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My 9 year old child's method

We must convert b in the denominator to 100. Thus we must multiply numerator and denominator by $100/b$, because we can't change $a/b$. Then $\color{violet}{\dfrac{a}{b}} \equiv \color{violet}{\dfrac{a}{b}} \times \dfrac{\frac{100}{b}}{\frac{100}{b}} \equiv \dfrac{ {\frac ab \times 100}}{100} \equiv \dfrac ab \times \color{limegreen}{100} \times \color{red}{\dfrac1{100}} \equiv \dfrac ab \times \color{limegreen}{100} \color{red}{\%}$.

But why does her answer differ from Mohd Saad's answer?

We're NOT asking about Mohd Saad's method. We both know $\dfrac{a}{b} = \dfrac{n}{100} \iff \dfrac{a}{b} \times 100 = n$, because you simply multiply both sides by 100.

But Mohd Saad's answer final answer is merely $\frac{a}{b} • 100 = n =$ numerator. Who made a mistake, Mohd Saad or my daughter? Why aren't Mohd's and my daughter's answers selfsame?

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    Your child's method makes complete sense. The reason that only the numerator is used for the percentage is because the "per cent" in percent means "out of a hundred" in Latin, so $...%$ really means $\frac{...}{100}.$ So, to convert from your last fraction to a percentage, ditch the $/100$ and switch it to a percentage sign. – Stephen Donovan Dec 19 '21 at 21:06
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    There is no difference, although your daughter's approach may confuse her if $b$ does not divide $100$ evenly. The point is that, as the previous comment said, $n/100$ IS $n$ percent because of the meaning of "percent." So the equation $\frac ab = \frac n{100}$ IS the statement that $\frac ab$ IS $n$ percent. – Ted Shifrin Dec 19 '21 at 22:01
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    100% =1......... – Wakem Dec 19 '21 at 22:14
  • are you online? can I talk to you? –  Dec 29 '21 at 14:55

1 Answers1

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Your daughter and Mohd Saad are asking related but slightly different questions. Your daughter is asking what is the value of $\frac{a}{b}$ expressed as a percentage. Mohd Saad is saying to express $\frac{a}{b}$ as $n\%$ and asking what $n$ is.

So neither is wrong. When you talk about the "answer" you mean something that equals $\frac{a}{b}$. But Mohd Saad's $n$ does not equal $\frac{a}{b}$; it is $n\%$ that equals $\frac{a}{b}$. So $n\%$ is the answer in your sense of the word "answer".

Mohd Saad, in essence, says $\frac{a}{b}=n\%=\frac{n}{100}$ and solves for $n$. Now $n=100\frac{a}{b}$ and one concludes that $\frac{a}{b}=100\frac{a}{b}\%$, the same as your daughter's answer.

Will Orrick
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    If this answer is still not doing it for you, think of it this way: your daughter and Mohd Saad are asking related but slightly different questions. Your daughter is asking what is the value of $\frac{a}{b}$ expressed as a percentage. Mohd Saad is saying to express $\frac{a}{b}$ as $n%$ and asking what $n$ is. – Will Orrick Dec 20 '21 at 02:17
  • thanks for your comment! I admit your comment assisted us more than your answer. Any plans to embody your comment into your answer? –  Dec 27 '21 at 21:59
  • Rereading my first paragraph, I'm not longer very happy with it, especially the second sentence ("They are the same."). The rest of that paragraph I still agree with, but I can see how it might not be easy to follow. Perhaps I can replace the first paragraph with my comment? Would that be better? – Will Orrick Dec 27 '21 at 22:26
  • OK! thanks. edit your answer as you like! –  Dec 27 '21 at 22:40
  • @WillOrrick I wrote that answer but you explained the difference better that I would. – Mohd Saad Feb 20 '22 at 11:45