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

Norbert buys some new clothes.

When he gets home, he decides to work out how much each outfit cost him on average.

What answer should he come up with?

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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