Worksheet: Applications on Systems of Linear Equations in Three Variables

In this worksheet, we will practice solving real-world problems using a system of linear equations in three variables.

Q1:

An animal shelter has a total of 350 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than half the number of cats, and there are 20 more cats than dogs, how many of each are at the shelter?

  • A70 rabbits, 150 cats, 130 dogs
  • B78 rabbits, 146 cats, 126 dogs
  • C180 rabbits, 95 cats, 75 dogs
  • D120 rabbits, 125 cats, 105 dogs

Q2:

Three numbers sum up to 147. The smallest number is half the middle number, which is half the largest number. What are the three numbers?

  • A29.4,58.8,117.6
  • B21,42,84
  • C147,294,588
  • D11,45,90

Q3:

Three even numbers add up to 108. The smaller is half the larger and the middle number is 34 of the larger. What are the three numbers?

  • A30, 38, 40
  • B24, 36, 48
  • C20, 40, 48
  • D25, 33, 50
  • E12, 18, 24

Q4:

If the measure of one angle of a triangle is half the measure of the second angle and less than the measure of the third one by 92, then what are the measures of the three angles?

  • A22,34,124
  • B32,24,124
  • C27,54,99
  • D44,22,114

Q5:

400 pets were entered into a pet shelter for dogs, cats, and rabbits. There were twice as many cats as rabbits, and 50 more dogs than cats. How many of each type of pet were there?

  • A138 dogs, 175 cats, 87 rabbits
  • B138 dogs, 87 cats, 175 rabbits
  • C190 dogs, 70 cats, 140 rabbits
  • D190 dogs, 140 cats, 70 rabbits

Q6:

Your roommate, Amelia, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to keep the receipts but remembered that your groceries were $0.05 cheaper than half of her groceries and that your other roommate’s groceries were $2.10 more than your groceries. How much was each person's share of the groceries?

  • Amy share: $20, Amelia’s share: $39.90, other roommate’s share: $22.10
  • Bmy share: $21, Amelia’s share: $42.10, other roommate’s share: $18.90
  • Cmy share: $33.63, Amelia’s share: $16.84, other roommate’s share: $31.53
  • Dmy share: $19.95, Amelia’s share: $40, other roommate’s share: $22.05

Q7:

In a bag, a child has 325 coins worth $19.50. There are three types of coins: pennies, nickels, and dimes. If the bag contains the same number of nickels as dimes, how many of each type of coin are in the bag?

  • A75 pennies, 125 nickles, 125 dimes
  • B315 pennies, 5 nickles, 5 dimes
  • C25 pennies, 150 nickles, 150 dimes
  • D105 pennies, 110 nickles, 110 dimes

Q8:

William has lemon, orange, and pomegranate trees in his backyard. An orange weighs 8 oz, a lemon 5 oz, and a pomegranate 11 oz. William picked 142 pieces of fruit weighing a total of 70 lb10 oz. Given that he picked 15.5 times more oranges than pomegranates, how many of each of the three fruits did William pick?

  • A124 oranges, 10 lemons, 8 pomegranates
  • B93 oranges, 43 lemons, 6 pomegranates
  • C100 oranges, 20 lemons, 22 pomegranates
  • D110 oranges, 25 lemons, 7 pomegranates

Q9:

Sophia, Hannah, and Natalie weigh 370 lb in total. If Natalie weighs 20 lb more than Hannah, and Sophia weighs 1.5 times as much as Hannah, how much does each girl weigh?

  • ASophia’s weight is 150 lb, Hannah’s weight is 120 lb, Natalie’s weight is 100 lb
  • BSophia’s weight is 150 lb, Hannah’s weight is 100 lb, Natalie’s weight is 120 lb
  • CSophia’s weight is 120 lb, Hannah’s weight is 80 lb, Natalie’s weight is 170 lb
  • DSophia’s weight is 100 lb, Hannah’s weight is 150 lb, Natalie’s weight is 120 lb

Q10:

A clothing store has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at $13.99, the beanie at $7.99, and the cowboy hat at $14.49. If 100 hats were sold this past quarter, $1,119 was taken in by sales, and the amount of beanies sold was 10 more than cowboy hats, how many of each did the clothing store sell?

  • A20 straw hats, 45 beanies, 35 cowboy hats
  • B30 straw hats, 40 beanies, 30 cowboy hats
  • C10 straw hats, 50 beanies, 40 cowboy hats
  • D5 straw hats, 55 beanies, 40 cowboy hats

Q11:

Using the following facts, is it possible to calculate the annual salaries of the three workers? Why?

  • The sum of the annual salaries of the warehouse manager and the office manager is $82,000.
  • The office manager makes $4,000 more than the truck driver annually.
  • The annual salaries of the warehouse manager and the truck driver total $78,000.
  • AYes, because the system is independent.
  • BNo, because there are infinitely many solutions.
  • CThere are no solutions.

Q12:

Three roommates shared a package of 12 ice cream bars. If Michael ate twice as many ice cream bars as David, and Liam ate three less than Michael, how many ice cream bars did each roommate eat?

  • AMichael: 5, David: 5, and Liam: 2
  • BMichael: 4, David: 2, and Liam: 6
  • CMichael: 6, David: 3, and Liam: 3
  • DMichael: 4, David: 5, and Liam: 3

Q13:

A bag of mixed nuts contains cashews, pistachios, and almonds. There are 1,000 nuts in the bag in total, and there are 100 fewer almonds than pistachios. Each cashew weighs 3 grams, each pistachio weigh 4 grams, and each almond weighs 5 grams. If the bag weighs 3.7 kg, how many of each nut are in the bag?

  • A300 almonds, 500 cashews, 200 pistachios
  • B200 almonds, 500 cashews, 300 pistachios
  • C300 almonds, 300 cashews, 400 pistachios
  • D400 almonds, 100 cashews, 500 pistachios

Q14:

A movie theater sold tickets for three different movies at three different prices: $5 for the first movie, $11 for the second movie, and $12 for the third movie. 100 tickets were sold for the first movie. The total number of tickets sold was 642, for a total revenue of $6,774. How many tickets were sold for each movie?

  • A100 for the first movie, 342 for the second movie, and 200 for the third movie
  • B100 for the first movie, 200 for the second movie, and 342 for the third movie
  • C100 for the first movie, 230 for the second movie, and 312 for the third movie
  • D100 for the first movie, 312 for the second movie, and 230 for the third movie

Q15:

Last year, at a women’s prison, the total number of inmates aged 2049 totaled 5,525. This year, the 2029 age group increased by 10%, the 30–39 age group decreased by 20%, and the 4049 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 3039 age group than in the 2029 age group. Determine the prison population for each age group last year.

  • AThe 2029 age group: 2,100, the 3039 age group: 825, the 4049 age group: 2,600
  • BThe 2029 age group: 2,600, the 3039 age group: 2,100, the 4049 age group: 825
  • CThe 2029 age group: 2,100, the 3039 age group: 2,600, the 4049 age group: 825
  • DThe 2029 age group: 825, the 3039 age group: 2,100, the 4049 age group: 2,600

Q16:

A farmer constructed a chicken coop out of chicken wire, wood, and plywood. The chicken wire cost $2 per square foot, the wood $10 per square foot, and the plywood $5 per square foot. Overall, the farmer spent $51 on 14 square feet of materials. Given that he used 3 square feet more chicken wire than plywood, how much of each material did the farmer use?

  • Achicken wire: 5 ft2, plywood: 2 ft2, wood: 7 ft2
  • Bchicken wire: 6 ft2, plywood: 3 ft2, wood: 5 ft2
  • Cchicken wire: 7 ft2, plywood: 4 ft2, wood: 3 ft2
  • Dchicken wire: 8 ft2, plywood: 5 ft2, wood: 1 ft2

Q17:

One year ago, you inherited one million dollars, and the following bank accounts were available:

  • Account A pays 3% compounded annually.
  • Account B pays 4% compounded annually.
  • Account C pays 2% compounded annually.

You invested all of the money into these three accounts and put four times as much into account A than account C. Given that you earned $34,000 in interest in a year, how much did you invest in each account?

  • Aaccount A: $400,000, account B: $500,000, account C: $100,000
  • Baccount A: $66,666.7, account B: $266,667, account C: $66,667
  • Caccount A: $66,666.7, account B: $66,667, account C: $266,667
  • Daccount A: $500,000, account B: $400,000, account C: $100,000

Q18:

A local band sold all tickets for their concert. Ticket prices are shown in the table and ticket sales totaled $28,112.50. If the band sold twice as many adult tickets as child tickets, how many of each type of ticket were sold?

Ticket Type Child Student Adult
Price$22.50$20$29
  • Aadult: 329, child: 659, student: 187
  • Badult: 659, child: 329, student: 187
  • Cadult: 450, child: 500, student: 225
  • Dadult: 450, child: 225, student: 500

Q19:

One year ago, you inherited one hundred thousand dollars, and the following bank accounts were available.

  • Account A, which pays 4% compounded annually.
  • Account B, which pays 3% compounded annually.
  • Account C, which pays 2% compounded annually.

You invested all of the money in these three accounts and put five times as much in account A than in account B. Given that you earned $3 650 in interest in a year, how much did you invest in each account?

  • Aaccount A: $75 000, account B: $15 000, account C: $10 000
  • Baccount A: $80 000, account B: $15 000, account C: $5 000
  • Caccount A: $70 000, account B: $17 500, account C: $12 500
  • Daccount A: $75 000, account B: $10 000, account C: $15 000

Q20:

Last year, at Haven’s Pond Car Dealership, a BMW, a Jeep, and a Toyota could be bought for a total of $140,000. This year, due to inflation, the same three cars would cost $151,830. The cost of the BMW increased by 8%, the Jeep by 5%, and the Toyota by 12%. If the price of last year’s Jeep was $7,000 less than the price of last year’s BMW, what was the price of each of the three cars last year?

  • ABMW: $49,636.40, Jeep: $42,636.40, Toyota: $47,727.30
  • BBMW: $47,636.40, Jeep: $40,636.40, Toyota: $47,727.30
  • CBMW: $42,636.40, Jeep: $49,636.40, Toyota: $47,727.30
  • DBMW: $54,727.30, Jeep: $47,727.30, Toyota: $42,636.40

Q21:

Jacob, Michael, and Charlotte each had 1 pound to spend in the candy shop. Jacob bought a packet of Stars and a bar of Coolies and got 13 pence in change. Michael bought a packet of Stars and a Munch bar and got 17 pence in change. Charlotte bought a bar of Coolies and a Munch bar and got 40 pence in change. Their friend Madison wanted to buy three packets of Stars, four bars of Coolies, and five Munch bars for her family. How much would it cost her?

Q22:

Three bands performed at a concert venue. The first band charged $15 per ticket, the second $45 per ticket, and the third $22 per ticket. Across all the performances, there were 510 tickets sold for a total of $12,700. If the first band had 40 more audience members than the second one, how many tickets did each band sell?

  • AFirst band: 35 tickets, second band: 75 tickets, third band: 400 tickets
  • BFirst band: 150 tickets, second band: 250 tickets, third: 110 tickets
  • CFirst band: 110 tickets, second band: 150 tickets, third band: 250 tickets
  • DFirst band: 150 tickets, second band: 110 tickets, third band: 250 tickets

Q23:

A recent college graduate took advantage of his business education and invested in three investments immediately after graduating. He invested $80,500 into the following three accounts:

  • Account A with 4% annual simple interest
  • Account B with 3.125% annual simple interest
  • Account C with 2.5% annual simple interest

He earned $2,670 interest at the end of the first year. If the amount of money invested in account B was four times the amount invested in account C, how much was invested in each of the three accounts?

  • AAccount A: $44,000, account B: $25,500, account C: $11,000
  • BAccount A: $40,500, account B: $32,000, account C: $8,000
  • CAccount A: $40,500, account B: $8,000, account C: $32,000
  • DAccount A: $25,500, account B: $44,000, account C: $11,000

Q24:

Scarlett bought 3 plastic cars and 2 plastic boats for £1.15. Sophia spent 4p less on 1 car, 1 boat, and 2 bikes. James got 2 boats and a bike for £1.06. How much does one plastic bike cost?

Q25:

Three numbers add up to 106. The first number is 3 less than the second number. The third number is 4 more than the first number. What are the three numbers?

  • A34, 35, 37
  • B33, 35, 38
  • C33, 36, 37
  • D32, 35, 39
  • E33, 34, 39

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