more than twice as much as Mrs. Platcheck has left to spend on school supplies this year. Together they have $150. How much money does each have?


5. A rectangle is 5 times long as it is wide. If it were 24 cm shorter and 24 cm wider, it would be a square. What are its dimensions?


6. The sum of two numbers is 22. Five times one number is equal to six times the second number. Find the two numbers.


7. The difference between the length and width of a rectangle is 13 inches. The perimeter of the rectangle is 182 inches. Find the length and width.



Part 3 – Practice with Angle, Age and Mixture Problems
Supplementary Angles – Complementary Angles –
Hints: Mixture problems are VERY similar to quantity/worth problems

Age problems are VERY similar to number problems


EX 1. Two angles are complementary. If one angle is 14 more than the other angle, find the measurement of both angles.

EX 2. Two angles are supplementary. If one angle is 3 more than twice the other, find the measurement of both angles.


EX 3 Wheat costing $0.80 per kg is mixed with rye costing $0.60 per kg. The number of kg of rye is 30 less than the number of kg of wheat. The mixture is worth $52.00. How many kg of each grain should be used?


EX 4. A laboratory technician is mixing 10% saline solution with a 4% saline solution. How much of each solution is needed to make 500mL of a 6% saline solution?

EX 5. Patty is twice as old as Phoebe. Five years from now the sum of their ages will be 64. How old is each now?


TRY:

1. One angle is three times another. If the angles are complementary, find their measurements.


2. One angle is 6 less than another. If the two angles are supplementary, find the measurement of the smaller angle.

3. Two angles are complementary. One angle is 12 less than 3 times the other. Find the measurements.

4. Maureen is 22 years younger than Bob. Eight years from now Bob will be 3 times as old as Maureen. How old is each now?

5. Some $0.25 Ticonderoga pencils are mixed with $0.40 fancy pencils to make a Back-To-School present. There are 3 times as many Ticonderoga pencils than fancy ones. The present is worth $5.75. How many of each type of pencil will be in the package?

6. Mrs. Bisio wants to take her chemistry class to the science lab to do an experiment. The students need to create 90mL of a 3% acid solution. The have a 1% acid solution and a 10% acid solution available. How much of each should they put in to create the 3% acid solution?


Part 4 – Motion Word Problems in Two Variables
Very important formula!! (r)(t) = d aka (rate)(time)=distance

Most times, when we do rate, we will be taking into account a current (wind/water/etc).

Moving WITH the current, we seemingly go faster, so we use _________

Moving AGAINST the current, we seemingly go slower, so we use _________

Be sure to answer the question that is being asked!!
In ROUNDTRIP problems, the distances traveled: to and from, up(stream/wind/hill/etc) and down(stream/wind/hill/etc), left and right, there and back, are ALWAYS EQUAL!
EX 1. In a crew race, a team rows downstream 480 m in 60 s. The same team makes the trip upstream in 80 s. Find the team’s rate in still water and the rate of the current.

EX 2. It takes an airplane 1 h 30 min to fly 600 km against the wind. The return trip with the wind takes only 1 h. Find the total flying time for the round trip if there was no wind.


EX 3. A motorboat travels 18km/hr downstream and 12 km/hr upstream. If the roundtrip took 8 hours, how far out did it travel?


TRY:

1. A sailboat travels 12 mi downstream in only 2 h. The return trip upstream takes 3 h. Find the speed of the sailboat in still water and the rate of the current.


2. The 4200 km trip from New York City to San Francisco takes 6 h flying against the wind but only 5 h returning. Find the speed of the plane in still air and the speed of the wind.

3. The 1080 km trip from Madrid to Paris takes 2 h flying against the wind and 1.5 h flying with the wind. Find the speed of the plane in still air and the speed of the wind.

4. A jet ski has a 4-hour supply of gas. How far from the marina can it travel if the rate going out against the current is 20 mph and the rate coming back to 30mph?


5. It takes 6 h for a plane to travel 720 km with a tail wind and 8 h to make the return trip with a head wind. Find the air speed of the plane and the speed of the current.




6. Amy walks to the repair shop at 2 mph, picks up her bicycle, and rides home at 10 mph. If the round trip took hours, how far is the shop from Amy’s house?

7. With a tail wind, a helicopter traveled 300 mi in 1 h 40 min. The return trip against the same wind took 20 min longer. Find the wind speed and also the air speed of the helicopter.

8. An airplane left Philadelphia at 1 PM and flew directly east at 340 mph. At 2:30, a jet plane traveling at 595 mph started east from the same airport. In how many hours did the jet plane overtake the first plane?
Part 5 – Digit and Fraction Word Problems in Two Variables

In the 3-digit number 123:

1 is in the ____________________ place

2 is in the ____________________ place

3 is in the ____________________ place

Note: The number 45 can be written: 10(4) + 1(5) = 45.



Therefore, if we want to represent any two-digit number, let x represent the tens digit and y represent the ones/units digit. The number can now be written as the expression 10x + y
EX 1: The sum of the digits of a two-digit number is 8. If the digits are reversed, the number is increased by 18. What is the original number?
EX 2: A number is 6 times the sum of its digits. The tens digit is 1 greater than the units digit. Find the number.
EX 3: The denominator of a fraction is 12 more than the numerator. If 16 is added to the numerator and 16 is subtracted from the denominator, the value of the resulting fraction is equal to 2. Find the original fraction.
EX 4: The denominator of a fraction is 7 more than the numerator. If 5 is added to both the numerator and denominator, the value of the resulting fraction is . What is the original fraction?
TRY:

  1. The denominator of a fraction is 1 more than the numerator. If the numerator is decreased by 1, the value of the resulting fraction is .




  1. The sum of the digits in a two-digit number is 7. The new number obtained when the digits are reversed is 27 less than the original number. Find the original number.



  1. A two-digit number is seven times the sum of its digits. The tens digit is 3 more than the units digit. What is the number?


  1. If 1 is added to both the numerator and the denominator of a certain fraction, the value of the fraction becomes
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