At the Boxer carnival, you can exchange 56 tickets for various assortments of prizes. Some of those assortments are below. How many tickets does it take to get a single goldfish? 1... At the Boxer carnival, you can exchange 56 tickets for various assortments of prizes. Some of those assortments are below. How many tickets does it take to get a single goldfish? 1 goldfish, 6 tacky mirrors, 2 stuffed tiger; 3 goldfish, 2 stuffed tiger; 4 goldfish, 2 tacky mirrors. Read more

Steps to Solve

1. Set Up the Variables Let:

• $g$ = number of tickets for 1 goldfish
• $m$ = number of tickets for 1 tacky mirror
• $t$ = number of tickets for 1 stuffed tiger

2. Create the Equations From the information provided:

• For 1 goldfish, 6 tacky mirrors, and 2 stuffed tigers: $$ g + 6m + 2t = 56 $$

• For 3 goldfish and 2 stuffed tigers: $$ 3g + 2t = 56 $$

• For 4 goldfish and 2 tacky mirrors: $$ 4g + 2m = 56 $$

3. Express One Variable in Terms of Another Using the second equation, we can express $t$ in terms of $g$: $$ t = \frac{56 - 3g}{2} $$

4. Substitute Into Other Equations Substitute $t$ from the previous step into the first equation: $$ g + 6m + 2 \left( \frac{56 - 3g}{2} \right) = 56 $$

This simplifies to: $$ g + 6m + 56 - 3g = 56 $$ Thus, $$ -2g + 6m = 0 , \Rightarrow , m = \frac{g}{3} $$

5. Substitute into the Third Equation Now substitute the expression for $m$ back into the third equation: $$ 4g + 2 \left( \frac{g}{3} \right) = 56 $$

This simplifies to: $$ 4g + \frac{2g}{3} = 56 $$ Multiplying through by 3 to eliminate the fraction: $$ 12g + 2g = 168 $$ $$ 14g = 168 $$

6. Solve for Tickets of Goldfish Dividing both sides by 14 gives: $$ g = 12 $$

Thus, it takes 12 tickets to get one goldfish.