You have a package of candy. Your friend takes six pieces. You're left with a third of what you started with. How much candy did you start with?

Question

You have a package of candy. Your friend takes six pieces. You're left with a third of what you started with. How much candy did you start with?

Answer 1

Answer:

Let n be the number of candies at start.

After my friend takes 6, we are left with ( n - 6) candies.

But now I have 1/3 rd

So, n - 6 = n/3

Or, 3n - 18 = n

After rearranging terms:

2n = 18

Or, n = 9

Answer, there were 9 candies at start.

Answer 2

To answer this question, it is best to try to represent it as an equation, which we can understand as a statement of equivalence. Something that we know is equal to something else we know. This "something" can be a value (represented as any number, like 3, -4 or 3/7, etc.) or as a relationship.

In this problem, we want to figure out how many pieces of candy we started with. That's what we don't know. What we do know is that our friend took six pieces of our candy, so what we have less is six less than the original total. We also know that what we ended up with is one third of what it was originally. Note that these aren't values, but relationships like "six less than" or "one third of."

It sure would be nice to know a value like the number of candies left after six were taken, but we can still use these relationships in order to solve for the value we want, the original number of candies. For instance, note that both of the things we know describe what would be the same value (if we had a value to work with!). Both describe the amount that is left, and we can represent them mathematically as follows, where x refers to the original number of candies:

Six less than the original = x - 6

One third of the original = x/3

Because these represent the same value (as is made clear from the problem statement), we can say that they are equivalent, placing them on both sides of an equation:

x - 6 = x/3

No we just need to solve for x and see what we get! Personally, I'd start by multiplying both sides of the equation by 3 to cancel out the fraction on the right, since fractions can be tricky to deal with:

3(x - 6) = 3(x/3)

3x - 18 = x

Note that this technically gives us an answer: x, the original, is equal to three times the original, minus 18! Unfortunately, this isn't very helpful (it's still a relationship, not a value), so let's continue by combining like terms, subtracting x from both sides to restrict x to one term:

3x - 18 - x= x - x

2x - 18 = 0

Now we just need to isolate x and place a value on the other side to get our answer:

2x - 18 + 18 = +18 => 2x = 18

2x/2 = 18/2 => x = 9

Now we have what we want: an equivalence statement (i.e. an equation) that gives a value for x, the original number of candies: Our answer is 9, there were 9 candies originally.

Does this make sense? Well, we can check just to be sure: if our friend took six candies, we would be left with just three. Is three one third of the original? It is, so it seems like we solved this correctly.