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For example, suppose I make a purchase that is $\$5.36$. Now there is tax added. So my total price is actually $$(5.36 + (5.36 * 0.13))=\$6.06$$

But really, I can just do this and get the same answer:

$$5.36*1.13=\$6.06$$

Why does the second way work? Why does it make sense?

As mentioned in the answers section, this is very easily mathematically shown (factoring and such), but I am looking for the actual "intuition" behind it. How would you explain it to someone without using math?

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because multiplication is distributive, 5.36 is 100% of 5.36, 13% tax is on top of that is 13%, so you have 113% after tax. which is represented by 1.13 in decimal form.

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Because

$$ 5.36 + (5.36\times0.13) = 5.36(1 + 0.13) = 5.36 \times 1.13$$

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  • $\begingroup$ Using math it makes sense, but if you were to verbally explain to someone why it makes sense, then what would you say? $\endgroup$ – K Split X Jul 29 '17 at 18:20
  • $\begingroup$ This answer could arguably be improved by adding another step introducing the $\times1$. $\endgroup$ – Mark S. Jul 29 '17 at 18:21
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    $\begingroup$ I see what you mean-perhaps re-word your question so its clear what you're asking? $\endgroup$ – John_dydx Jul 29 '17 at 18:21
  • $\begingroup$ @MarkS. This is a fine answer, and I doubt K.Split needs such obvious a step. $\endgroup$ – Namaste Jul 29 '17 at 18:43
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If there's 13% sales tax on something, that means you have to pay 13% extra.

In other words, you have to pay the entire price, and then another 13% on top of that.

In other words, you have to pay 100% of the price, plus another 13% of the price.

In other words, you have to pay 100% plus 13%.

In other words, you have to pay 113%.

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The tax is the extra money you pay on top of the total. The two ways to think about this are:

  • Find $13\%$ of the price and add it to the total, or
  • Multiply the price by $1.13$.

Why? As others have said, there's the distributive property, but you know that the tax is going to increase the price, hence multiplying by something greater than $1$.

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