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1. Kabir paid an entrance fee of [tex]$5[/tex] plus an additional [tex]$0.75[/tex] per game at a local arcade. The next week, he and a friend went to another arcade where Kabir paid [tex]$1.25[/tex] per game, but there was no entrance fee. How many games will Kabir have to play at each arcade to spend the same amount of money at each arcade?

2. Given three consecutive even integers, six more than the smallest integer is the sum of the two larger integers. What are the three consecutive even integers?

Answer :

Sure! Let's solve each part of the question step-by-step.

### Part 1: Kabir's Arcade Expenses

Kabir visits two different arcades. We want to find out how many games Kabir needs to play at each arcade so that he spends the same amount of money at both places.

1. First Arcade:
- Entrance fee: [tex]$5
- Cost per game: $[/tex]0.75

2. Second Arcade:
- No entrance fee
- Cost per game: $1.25

To find the number of games that make the total cost the same, set up an equation where the total cost is equal at both arcades:

[tex]\[ \text{Total cost at First Arcade} = \text{Total cost at Second Arcade} \][/tex]

[tex]\[ 5 + 0.75x = 1.25x \][/tex]

Here, [tex]\( x \)[/tex] represents the number of games played.

- Rearrange the equation:
[tex]\[ 5 + 0.75x = 1.25x \][/tex]

- Subtract [tex]\( 0.75x \)[/tex] from both sides:
[tex]\[ 5 = 1.25x - 0.75x \][/tex]

- Simplify:
[tex]\[ 5 = 0.5x \][/tex]

- Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{5}{0.5} \][/tex]
[tex]\[ x = 10 \][/tex]

So, Kabir needs to play 10 games at each arcade to spend the same amount of money.

### Part 2: Consecutive Even Integers

We need to find three consecutive even integers. Let's represent these integers as:
- The smallest integer: [tex]\( x \)[/tex]
- The next integer: [tex]\( x + 2 \)[/tex]
- The largest integer: [tex]\( x + 4 \)[/tex]

The problem states: six more than the smallest integer is the sum of the two larger integers. This can be expressed as:

[tex]\[ x + 6 = (x + 2) + (x + 4) \][/tex]

- Simplify the right side:
[tex]\[ x + 6 = x + 2 + x + 4 \][/tex]

- Combine like terms:
[tex]\[ x + 6 = 2x + 6 \][/tex]

- Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 6 = x + 6 \][/tex]

- Subtract 6 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \][/tex]

Now, find the three integers:
- Smallest integer: [tex]\( x = 0 \)[/tex]
- Second integer: [tex]\( x + 2 = 2 \)[/tex]
- Third integer: [tex]\( x + 4 = 4 \)[/tex]

Thus, the three consecutive even integers are 0, 2, and 4.

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