Calculate Number of Apartments with 5 Tenants: Solving y-2 Average Equation

Q: Why do we get a fractional answer when counting apartments?

The fractional result $ 70.5 - 10y $ represents a relationship, not an actual count! For any specific value of y, this expression must give a whole number between 0 and 9 for the problem to have a real solution.

Q: How do I know which apartments have 5 tenants vs 7 tenants?

Let $ x_1 $ = apartments with 5 tenants and $ x_2 $ = apartments with 7 tenants. Since there are 9 remaining apartments total, we know $ x_1 + x_2 = 9 $.

Q: What does the average formula really mean here?

The average $ y-2 $ means: Total tenants ÷ 20 apartments = y-2. So total tenants = $ 20(y-2) $. This connects our counting to the algebraic expression!

Q: Why substitute before solving the equation?

Substituting $ x_2 = 9 - x_1 $ reduces our system from two equations with two unknowns to one equation with one unknown. This makes solving much easier!

Q: How can I check if my final answer makes sense?

The number of 5-tenant apartments must be between 0 and 9Total tenants should equal $ 38 + 5x_1 + 7x_2 $Average should equal $ y-2 $ when calculated ✓

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

In a apartment block there are 20 apartments.

5 apartments house 4 tenants each.

6 apartments house 3 tenants each.

The rest of the apartments house 5 or 7 tenants.

On average each apartment houses $y-2$ tenants.

How many apartments are there where 5 tenants live?

2

Step-by-step solution

To solve this problem, we first note the setup: 5 apartments with 4 tenants and 6 apartments with 3 tenants are explicitly mentioned. That accounts for:

$5 \times 4 = 20$ tenants from the 4-tenant apartments.
$6 \times 3 = 18$ tenants from the 3-tenant apartments.

Now, the remaining $9$ apartments (since $20 - (5 + 6) = 9$ ) can house either 5 or 7 tenants. Let $x_1$ be the number of 5-tenant apartments and $x_2$ be the number of 7-tenant apartments:

$x_1 + x_2 = 9$
Total number of tenants = $20 + 18 + 5x_1 + 7x_2$

The average number of tenants per apartment is given by $y - 2$ . We express the total tenant number equation:

$\frac{20 + 18 + 5x_1 + 7x_2}{20} = y - 2$

Substitute $x_2 = 9 - x_1$ into the tenant equation:

Total number of tenants = $38 + 5x_1 + 7(9 - x_1)$

= $38 + 5x_1 + 63 - 7x_1$

= $101 - 2x_1$

Average tenants = $\frac{101 - 2x_1}{20} = y - 2$

Multiplying throughout by 20:

101 - 2x_1 = 20(y - 2)

101 - 2x_1 = 20y - 40

Solving for $x_1$ :

$-2x_1 = 20y - 40 - 101$

$-2x_1 = 20y - 141$

$x_1 = \frac{141 - 20y}{2}$

Therefore, $x_1 = 70.5 - 10y$ .

Thus, the correct answer is $\boxed{70.5 - 10y}$ .

3

Final Answer

$70.5-10y$

Key Points to Remember

Essential concepts to master this topic

Setup: Identify known quantities before setting up variables
Technique: Use substitution: $x_2 = 9 - x_1$ to eliminate one variable
Check: Verify $x_1 + x_2 = 9$ and total tenants equals $20(y-2)$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to account for all apartments
Don't set up equations using only the variable apartments = incomplete system! Students often forget the 5 apartments with 4 tenants and 6 apartments with 3 tenants contribute 38 total tenants. Always include all given information in your total count before setting up equations.

Practice Quiz

Test your knowledge with interactive questions

A hotel's overall rating is determined according to a weighted average of several categories. Each category is given a rating and a weighted factor. Below are the ratings for the "Happy Tourist" hotel:

Determine the hotel's overall rating?

FAQ

Everything you need to know about this question

Why do we get a fractional answer when counting apartments?

+

The fractional result $70.5 - 10y$ represents a relationship, not an actual count! For any specific value of y, this expression must give a whole number between 0 and 9 for the problem to have a real solution.

How do I know which apartments have 5 tenants vs 7 tenants?

+

Let $x_1$ = apartments with 5 tenants and $x_2$ = apartments with 7 tenants. Since there are 9 remaining apartments total, we know $x_1 + x_2 = 9$ .

What does the average formula really mean here?

+

The average $y-2$ means: Total tenants ÷ 20 apartments = y-2. So total tenants = $20(y-2)$ . This connects our counting to the algebraic expression!

Why substitute before solving the equation?

+

Substituting $x_2 = 9 - x_1$ reduces our system from two equations with two unknowns to one equation with one unknown. This makes solving much easier!

How can I check if my final answer makes sense?

+

The number of 5-tenant apartments must be between 0 and 9
Total tenants should equal $38 + 5x_1 + 7x_2$
Average should equal $y-2$ when calculated ✓

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