Let’s solve the problem step by step.
(a) Write an equation.
Let \( x \) represent the cost of one of Tanya's items. Then, the cost for Tanya's 3 items is:
\[ 3x \]
As for Tony, the cost of one of his items can be expressed as \( x - 2.25 \). Since Tony bought 4 items, the total cost for his items is:
\[ 4(x - 2.25) \]
Since both Tanya and Tony paid the same amount of money, we can set their total costs equal to each other:
\[ 3x = 4(x - 2.25) \]
(b) Solve the equation. Show your work.
Now, we will solve the equation:
\[ 3x = 4(x - 2.25) \]
First, distribute the 4 on the right side:
\[ 3x = 4x - 9 \]
Next, isolate \( x \) by subtracting \( 4x \) from both sides:
\[ 3x - 4x = -9 \] \[ -x = -9 \]
Now, multiply both sides by -1:
\[ x = 9 \]
(c) Check your solution. Show your work.
We have found that \( x = 9 \). Now, let's calculate the total costs for both Tanya and Tony:
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Tanya’s total cost: \[ 3x = 3(9) = 27 \]
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Cost for one of Tony's items: \[ x - 2.25 = 9 - 2.25 = 6.75 \]
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Tony’s total cost: \[ 4(x - 2.25) = 4(6.75) = 27 \]
Since both totals are equal (27 for Tanya and 27 for Tony), our solution is correct.
(d) State the solution in complete sentences.
Tanya bought 3 items at a cost of $9 each, resulting in a total expenditure of $27. Tony purchased 4 items, each costing $6.75, which also totaled $27. Therefore, the individual cost of each person's items is $9 for Tanya and $6.75 for each of Tony's items.