For what temperature the Fahrenheit and Centigrade scale will have same value?

(273 - 32)^{o} C

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- 40^{o} C

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(273 + 32)^{o} C

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{574.25}^{o} C

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Solution

The correct option is B $- 40^{o} C$
The temperature scales are given by:
$\frac{θ - θ_{f r e e z i n g p o i n t}}{θ_{b o i l i n g p o i n t} - θ_{f r e e z i n g p o i n t}} = c o n s t a n t$
In the Centigrade scale:
$θ_{f r e e z i n g p o i n t} = 0^{\circ} C$
$θ_{b o i l i n g p o i n t} = 100^{\circ} C$
In the Fahrenheit scale:
$θ_{f r e e z i n g p o i n t} = 32^{\circ} F$
$θ_{b o i l i n g p o i n t} = 212^{\circ} F$
i.e.
$\frac{θ_{c e n t i g r a d e} - θ_{f r e e z i n g p o i n t}}{θ_{b o i l i n g p o i n t} - θ_{f r e e z i n g p o i n t}} = \frac{θ_{f a h r e n h e i t} - θ_{f r e e z i n g p o i n t}}{θ_{b o i l i n g p o i n t} - θ_{f r e e z i n g p o i n t}}$

If they have the same reading on both scales.
$θ_{c e n t i g r a d e} = θ_{f a h r e n h e i t}$
$\frac{θ - 0}{100 - 0} = \frac{θ - 32}{212 - 32}$
i.e. $θ = - 40^{\circ} C$

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