what is the constant of proportionality?

Question

I have been learning ratio and proportion lately and my teacher gave me this sum.

If 7 oranges cost 700$, then how much will 60 oranges cost.

She made a table which had 2 headings : Number of oranges and cost.

Then she told me let the number of oranges be x and the cost be y.

Then she came to the conclusion that x / y = k where k is the constant of proportionality.

Can anyone please tell me what is the constant of proportionality and how did she derive the formula x/y = k

All help is appreciated!

Answer 1

The table should look like this one:

$\begin{array}{|c|c|} \hline \texttt{x} & \texttt{y} & \texttt{x/y} \\ \hline 7 & 700 & 0.01 \\ \hline 60 & \color{red}z & 0.01 \\ \hline \end{array}$

The unknown value is $\color{red}\bf z$. The ratio is $\texttt{constant}$ for all amounts of oranges.

Therefore $\frac{60}{\color{red}\bf z}=0.01$

This can be solved by multiplying both sides by $\color{red}z$.

Therefore $60=0.01\cdot \color{red}z$

Dividing both sides by 0.01.

$\frac{60}{0.01}=\color{red}z$

And $0.01=\frac{1}{100}$ Inverting the fractions $\Rightarrow \frac{1}{0.01}=100$

$100\cdot 60=\color{red}z$

$\color{red}z=6,000$

Answer 2

$k$ is the number of oranges per \$. If you put more money, you get more oranges. If you double the cash, you get twice more fruit. And so on, things are proportional.

In this case $7$ oranges for $700\$$ is $0.01$ orange per \$. $x/y=7/100=0.01$.

If you want $60$ oranges, put $6000\$$ on the desk. $x/y=60/6000=0.01$.

Answer 3

Maybe using the Rule of three will be more illuminating – at leats that's the way I learnt proportionality in elementary school:

If $7$ oranges cost $\$\,700$, then $1$ orange will cost $7$ times less, i.e. $\$\,100$, and $60$ oranges $60$ times more than $1$ orange, i.e. $\$\,6000$.

The constant of proportionality is the constant factor by which you multiply the number of items to get the total price of the items. The price per item, may I say.