Question

6-46. mixing candy
renard thought that writing two equations for problem 6-44 was easy. he wants to use two equations with two variables to solve this problem:
ariel bought several bags of caramel candy and several bags of taffy. the number of bags of taffy was 5 more than the number of bags of caramels. taffy bags weigh 8 ounces each, and caramel bags weigh 16 ounces each. the total weight of all the bags of candy was 400 ounces. how many bags of candy did she buy?
a. renard lets $t=$ the number of taffy bags and $c=$ the number of caramel bags. help him write two equations to represent the information in the problem.
b. now renard is stuck. he says, \if both of the equations were in the form t= something, i could set the two expressions equal to each other to find the solution.\ help him rewrite the equations into a form he can use to solve the problem.

Explanation:

Step1: Define variables and first equation

Let $t$ = number of taffy bags, $c$ = number of caramel bags. The number of taffy bags is 5 more than caramel bags:
$t = c + 5$

Step2: Define second weight equation

Taffy bags are 8 oz, caramel 16 oz, total 400 oz:
$8t + 16c = 400$

Step3: Substitute $t$ into weight equation

Replace $t$ with $c+5$ in the second equation:
$8(c + 5) + 16c = 400$

Step4: Expand and simplify the equation

$8c + 40 + 16c = 400$
$24c + 40 = 400$

Step5: Solve for $c$

$24c = 400 - 40$
$24c = 360$
$c = \frac{360}{24} = 15$

Step6: Solve for $t$

Use $t = c + 5$:
$t = 15 + 5 = 20$

Step7: Rewrite equations for part b

Rearrange the first equation to $t = c + 5$. Simplify the second equation by dividing by 8:
$t + 2c = 50$
Then rewrite the second equation to $t = 50 - 2c$

Answer:

a. The two equations are:
$t = c + 5$
$8t + 16c = 400$
Ariel bought 20 bags of taffy and 15 bags of caramel.
b. The equations rewritten in the form $t = \text{something}$ are:
$t = c + 5$
$t = 50 - 2c$