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Difficulty: HardLinear Equations in One Variable

At a charity fundraising event, standard tickets were sold for $45\$45 each and VIP tickets were sold for $80\$80 each. The number of standard tickets sold was 1212 more than twice the number of VIP tickets sold. If the total revenue generated from standard tickets exceeded the total revenue from VIP tickets by $1,830\$1,830, how many VIP tickets were sold?

Answer: 129

Answer

129 VIP tickets were sold.
Letting vv represent the number of VIP tickets sold, the number of standard tickets sold is 2v+122v + 12. Expressing the revenue condition yields the linear equation 45(2v+12)−80v=183045(2v + 12) - 80v = 1830. Distributing 45 gives 90v+540−80v=183090v + 540 - 80v = 1830. Combining like terms results in 10v+540=183010v + 540 = 1830. Subtracting 540 from both sides gives 10v=129010v = 1290, which yields v=129v = 129.

Step-by-Step Solution

1
Define the unknown variable
Let vv be the number of VIP tickets sold.
The question asks for the number of VIP tickets, making vv a direct choice for the variable.
2
Translate the relationship between ticket quantities into an algebraic expression
Standard tickets sold =2v+12= 2v + 12
'12 more than twice the number of VIP tickets' translates directly to 2v+122v + 12.
3
Formulate total revenue expressions and construct the single-variable linear equation
45(2v+12)−80v=183045(2v + 12) - 80v = 1830
Total standard revenue minus total VIP revenue equals the given excess of $1,830\$1,830.
4
Distribute and combine like terms to solve for vv
90v+540−80v=1830  ⟹  10v+540=1830  ⟹  10v=1290  ⟹  v=12990v + 540 - 80v = 1830 \implies 10v + 540 = 1830 \implies 10v = 1290 \implies v = 129
Applying standard algebraic operations isolates vv on one side of the equation.

Key Concept

Formulating and solving a linear equation in one variable from a multi-step word problem context
Estimated Time:2m 0s
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Topics

Linear Equations in One Variable

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