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Worksheet: Applications of Systems of Equations

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:

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

Q3:

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

Q4:

Jacob 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 Jacob 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 %

Q5:

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?

Q6:

Sophia 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

Q7:

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

Q8:

A Jeep and a BMW entered a highway at the same interchange, heading in opposite directions. The Jeep entered the highway 30 minutes before the BMW and traveled 7 mph slower than the BMW. Two hours after the BMW entered the highway, the two cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.

  • Athe Jeep: 72 mph, the BMW: 80 mph
  • Bthe Jeep: 72 mph, the BMW: 65 mph
  • Cthe Jeep: 57 mph, the BMW: 61 mph
  • Dthe Jeep: 65 mph, the BMW: 72 mph

Q9:

The difference between the measures of the acute angles in a right-angled triangle is . Find the measure of each acute angle.

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q10:

If ( 𝑥 + 9 , 7 ) = ( 4 , 1 0 𝑦 + 𝑥 ) , find the values of 𝑥 and 𝑦 .

  • A 𝑥 = 1 3 , 𝑦 = 2 0
  • B 𝑥 = 5 , 𝑦 = 7
  • C 𝑥 = 5 , 𝑦 = 2
  • D 𝑥 = 1 3 , 𝑦 = 2

Q11:

You invested $ 8 0 0 0 0 in two accounts: $ 2 2 0 0 0 in account A and $ 5 8 0 0 0 in account B. Both accounts earn simple interest and, after one year, you have earned $ 2 4 7 0 in interest. Given that account B earns half a percent less than twice the interest account A earns, what are the interest rates for your accounts?

  • Aaccount A: 1 . 5 % , account B: 2 . 5 %
  • Baccount A: 3 . 5 % , account B: 2 %
  • Caccount A: 2 . 9 % , account B: 1 . 7 %
  • Daccount A: 2 % , account B: 3 . 5 %

Q12:

Madison 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?

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:

At the beginning of a year, you invested $10 000 into two accounts, A and B, which receive 8 % simple interest and 5 % simple interest respectively. At the end of that year, you had $10 710 in your combined accounts. How much was invested in each account?

  • Aaccount A: $5 000, account B: $5 000
  • Baccount A: $3 000, account B: $7 000
  • Caccount A: $8 000, account B: $2 000
  • Daccount A: $7 000, account B: $3 000

Q15:

Noah buys 5 apples and 3 bananas from a grocery store and pays $3.40. Madison buys 3 apples and 2 bananas from the same store and pays $2.10. Work out the price of a single apple and a single banana.

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

Q16:

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

Q17:

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

Q18:

Four times the weight of Michael is 150 pounds more than the weight of Victoria. Four times the weight of Victoria is 660 pounds less than seventeen times the weight of Michael. Four times the weight of Michael plus the weight of Benjamin equals 290 pounds. Liam would balance all three of the others. Find the weights of the four people.

  • AMichael’s weight = 60 lb, Victoria’s weight = 90 lb, Liam’s weight = 50 lb, Benjamin’s weight = 200 lb
  • BMichael’s weight = 90 lb, Victoria’s weight = 60 lb, Liam’s weight = 200 lb, Benjamin’s weight = 50 lb
  • CMichael’s weight = 90 lb, Victoria’s weight = 60 lb, Liam’s weight = 50 lb, Benjamin’s weight = 200 lb
  • DMichael’s weight = 60 lb, Victoria’s weight = 90 lb, Liam’s weight = 200 lb, Benjamin’s weight = 50 lb
  • EMichael’s weight = 50 lb, Victoria’s weight = 60 lb, Liam’s weight = 90 lb, Benjamin’s weight = 200 lb

Q19:

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

Q20:

In a test with 20 questions, 𝑥 marks are awarded for each correct answer and 𝑦 marks are deducted for each incorrect answer. Benjamin answered 12 questions correctly and 8 questions incorrectly, and he scored 44 points. Emma 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

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 ¢ per day and per item, the fine is 2 0 ¢ for one day after the return date, 4 0 ¢ for two days, 8 0 ¢ 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 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

Q23:

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 = 𝑚 𝑛 + 𝑏 that gives the yield when 𝑛 stalks are planted.

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

Q24:

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 ) = 𝑚 𝑛 + 𝑏 that gives the price they can charge if they want to sell 𝑛 shirts.

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

Q25:

William buys 4 burgers and 3 hot dogs for $16 from a fast-food vendor. Matthew 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.