Explain to 9 year old — Why multiply by 100 and add %, to convert to percent?

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?

Solution 1:

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.