How many more families have to join to get the vacation home to be manageable for your family?

Question

Your family plans to take a vacation this summer. For the cost of the vacation to be manageable, you determine that it will be in the best interest of your family to share a vacation home with other families. The cost c(in dollars) per family of the vacation home is inversely proportional to the number t of families sharing the house. Your family wants the cost of the house per family to be $600. Currently, with three families (including your family), the vacation home rental cost is $1000 per family.

Answer 1

The key sentence in this problem is "The cost c(in dollars) per family of the vacation home is inversely proportional to the number t of families sharing the house". Let's translate that into an equation by breaking it down. This is a fairly straightforward question but I will go through it in small steps as learning to translate word problems can be tricky!

First remove the descriptive words that do not translate to variables, constants, or operators. This leaves "cost per family is inversely proportional to number of families per cost of house". I rewrote "sharing the house" as "cost of house" since the families are sharing the cost of the house, not the actual house itself simultaneously. Proportionality in math is represented by the equation:

a₁ / b₁ = a₂ / b₂

In math "is" implies "=" or equality, and "per" implies "/" or division. So let's rewrite the sentence with that information.

"cost / family = number of families / cost of house"

Inversely proportional tells us that as c grows, t will decrease and vice-versa. Thus, we know c and t are on opposite side of the division sign. Another way you can look at it, is that we must flip the left side of the equation as it is inversely proportional (what we have now is directly proportional).

"cost / family = cost of house / number of families"

Now to get our final equation we just need to replace the remaining words with variables and constants. I will use C_h for cost of house and f for family, we already were given t for number of families, and c for cost.

c / f = C_h / t

We do not know C_h yet, but we do know f = 1 as it represents 1 family's associated cost. We can solve for C_h if we plug in the c = 1000, t = 3 pair of values we were given. I will omit f as division by 1 simplifies to the numerator (in this case c).

1000 = C_h/3

3000 = C_h Multiplying both sides by 3 and dividing simplifying division by 1

We can now rewrite our final proportion equation: c = 3000/t. Remember we want the cost to be 600 per family and are looking for t, the number of families we need.

600 = 3000 / t

600 * t = 3000 Multiplying both side by t

t = 3000 / 600 Divide both sides by 600

t = 5

So 5 families will result in each family contributing $600. The cool part about proportionality equations, is once we have solved for the equation constants we can figure out the resulting variable from any other variable. For example if I knew 8 families had signed up to rent my beach house this summer then I would be able to plug in t = 8 and get c = 3000/8 or c = 375. Thus each family would only pay $375. I hope this helps your understanding of proportionality and translating word problems into equations.