Again, easy but desperate Sweet Springs Candy Shop sold 2 pounds of chocolates for $9.98 and 5 pounds of chocolate for $24.95.

A. Is the relationship between the number of pounds of chocolate and the total cost proportional?

B. Why or why not?

Answers:

Let,

x = number of pounds of chocolate

y = cost of buying x pounds of chocolate

If you buy x = 2 pounds, then you spend y = 9.98 dollars.

So k = y/x = 9.98/2 = 4.99 is the cost per pound. This is known as the unit price or unit cost. It's the cost of one pound of chocolate.

Now if you buy x = 5 pounds, then you'll be charged a total of y = 24.95

So, k = y/x = 24.95/5 = 4.99

We get the same constant of proportionality here, and therefore, this is a direct proportional relationship. Each time x goes up by 1, y goes up by 4.99; in other words, each time you buy 1 extra pound of chocolate, you pay $4.99 more. This is to be expected whenever you're at the grocery store as long as you ignore things like coupons or bulk discounts or things of that nature.

The equation of this line is y = 4.99x

This shows the slope is 4.99 and the y intercept is 0.

An alternative approach:

Again we define x and y as stated above. We see that the two points are (2,9.98) and (5,24.95)

Computing the slope gets us

m = (y2-y1)/(x2-x1)

m = (24.95-9.98)/(5-2)

m = 14.97/3

m = 4.99

So this matches what we got earlier.

Now turn to y = mx+b. Plug in that value of m along with (x,y) = (2,9.98)

Doing so gets us 9.98 = 4.99*2+b

Let's solve for b

9.98 = 4.99*2+b

9.98 = 9.98+b

0 = b

b = 0

I subtracted 9.98 from both sides to isolate b. The fact we get the y intercept is 0 helps show that we have a direct proportion. All direct proportion linear equations go through the origin (0,0). This is to reflect that buying 0 pounds of chocolate means you pay 0 dollars.

We get the same result as the previous section.

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