Question

A store sells candy at $0.50, $1, $1.50, $2, and $3 per kilogram. How many kilograms of each kind of vandy does $3 buy?
__ kg at $.50 per kilogram could be bought.
__ kg at $1 per kilogram could be bought.
__ kg at $1.50 per kilogram could be bought.
__ kg at $2 per kilogram could be bought.
__ kg at $3 per kilogram could be bought.

Answer 1

A store sells candy at $0.50, $1, $1.50, $2, and $3 per kilogram. You can see the unit price of candies that $3 buys vary inversely. What is the constant of variation?

Answer 2

Let's solve this problem step by step.

1. We will start by calculating how many kilograms of candy can be bought at $0.50 per kilogram.

Since each kilogram costs $0.50 and we have $3, we can buy 3/0.50 = 6 kilograms of candy at this price.

2. Next, let's calculate how many kilograms of candy can be bought at $1 per kilogram.

Similarly, since each kilogram costs $1 and we have $3, we can buy 3/1 = 3 kilograms of candy at this price.

3. Moving on to the candy that costs $1.50 per kilogram.

Once again, since each kilogram costs $1.50 and we have $3, we can buy 3/1.50 = 2 kilograms of candy at this price.

4. Now, let's calculate how many kilograms of candy can be bought at $2 per kilogram.

Following the same logic as before, since each kilogram costs $2 and we have $3, we can buy 3/2 = 1.5 kilograms of candy at this price. However, since we can't buy a fraction of a kilogram, we round down to the nearest whole number. Therefore, we can only buy 1 kilogram of candy at this price.

5. Lastly, we will calculate how many kilograms of candy can be bought at $3 per kilogram.

Once again, since each kilogram costs $3 and we have $3, we can buy 3/3 = 1 kilogram of candy at this price.

To summarize:
- We can buy 6 kilograms of candy at $0.50 per kilogram.
- We can buy 3 kilograms of candy at $1 per kilogram.
- We can buy 2 kilograms of candy at $1.50 per kilogram.
- We can buy 1 kilogram of candy at $2 per kilogram.
- We can buy 1 kilogram of candy at $3 per kilogram.

Therefore, $3 can buy 6 kg at $0.50 per kilogram, 3 kg at $1 per kilogram, 2 kg at $1.50 per kilogram, 1 kg at $2 per kilogram, and 1 kg at $3 per kilogram.

Answer 3

To find the constant of variation, we can set up a proportion using the price and quantity of candy that $3 can buy at each price point.

Let's label the quantities of candy that $3 can buy at each price as x, y, z, w, and v, respectively, where x is for $0.50 per kilogram, y is for $1 per kilogram, z is for $1.50 per kilogram, w is for $2 per kilogram, and v is for $3 per kilogram.

We know that the total cost is $3, so we can set up the following proportion:

x * $0.50 + y * $1 + z * $1.50 + w * $2 + v * $3 = $3

Simplifying this equation, we get:

0.50x + y + 1.50z + 2w + 3v = 3

Next, we can set up another proportion relating the quantities of candy bought at each price:

x/y = y/x = z/y = w/z = v/w = k

Here, k represents the constant of variation that we want to find.

Let's take the first ratio, x/y:

x/y = k

Rearranging this equation, we get:

x = ky

Similarly, we can find the other quantities in terms of y:

y = kx
z = ky
w = kz
v = kw

We can substitute these values back into the earlier equation:

0.50(ky) + y + 1.50(ky) + 2(kz) + 3(kw) = 3

Simplifying this equation, we get:

0.50ky + y + 1.50ky + 2kz + 3kw = 3

Now, we can factor out k:

k(0.50y + y + 1.50y + 2z + 3w) = 3

Simplifying further, we get:

k(3y + 2z + 3w) = 3

Finally, we can isolate k by dividing both sides by (3y + 2z + 3w):

k = 3 / (3y + 2z + 3w)

Therefore, the constant of variation is 3 / (3y + 2z + 3w).