Understanding Temperature Scales: Celsius and Fahrenheit

This question asks us to find a specific temperature where the number representing the temperature in Fahrenheit is exactly double the number representing it in Celsius. To solve this, we need the formula that converts Celsius to Fahrenheit.

The standard formula for converting a temperature from Celsius ($C$) to Fahrenheit ($F$) is:

$F = \frac{9}{5}C + 32$

Setting Up the Equation for the Specific Condition

The problem states a condition: the numerical value on the Fahrenheit scale is exactly twice the numerical value on the Celsius scale. We can write this relationship as an equation:

$F = 2C$

Solving for the Temperature in Celsius

Now we have two equations:

1. $F = \frac{9}{5}C + 32$
2. $F = 2C$

We can solve for $C$ by substituting the second equation into the first one. This means we replace $F$ in the first equation with $2C$:

$2C = \frac{9}{5}C + 32$

To solve for $C$, we need to get all the terms involving $C$ on one side of the equation. Let's subtract $\frac{9}{5}C$ from both sides:

$2C - \frac{9}{5}C = 32$

To subtract these terms, we need a common denominator, which is 5. We can rewrite $2C$ as $\frac{10}{5}C$:

$\frac{10}{5}C - \frac{9}{5}C = 32$

Now, subtract the fractions:

$\frac{1}{5}C = 32$

Finally, to isolate $C$, multiply both sides by 5:

$C = 32 \times 5$

$C = 160$

So, the temperature is 160 degrees Celsius.

Verifying the Fahrenheit Value

Let's check if this temperature satisfies the condition. If $C = 160$, then the Fahrenheit value should be twice this, which is $2 \times 160 = 320$.

Now, let's use the conversion formula to see what $160^\circ C$ is in Fahrenheit:

$F = \frac{9}{5}C + 32$

$F = \frac{9}{5}(160) + 32$

$F = 9 \times (160 \div 5) + 32$

$F = 9 \times 32 + 32$

$F = 288 + 32$

$F = 320$

The calculated Fahrenheit value is 320, which is indeed twice the Celsius value of 160. Thus, the condition is met at $160 ^\circ C$.

Conclusion

The temperature at which the numerical value on the Fahrenheit scale is exactly twice the numerical value on the Celsius scale is $160 ^\circ C$.