Temperature

-15C is 5 F
-65C is -85F
 
 
 
The answer to this problem is yes, there IS a temperature where Celsius and Fahrenheit are equal, and it is $-40$ degrees.

All the solutions to this problem took one of three forms: trial-and-improvement, graphical or algebraic:

This is how Samuel from Long Buckby Junior School reasoned:

There is a temperature at which Celsius and Farenheit are the same.
It is $-40$ degrees, because $9/5$ of $-40$ is $-72$ and $-72 + 32 = -40$.

I decided to look at negative numbers because starting with a positive number and multiplying it by $9/5$ is going to increase it and so is adding 32 so you're always going to end up with a number greater than the number you started with.

However, if you start with a negative number, multiplying it by $9/5$ decreases it, and adding $32$ increases it, so I realised that with the correct number, Celsius and Fahrenheit might be the same.

I decided to go down in tens:
$9/5$ of $-10 = -18$
and $-18 + 32 = 14$,
so that doesn't work;

$9/5$ of $-20 = -36$
and $-36 + 32 = -4$,
so that doesn't work;

$9/5$ of $-30 = -54$
and $-54 + 32 = -22$,
so that doesn't work.

But $9/5$ of $-40 = -72$
and $-72 + 32 = -40$ so it works.

The reason it works is because multiplying by $9/5$ is equivalent to adding $4/5$ of it, and for $-40$ adding $32$ is equivalent to subtracting $4/5$ of it (because $32$ is $4/5$ of $40$).

Because of this, Farenheit and Celsius are equivalent ONLY at $-40$ degrees.

The Four Mathemateers from Brocks Hill Primary School also used a trial and error approach, as displayed here:

First we started going down in tens of Celsius from $0$, and we found out a pattern:
the difference between F and C was getting closer by eights every time.

When we got to $-30C$ the difference was only $8$. So $-30$C is equal to $-22$F.
Then we tried $-40$C and found out that $-40$C was the same as $-40$F.

So the answer is $-40$.

Yesuhei used a similar strategy:

Others who found the correct answer by this method are Emma and Chloe from The Mount School and Michael from Bilton School.

Beatrice fron Raffles Girls' School and Michael used a graphical approach.

Michael's answer is shown here:

I plotted the lines of the simultaneous equations against each other and found where they crossed.
In the graphs $y =$ F and $x = $C.



The quickest way to solve this problem is with an algebraic approach, and both of the people who used graphs used this approach as well. The other people that obtained the correct answer by this method include Sugam and Fiona from The Mount School, Chris from CCSN, Samantha from The Steele School, Gemma, Griselda and Charlie from Colyton Grammar School, Jasvir, Matt and Christian from Kingshill, Ed from Tunbridge Wells Grammar School for Boys, Stephen and Joe from Singapore International School, Kieran from Alcester Grammar School and Pradeesha:

Here is Sugam's working:

Let x be the temperature where Fahranheit and Celsius are equal.

$x=\frac{9}{5} x+32$

$5x = 9x + 160 $

$-4x = 160 $

$x = -40 $

Therefore $-40$ Celsius $= -40$ Farenheit

And here is Michael's solution:

To solve it algebraically I can create two simultaneous equations:

$F = C$
$F = 1.8C + 32 $

Therefore
$C = 1.8C + 32 $
$C = -32 / 0.8 = -40 $

And here is Kieran's solution:

Using the equation, $F=\frac{9}{5} C+32$,
we can remove the fraction by multiplying both sides by five.

Doing so produces $5F = 9C + 160$, and thus, using the sought after equation of $F = C$,
we may further deduce that, since $5F and 5C$ are one and the same, subtracting the two equal amounts from either side leaves

$0 = 4C + 160$
or $4C = -160$
or $C = -40$

Consequently, $-40$ Celcius is the same as $-40$ Fahrenheit.

Oliver remembered to check that his solution worked:

We can substitute $-40$ as $C$ in $F = 9/5C + 32$ to check our answer
As $-40 = -72 + 32$, our answer is correct

Beatrice combined an algebraic and graphical approach:

This is a linear equation, because it follow the structure $y = mx + c$

Now let us make C the subject:
$F = (9/5)C + 32 $
$F - 32 = (9/5)C$
$C = (5/9)(F - 32)$ [equation 2]

Now plot equations 1and 2 on Graphmatica.
They will intersect at the point $(-40,-40)$.
So we know that $-40F = -40C$.




Well done to all of you who solved this problem correctly.

Short solution: 

1. Is there a temperature at which Celsius and Fahrenheit readings are the same?

Yes, -40°C = -40°F

2. Can you describe a way of converting Fahrenheit readings into Celsius?

$C = (F-32):1.8$ where C is a temperature in Celsius and F is the temperature in Fahrenheit.

3. Can you describe a way of converting Celsius readings into Fahrenheit?

$F = 32+1.8C$

4. Is there a temperature at which the Fahrenheit reading is  20 degrees higher than the Celsius reading? Yes, -15°C = 5°F

5. Is there a temperature at which the Celsius reading is 20 degrees higher than the Fahrenheit reading?

Yes, -65°C = -85°F

6. Is there a temperature at which Kelvin and Fahrenheit readings are the same?

Yes, 574.6°K = 574.6°F

7. Is there a temperature at which Kelvin and Celsius readings are the same?

No

8. Can you describe ways of converting Kelvin readings into Fahrenheit and Celsius readings?

F = 32 + 1.8(K-273.15) and C = K - 273.15 where K is the temperature in Kelvin.