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On buying a pen, the shopkeeper gives three refills free. On buying a pen and six refills, the shopkeeper gives additional four refills free. If the equivalent discount is the same in both cases, then how many refills will be equal in value to a pen?

The solution provided for this problem is as follows :-
Let the cost of pen be $p$ and that of refill be $r$.
Then $\frac{3r}{p}=\frac{4r}{p+6r} \Rightarrow p=18r$

I am not able to understand how did the first fraction relation came up. Please help me understand that.

Thanks in advance !!!

Ganit
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1 Answers1

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The logic is that $\frac{3r}{p}$ and $\frac{4r}{p+6r}$ both measure $$\frac{\text{value of free stuff you got}}{\text{amount you paid}}$$ in the two transactions. If you buy a pen and get three free refills, you paid $p$ and got $3r$ worth of free stuff. If you buy a pen plus six refills, and get four free refills, you paid $p+6r$ and got $4r$ worth of free stuff.

This ratio indirectly corresponds to the "equivalent discount" in percentages.

Here's maybe a more intuitive solution. Assume equivalent discounts get you:

  • $3$ free refills, if you buy a pen;
  • $4$ free refills, if you buy a pen and $6$ refills.

Then the extra $6$ refills you bought in the second case are getting you $1$ extra free refill. So we immediately conclude that the extra free stuff is worth $\frac16$ of what you paid for.

If $3$ free refills are $\frac16$ of the price of a pen, then $3 \cdot 6 = 18$ refills are equal to the price of a pen.

Misha Lavrov
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