Worksheet: Applications of Systems of Equations

In this worksheet, we will practice translating a real-life problem into a system of equations and finding its solution(s).

Q1:

Two cousins were born 8 years apart. How old will the elder be when the sum of their ages is equal to 20 years?

  • A 6 years
  • B 12 years
  • C 8 years
  • D 14 years
  • E 16 years

Q2:

A man’s age is 4 times his son’s age. In five years, the sum of their ages will be 105. How old are they now?

  • A 20 years, 62 years
  • B 23 years, 60 years
  • C 25 years, 43 years
  • D 19 years, 76 years
  • E 21 years, 84 years

Q3:

I am thinking of two numbers. Use the clues to determine what the numbers are.

  • The product of the two numbers is 48.
  • The difference between the two numbers is 2.
  • A 9 and 7
  • B 6 and 4
  • C 5 and 3
  • D 8 and 6
  • E7 and 5

Q4:

In a test with 20 questions, π‘₯ marks are awarded for each correct answer and 𝑦 marks are deducted for each incorrect answer. Maged answered 12 questions correctly and 8 questions incorrectly, and he scored 44 points. Sarah answered 14 questions correctly and 6 questions incorrectly, and she scored 58 points. How many points were deducted for each incorrect answer?

  • A 6 points
  • B 5 points
  • C 3 points
  • D 2 points
  • E 4 points

Q5:

Fady bought 3 muffins and 2 cookies for $3.30, while Ramy bought 2 muffins and 5 cookies for $5.50. Work out the price of a single muffin and a single cookie.

  • Amuffin: $0.50, cookie: $0.75
  • Bmuffin: $0.90, cookie: $0.50
  • Cmuffin: $0.60, cookie: $0.75
  • Dmuffin: $0.50, cookie: $0.90
  • Emuffin: $0.75, cookie: $0.90

Q6:

A 30-foot-long ribbon was cut into three pieces. The first piece is one-third as long as the second piece, and the third piece is 4 feet longer than three times the length of the second piece. How long is the longest ribbon piece?

Q7:

A store clerk sold 60 pairs of sneakers. The high-tops sold for $98.99 and the low-tops sold for $129.99. If the receipts for the two types of sales totaled $ 6 4 0 4 . 4 0 , how many of each type of sneaker were sold?

  • Ahigh-tops: 30 pairs, low-tops: 30 pairs
  • Bhigh-tops: 15 pairs, low-tops: 45 pairs
  • Chigh-tops: 40 pairs, low-tops: 20 pairs
  • Dhigh-tops: 45 pairs, low-tops: 15 pairs

Q8:

A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $1 and the chocolate chip cookies at $0.75. They raised $700 and sold 850 items. How many brownies and how many cookies were sold?

  • Abrownies: 300, chocolate chip cookies: 550
  • Bbrownies: 350, chocolate chip cookies: 500
  • Cbrownies: 200, chocolate chip cookies: 650
  • Dbrownies: 250, chocolate chip cookies: 600

Q9:

The lines π‘Ž π‘₯ + 𝑏 𝑦 = 4 3 and 6 π‘Ž π‘₯ + 𝑏 𝑦 = 2 4 3 intersect at the point ( 5 , 1 ) . Find the values of π‘Ž and 𝑏 .

  • A π‘Ž = 3 , 𝑏 = 8
  • B π‘Ž = βˆ’ 8 , 𝑏 = βˆ’ 3
  • C π‘Ž = 4 0 , 𝑏 = 3
  • D π‘Ž = 8 , 𝑏 = 3
  • E π‘Ž = βˆ’ 4 0 , 𝑏 = βˆ’ 3

Q10:

I am thinking of two numbers. Use the clues to determine what the numbers are.

  • When you divide one by the other, the quotient is 8.
  • The sum of the two numbers is 81.
  • A 71 and 10
  • B 70 and 11
  • C 4 and 8
  • D 72 and 9
  • E 75 and 6

Q11:

A farmer finds there is a linear relationship between the number of bean stalks, 𝑛 , she plants and the yield, 𝑦 , each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationship in the form 𝑦 = π‘š 𝑛 + 𝑏 that gives the yield when 𝑛 stalks are planted.

  • A 𝑦 = βˆ’ 2 𝑛 + 9 0
  • B 𝑦 = βˆ’ 𝑛 2 βˆ’ 4 5
  • C 𝑦 = 𝑛 2 βˆ’ 3 0
  • D 𝑦 = βˆ’ 𝑛 2 + 4 5
  • E 𝑦 = βˆ’ 2 𝑛 + 3 0

Q12:

A clothing business finds there is a linear relationship between the number of shirts, 𝑛 , it sells and the price, 𝑝 , that it charges per shirt. In particular, historical data shows that, when the price was $30, they sold 1 0 0 0 shirts, whereas, when the price was $22, they sold 3 0 0 0 . Find a linear equation in the form 𝑝 ( 𝑛 ) = π‘š 𝑛 + 𝑏 that gives the price they can charge if they want to sell 𝑛 shirts.

  • A 𝑝 ( 𝑛 ) = βˆ’ 𝑛 2 5 0 βˆ’ 3 4
  • B 𝑝 ( 𝑛 ) = 𝑛 2 5 0 + 3 4
  • C 𝑝 ( 𝑛 ) = 2 5 0 𝑛 βˆ’ 6 5 0 0
  • D 𝑝 ( 𝑛 ) = βˆ’ 𝑛 2 5 0 + 3 4
  • E 𝑝 ( 𝑛 ) = 𝑛 2 5 0 βˆ’ 3 4

Q13:

A chemistry teacher needs to mix a 3 0 % salt solution with a 7 0 % salt solution to make 20 qt of a 4 0 % salt solution. How many quarts of each solution should the teacher mix to get the desired result?

  • A 12 qt of the 3 0 % solution, 8 qt of the 7 0 % solution
  • B 5 qt of the 3 0 % solution, 15 qt of the 7 0 % solution
  • C 8 qt of the 3 0 % solution, 12 qt of the 7 0 % solution
  • D 15 qt of the 3 0 % solution, 5 qt of the 7 0 % solution
  • E 10 qt of the 3 0 % solution, 10 qt of the 7 0 % solution

Q14:

Bassem has $ 2 0 0 0 0 to invest. His intent is to earn 1 1 % interest on his investment. He can invest part of his money at 8 % interest and part at 1 2 % interest. How much does Bassem need to invest in each option to get a total 1 1 % return on his $ 2 0 0 0 0 ?

  • A $ 1 8 0 0 0 at 8 % , $ 2 0 0 0 at 1 2 %
  • B $ 1 5 0 0 0 at 8 % , $ 5 0 0 0 at 1 2 %
  • C $ 2 0 0 0 at 8 % , $ 1 8 0 0 0 at 1 2 %
  • D $ 5 0 0 0 at 8 % , $ 1 5 0 0 0 at 1 2 %
  • E $ 4 1 0 0 0 0 at 8 % , $ 1 0 0 0 0 at 1 2 %

Q15:

Mariam starts riding her bike at 20 mph. After a while, she slows down to 12 mph and maintains that speed for the rest of the trip. The whole trip of 70 mi takes her 4.5 h. For what distance did she travel at 20 mph?

Q16:

Four times the weight of Fady is 150 pounds more than the weight of Yara. Four times the weight of Yara is 660 pounds less than seventeen times the weight of Fady. Four times the weight of Fady plus the weight of Maged equals 290 pounds. Amir would balance all three of the others. Find the weights of the four people.

  • AFady’s weight = 60 lb, Yara’s weight = 90 lb, Amir’s weight = 50 lb, Maged’s weight = 200 lb
  • BFady’s weight = 90 lb, Yara’s weight = 60 lb, Amir’s weight = 200 lb, Maged’s weight = 50 lb
  • CFady’s weight = 90 lb, Yara’s weight = 60 lb, Amir’s weight = 50 lb, Maged’s weight = 200 lb
  • DFady’s weight = 60 lb, Yara’s weight = 90 lb, Amir’s weight = 200 lb, Maged’s weight = 50 lb
  • EFady’s weight = 50 lb, Yara’s weight = 60 lb, Amir’s weight = 90 lb, Maged’s weight = 200 lb

Q17:

A rectangle’s length is 16 cm less than four times its width. Given that its perimeter is equal to that of a square of side 12 cm, find the dimensions of the rectangle.

  • A 3 cm, 22 cm
  • B 4 cm, 8 cm
  • C 3 cm, 13 cm
  • D 8 cm, 16 cm

Q18:

Adel buys 5 apples and 3 bananas from a supermarket and pays $3.40. Mariam buys 3 apples and 2 bananas from the same supermarket and pays $2.10. Work out the price of a single apple and a single banana.

  • A An apple is 25Β’ and a banana is 15Β’.
  • B An apple is 26Β’ and a banana is 14Β’.
  • C An apple is 47Β’ and a banana is 6Β’.
  • DAn apple is 50Β’ and a banana is 30Β’.
  • E An apple is 45Β’ and a banana is 35Β’.

Q19:

Nabil buys 4 burgers and 3 hot dogs for $16 from a fast-food vendor. Sameh buys 3 burgers and 4 hot dogs for $15.50 from the same vendor. Work out the price of a single burger and a single hot dog.

  • A A burger is $4.42 and a hot dog is $0.56.
  • B A burger is $2 and a hot dog is $2.50.
  • C A burger is $0.56 and a hot dog is $4.42.
  • DA burger is $2.50 and a hot dog is $2.
  • E A burger is $3 and a hot dog is $2.50.

Q20:

A bank offers two savings account options. In the first one, a deposit of $ 4 0 is received every year for an initial capital of $ 1 0 0 0 . In the second option, an annual interest rate of 2 . 8 % is offered and it’s compounded monthly. Which option is more profitable after 30 years?

  • Aoption 2
  • Boption 1

Q21:

A library decides to put in place a new fine system as a stronger incentive for borrowers to return items on time. Instead of issuing a fine of 2 0 Β’ per day and per item, the fine is 2 0 Β’ for one day after the return date, 4 0 Β’ for two days, 8 0 Β’ for three days, and so on. How much more will a borrower pay for an item returned 7 days after the return date?

  • A $ 2 5 . 4 0
  • B $ 1 2 . 8 0
  • C $ 2 4 . 0 0
  • D $ 1 1 . 4 0
  • E $ 0

Q22:

A concert venue sells single tickets for $40 each and couple tickets for $65. If the total revenue was $ 1 8 0 9 0 and 321 tickets were sold, how many of each type of ticket did they sell?

  • A100 single tickets, 221 couple tickets
  • B210 single tickets, 111 couple tickets
  • C121 single tickets, 200 couple tickets
  • D111 single tickets, 210 couple tickets

Q23:

Two planes flying in opposite directions pass each other. One is flying at 450 mph and the other at 550 mph. How long will it take before they are 4000 miles apart?

Q24:

Amira invested 1.1 million dollars in two land investments. Her return on the first investment, Swan Peak, was a 1 1 0 % increase on the money she had invested. On her second investment, Riverside Community, she earned 5 0 % over what she had invested. Given that she earned $1 million in profit, how much did she invest in each of the land deals?

  • ASwan Peak: $750 000, Riverside: $250 000
  • BSwan Peak: $350 000, Riverside: $750 000
  • CSwan Peak: $650 000, Riverside: $350 000
  • DSwan Peak: $750 000, Riverside: $350 000

Q25:

A concert manager counted 350 ticket receipts the day after a concert. The price for a student ticket was $12.50, and the price for an adult ticket was $16.00. The register confirms that $ 5 0 7 5 were taken in. How many student tickets and adult tickets were sold?

  • Astudent tickets: 250, adult tickets: 100
  • Bstudent tickets: 200, adult tickets: 150
  • Cstudent tickets: 300, adult tickets: 50
  • Dstudent tickets: 150, adult tickets: 200

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