# Constructing an equation using proportionality

The question is:

If the rate of a certain chemical reaction doubles for every $10$ degree rise in temperature then which is greater:

a) Twice the rate at $10$ degrees

b) Half the rate at $30$ degrees

According to my text they are same. Here is how I am doing it

$$\rm Rate = 2\times Temp$$

For $10$ degrees: $\rm Rate = 2 \times 10 = 20$ so $2r$ (Twice The Rate) $=40.$

For $30$ degrees: we have $R = 2 \times 30 = 60$ so half would be $30.$

Could anyone tell me what I am doing wrong since they are different?

• Rate is not twice temperature. If the rate at $10$ is $r$, then rate at $20$ is $2r$, rate at $30$ is $4r$, rate at $40$ is $8r$, rate at $50$ is $16r$. – André Nicolas Aug 2 '12 at 19:45
• Oh okay that makes sense. Btw if we wanted to represent this in an equation form how would that look ? – MistyD Aug 2 '12 at 19:49
• If you know the rate $r_a$ at temperature $a$, then the rate at temperature $t$ is $r_a 2^{(t-a)/10}$. But I think using this formula would be more work, and potentially more confusing, than the answers and comments posted. – André Nicolas Aug 2 '12 at 19:53

Let $x$ be the rate at $10$ degrees. Then twice the rate at $10$ degrees is $2x$. We know that at $20$ degrees the rate will be $2x$ (since this is a $10$ degree rise). Similarly, the rate at $30$ degrees is $2 \times 2x = 4x$, and half of this is $2x$.