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Liz won a bunch of tickets at the arcade. She plans to spend most of them on one large item and use the rest to buy hair ties. An employee at the arcade tells Liz that if she buys the kickball for 2,2002,200 tickets, she will have exactly enough left for 1313 hair ties. However, if she buys the toy airplane for 2,5002,500 tickets, she will have exactly enough left for 77 hair ties.\newlineWhich equation can you use to find tt, the number of tickets Liz needs for each hair tie?\newlineChoices:\newline(A) 2,200+13t=2,500+7t2,200 + 13t = 2,500 + 7t\newline(B) 2,200+13t=2,5007t2,200 + 13t = 2,500 - 7t\newlineHow many tickets does Liz need for each hair tie?\newline____ tickets

Full solution

Q. Liz won a bunch of tickets at the arcade. She plans to spend most of them on one large item and use the rest to buy hair ties. An employee at the arcade tells Liz that if she buys the kickball for 2,2002,200 tickets, she will have exactly enough left for 1313 hair ties. However, if she buys the toy airplane for 2,5002,500 tickets, she will have exactly enough left for 77 hair ties.\newlineWhich equation can you use to find tt, the number of tickets Liz needs for each hair tie?\newlineChoices:\newline(A) 2,200+13t=2,500+7t2,200 + 13t = 2,500 + 7t\newline(B) 2,200+13t=2,5007t2,200 + 13t = 2,500 - 7t\newlineHow many tickets does Liz need for each hair tie?\newline____ tickets
  1. Equation Setup: Let's denote the total number of tickets Liz has by TT and the number of tickets needed for each hair tie by tt. According to the problem, if Liz buys the kickball, she will spend 2,2002,200 tickets and have enough left for 1313 hair ties. This can be represented by the equation T2,200=13tT - 2,200 = 13t. Similarly, if she buys the toy airplane, she will spend 2,5002,500 tickets and have enough left for 77 hair ties, which can be represented by the equation T2,500=7tT - 2,500 = 7t. Since the total number of tickets TT is the same in both scenarios, we can set the two expressions for the remaining tickets equal to each other to find the value of tt.
  2. Isolate Variable: Set up the equation based on the information given: 2,200+13t=2,500+7t2,200 + 13t = 2,500 + 7t. This equation represents the fact that the remaining tickets after buying either the kickball or the toy airplane are used to buy hair ties, and the cost in tickets for the hair ties is the same in both scenarios.
  3. Simplify Equation: To solve for tt, we need to isolate the variable on one side of the equation. We can do this by subtracting 7t7t from both sides of the equation to get the tt terms on one side: 2,200+13t7t=2,500+7t7t2,200 + 13t - 7t = 2,500 + 7t - 7t.
  4. Subtract Terms: Simplify the equation: 2,200+6t=2,5002,200 + 6t = 2,500. This simplification combines like terms by subtracting 7t7t from both sides.
  5. Divide and Calculate: Now, subtract 2,2002,200 from both sides of the equation to isolate the term with tt: 2,200+6t2,200=2,5002,2002,200 + 6t - 2,200 = 2,500 - 2,200.
  6. Divide and Calculate: Now, subtract 2,2002,200 from both sides of the equation to isolate the term with tt: 2,200+6t2,200=2,5002,2002,200 + 6t - 2,200 = 2,500 - 2,200. Simplify the equation again: 6t=3006t = 300. This step involves performing the subtraction on both sides of the equation.
  7. Divide and Calculate: Now, subtract 2,2002,200 from both sides of the equation to isolate the term with tt: 2,200+6t2,200=2,5002,2002,200 + 6t - 2,200 = 2,500 - 2,200. Simplify the equation again: 6t=3006t = 300. This step involves performing the subtraction on both sides of the equation. Finally, divide both sides of the equation by 66 to solve for tt: 6t6=3006\frac{6t}{6} = \frac{300}{6}.
  8. Divide and Calculate: Now, subtract 2,2002,200 from both sides of the equation to isolate the term with tt: 2,200+6t2,200=2,5002,2002,200 + 6t - 2,200 = 2,500 - 2,200.Simplify the equation again: 6t=3006t = 300. This step involves performing the subtraction on both sides of the equation.Finally, divide both sides of the equation by 66 to solve for tt: 6t/6=300/66t / 6 = 300 / 6.Calculate the value of tt: t=50t = 50. This is the number of tickets Liz needs for each hair tie.

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