$18=12 C+13 S$ $\newline$ A batch of hard candy needs $C$ gallons of corn syrup and $S$ gallons of sucrose to create $18$ pounds of sweetener as shown in the equation. What is the increase in the amount of corn syrup in gallons if the amount of sucrose decreases by $1$ gallon?

Full solution

18=12 C+13 S

\newline

A batch of hard candy needs

C

gallons of corn syrup and

S

gallons of sucrose to create

18

pounds of sweetener as shown in the equation. What is the increase in the amount of corn syrup in gallons if the amount of sucrose decreases by

1

gallon?

Understand Equation: Let's first understand the equation given: $18 = 12C + 13S$ . This equation represents the total pounds of sweetener made from $C$ gallons of corn syrup and $S$ gallons of sucrose.
Find C Increase: We need to find out how much C increases when S decreases by $1$ gallon. Let's call the new amount of sucrose $S - 1$ .
Substitute $S-1$ : Substitute $S - 1$ for $S$ in the equation: $18 = 12C + 13(S - 1)$ .
Distribute $13$ : Now, distribute the $13$ to both terms inside the parentheses: $18 = 12C + 13S - 13$ .
Maintain Total Weight: We know that the original equation was $18 = 12C + 13S$ , so if we compare it with the new equation $18 = 12C + 13S - 13$ , we can see that the total pounds of sweetener should remain the same, which is $18$ pounds.
Isolate $12C$ Term: To keep the total at $18$ pounds, we need to increase the amount of corn syrup to compensate for the decrease in sucrose. $\newline$ So, we add $13$ to both sides of the new equation to isolate the $12C$ term: $18 + 13 = 12C + 13S$ .