When considering when the special offer will become a better deal, we want to find out after how many days we will be spending less money than with the full price passes. For example, after 3 days, the full price offer will cost us \$21 dollars and the special offer will cost us \$39, so the special will cost more. We see that this is because of the initial fee of \$30.

Algebraically, we can determine exactly when the special offer costs the same amount as the standard price by setting our two expressions for $m$ equal to each other and solving for $d$.
$$4d+30=7d$$
$$30=3d$$
$$d=10.$$
So, when $d=10$, or on the 10th day, the special offer and the original will have cost us the same amount of money:
$$m=7(10)=70$$
and
$$m=4(10)+30=70.$$
Because on the 11th day, and every day thereafter, the special offer will cost less (\$4 a day versus \$7 a day), we can conclude that after the 10th day is when the special offer becomes a better deal.

Graphically, we can see that the graph of the special offer drops below the graph of the original offer after $d=10$, and as lower translates into less money spent, we can see our conclusion in the graph, as well.

We have a total of \$60 to spend, and so $m=60$. We can substitute this into the equation for each offer.

Special offer:
$$60=4d+30 \implies 30=4d \implies \frac{30}{4}=7.5=d$$
Since $d$ represents days, and we can only go for full days, $.5$ of a day does not make sense, and so we find that with \$60, we can go to the pool for 7 days if we take the special offer.

Original offer:
$$60=7d \implies \frac{60}{7}=8\frac{4}{7}=d$$
Again, we cannot visit the swim center for $\frac{4}{7}$ of a day, and so with \$60 we are able to go for a full 8 days if we take the full price offer.

Therefore, we should take the original offer if we only have \$60 to spend, because we can go for an additional day over the special offer.

## Comments

Log in to comment