The cost, $c$, of a ham sandwich at a deli varies directly with the number of sandwiches, $n$. If $c=$54 when $n$ is 9, what is the cost of the sandwiches when $n$ is 3?
A. $18
B. $21
C. $27
D. $48
Answer (2)
The cost of 3 ham sandwiches is $18. This is determined by calculating the constant of proportionality and applying it to the number of sandwiches. Based on the calculations, when n = 3 , the cost is $18.
;
Establish the direct variation relationship: c = kn , where c is the cost and n is the number of sandwiches.
Use the given values to find the constant of proportionality: $54 = k
9 , w hi c h g i v es k = 6$.
Substitute n = 3 and k = 6 into the equation: $c = 6
3$.
Calculate the cost: c = 18 . The cost of 3 sandwiches is $18 .
Explanation
Understanding the Problem We are given that the cost, c , of ham sandwiches varies directly with the number of sandwiches, n . This means we can write the relationship as c = kn , where k is the constant of proportionality. We are also given that $c = 54 w h e n n = 9 . W e n ee d t o f in d t h ecos t c w h e n n = 3$.
Finding the Constant of Proportionality First, we need to find the constant of proportionality, k . We know that c = 54 when n = 9 , so we can substitute these values into the equation c = kn to get $54 = k
9$.
Calculating k To solve for k , we divide both sides of the equation $54 = k
9 b y 9 : 9 54 = 9 k 9 6 = k S o , t h eco n s t an t o f p ro p or t i o na l i t y i s k = 6$.
Finding the Cost for n=3 Now that we have the value of k , we can find the cost c when n = 3 . We substitute n = 3 and k = 6 into the equation c = kn :
c = 63 c = 18 Therefore, the cost of the sandwiches when n = 3 is $18.
Final Answer The cost of the sandwiches when n = 3 is $18 .
Examples
Direct variation is a concept that shows up in many real-life situations. For example, the amount you earn at a job where you're paid hourly varies directly with the number of hours you work. If you earn $15 per hour, your total earnings are directly proportional to the hours you put in. Similarly, in cooking, if you're doubling a recipe, the amount of each ingredient varies directly with the original recipe. Understanding direct variation helps in scaling quantities and understanding proportional relationships in everyday scenarios.
