# Algebra Word Problem (Linear Equations)

I attempted this problem however I highly suspect that my answer is incorrect, any help on how to approach it efficiently will be appreciated. It proceeds as follows:

I bought a box of chocolates for myself last week. However, by the time I got home I had eaten $$\frac{7}{8}$$ of the chocolates. As I was putting the groceries away, I ate $$\frac{2}{3}$$ of what was left. There are now 22 chocolates left in the box. How many chocolates were in the box in the beginning?

My Attempt:

Let the original number of chocolates be $$x$$,

I ate $$\frac{7}{8}x$$, the number remaining = $$\frac{1}{8}x$$.

Then I ate $$\frac{2}{3}(\frac{1}{8}x)=\frac{1}{12}x$$, the number remaining = $$\frac{11}{12}x$$, it follows that $$\frac{11}{12}x=22$$, hence $$x=24.$$

• The number left was $\frac{1}{3}(\frac{1}{8}x)=\frac{1}{24}x=22$, so the original number of chocolates was $22\cdot 24=528$. Check: You ate $\frac{7}{8}\cdot 528=462$ chocolates first, $528-462=66$ were left, you ate $\frac{2}{3}\cdot 66=44$, then $66-44=22$ were left (after you having eaten $462+44=506$ chocolates altogether. They must have been very small. Good appetite!)
– user700480
Jan 5, 2021 at 20:38
• Perfect, thanks a lot! Jan 6, 2021 at 9:35

Everything looks good up until you said you had $$\frac{11}{12}x$$ remaining. You are correct that as you were putting groceries away you ate $$\frac{1}{12}x$$, but you need to subtract that from the number of chocolates you had when you got home instead of the total number you started with, so it should be $$\frac{1}{8}x - \frac{1}{12}x = \frac{1}{24}x$$ instead of $$x - \frac{1}{12}x = \frac{11}{12}x$$. After that you set $$\frac{1}{24}x = 22$$and solve as you did. That will give you $$x = 22\cdot 24 = 528.$$ Another way to do it would be to say that when you got home you had $$\frac{1}{8}$$ of the original amount left, and then after eating more you had $$\frac{1}{3}$$ of the new amount left, which gives you $$\frac{1}{3}\left(\frac{1}{8} x\right) = 22,$$ which gets you to the same place.

• Thanks a ton. This makes a lot of sense. Jan 6, 2021 at 9:37

You eat $$7/8x$$ then $$1/8x$$ is left.

Then you eat $$2/3$$ of what is left that is $$2/3 (1/8x)=1/12x$$

Then $$\frac{7}{8}x+\frac{1}{12}x+22=x$$ and finally $$x=528$$

Check: $$\frac{7}{8}\,528=462$$ and $$528-462=66$$ then $$\frac{2}{3}\,66=44$$ and $$66-44=22$$.

Notice that you ate $$506$$ chocolates!

• Nice catch! Thank you! Jan 6, 2021 at 9:36