test preparation 💆🏻‍♀️💆🏻‍♀️

Question

there are 9 bottles in devins refrigerator. he adds 3 full boxes of water bottles to the refrigerator, then he adds 2 more boxes that each has 1 fewer bottle than a full box, when he’s done there’s 67 bottles in the refrigerator

Answer 1

9 + 3b + 2(b - 1) = 67

Solve for b.

Answer 2

In order to turn a word problem like this into something we can solve, we want to give a name to the thing we do not know. Here the unknown is how many bottles are in a full box, so let us call that b.

Now we just walk through the story one sentence at a time and write down what each piece adds to the refrigerator.

He starts with 9 bottles already in there, so that is just 9.

He adds 3 full boxes. Each full box has b bottles, so three of them is 3b.

Then he adds 2 more boxes that each have 1 fewer bottle than a full box. A box with one fewer than a full box has b - 1 bottles in it, and there are two of those, so that is 2(b - 1). This is the part where people usually slip, since it is tempting to write 2b - 1, but the "1 fewer" applies to each box separately, not once overall.

When he is done there are 67 bottles, so everything we just listed has to add up to 67:

9 + 3b + 2(b - 1) = 67

Now we simplify. Distributing the 2 gives 2b - 2:

9 + 3b + 2b - 2 = 67

If you notice, 3b and 2b are like terms since they both have b, and 9 and -2 are like terms since they are both just numbers:

5b + 7 = 67

Remember, our goal is to get b by itself, so we subtract 7 from both sides and then divide by 5:

5b = 60

b = 12

Let us check it back against the story. A full box has 12, so 3 full boxes is 36, and the two short boxes have 11 each which is 22. Then 9 + 36 + 22 = 67, which matches.

So, our final answer is that a full box has 12 bottles.