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 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 -30C is equal to -22F.
Then we tried -40C and found out that -40C was the same as -40F.

So the answer is -40.

Yesuhei used a similar strategy:

First I tried different solutions for Celsius like -50 and that gave me -58 Fahrenheit .
Then I tried -45C because whenever you go down (negative increasing) the F and C's distance increases. That gave me -48 Fahrenheit which was very close, so I tried -40 Celsius and that gave me -40 Fahrenheit.

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.

graph

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 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= 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= 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 5Fand5C 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:

We know that F = (9/5)C + 32 [equation 1]
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 1 and 2 on Graphmatica.
They will intersect at the point (-40,-40).
So we know that -40F = -40C.

graph


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