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This is for a 12 year old who is not good with finding percentages as it is, but who needs to be able to work out the original price of some sold goods if they know the new price and the discount applied.

e.g. A hat now costs €6.30 after a 10% reduction, how much was it originally (X).

i.e. $$X=\displaystyle \frac{6.3}{1-\frac{10}{100}}=7$$

How can I explain that without any algebra or mention of such?

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    $\begingroup$ The formula you have there could be good way to explain it... (at least, in my opinion) $\endgroup$ – apnorton Jan 22 '14 at 17:24
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    $\begingroup$ Consider that your proscription against algebra might actually be making it harder to explain. $\endgroup$ – Hurkyl Jan 22 '14 at 17:47
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I can only report on how my sister does it. They defined 3 different variables. One for the original price, one for the new price and one for the percentage. They then learned different formulas, by heart, without understanding the meaning, to compute the missing value, given the other two values. Personally I find this whole approach very unsatisfying because they don't understand the underlying principle but that's how their teacher explained it.

Maybe you could try giving her/him at least a way to approximate the result. Imagine two people buying the product, one after the discount, one before. The difference between both spendings has to be 10 percent of the original price. Now let us try some value: If the original price was 8Euro then she should have saved 80 cents but this means the product would have costed 7.20, so 8 was too high, try again with 7.50, still to high, go lower, and so on, until you get to 7 where it magically works.

Or start teaching her/him how to solve terms for x by always doing the same on each side of the equation. I guess this would be a good approach in the long run but right now she/he might only get confused.

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    $\begingroup$ I think I'm going to do try both first and second option and see which they prefer. They do a lot of guestimation and correction these days so the second option would probably fit in well. $\endgroup$ – HCAI Jan 22 '14 at 20:24
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You can try to explain your formula for $X$ in terms of it being the ratio of the current price (after discount) with respect to the fraction of the original price that the hat is now selling for $(1 - 0.1) = 0.9$, where $1$ is taken to be $100\%$ of the original price. (Of course, this may need some "reminder" prompts about navigating between how to represent a fraction/decimal as a percentage, and vice versa.)

The ratio's value will not change: $$\dfrac{6.3}{0.9} = \dfrac{7}{1}$$

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  • $\begingroup$ with the word "ratio", one opens another can of worms entirely. ;) $\endgroup$ – HCAI Jan 22 '14 at 20:19
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A method I found useful for students starting out: (thinking in terms of "units")

Using the example,
After the 10% reduction we have 90%.
90% of the hat costs 6.3.
10% of the hat costs $\displaystyle \frac{6.3}{9}$.
Hence, 100% of the hat costs $\displaystyle \frac{6.3}{9} \cdot 10 = 7$.

Eventually we want to proceed to the more general formula (like what you have) to handle "weirder" numbers (3.142%?) but this method could be a good start. When I guide students towards the more general formula one common mistake is confusing between multiplying or dividing by a certain percentage. I found explaining in terms of whether the costs should be larger or smaller helped.

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I suggest that you offer the 12-year-old the regula falsi method. In this method, you guess, check if the guess is correct, and if it isn't, you adjust the guess.

Let's try a slightly different problem:

A hat now costs €6.39 after a 10% reduction. How much was it originally?

The original price must have been a little more than €6.39, so we begin by guessing that it was around €7 before. But 10% of €7 is €0.70, so had it been €7 before, the price after reduction would be $€7 - €0.70 = €6.30$. This is too small by €0.09. €0.09 is small, so we must be very close. What increase in the original price would cause the discounted price to increase by €0.09? It must be a bit more than €0.09, so try increasing the supposed original price by €0.10, to €7.10. On checking, this is in fact the answer.

This method is self-correcting. Suppose the 12-year-old tried increasing the guess by €0.20, to €7.20. This would give a discounted price of $€7.20 - €0.72 = €6.48$, which is too high. So the correct answer must be somewhere between €7.00 and €7.20.

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