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1.Assume that you construct a thermometer using gallium as the fluid instead of mercury, and that you define the melting point of gallium as 0 °G and the boiling point of gallium as 1000 °G. What is the melting point of sodium chloride (801 °C) on the gallium scale?

First we determine the ratio of $^{\circ}\mathrm G/^{\circ}\mathrm C$ by dividing the melting points. This states that $0.4599\ ^{\circ}\mathrm G=1\ ^{\circ}\mathrm C$.

$$\dfrac{1000\ ^{\circ}\mathrm G-0\ ^{\circ}\mathrm G}{2204\ ^{\circ}\mathrm C-29.78\ ^{\circ}\mathrm C}=\dfrac{1000\ ^{\circ}\mathrm G}{2174.22\ ^{\circ}\mathrm C}=0.4599\ ^{\circ}\mathrm G/^{\circ}\mathrm C$$

To calculate the boiling point, this is done: $$T=0.4599\ ^{\circ}\mathrm G/^{\circ}\mathrm C\times\left(801\ ^{\circ}\mathrm C-29.78\ ^{\circ}\mathrm C\right)=355\ ^{\circ}\mathrm G$$

My question is, why are we subtracting by $29.78\ ^{\circ}\mathrm C$?

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Unlike an absolute temperature scale like degrees Kelvin, Celsius and your hypothetical °G are relative scales, where the zero points are set to the melting points of water and gallium, respectively. In your first step, you've accounted for the difference in the size of one degree Celsius and one °G, but not the difference in the zero points.

Instead of sodium chloride, think about what the melting point of gallium is on each scale. You've defined it as 0 °G, but you know it's not 0 °C. What do you have to do to correct for this to get a value in Celsius?

Incidentally, this is also why the conversion between Celsius and Fahrenheit is not just multiplying by a constant.

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It zeroes things out. It is analogous to the use of "32" in the equation to convert Fahrenheit to Celsius, where "32" represents the melting point of ice on the Fahrenheit scale. In your G scale you must subtract the melting point of gallium in the Centigrade scale in order to get 0G at the melting point of gallium $$\ce{^{o}C = (^{o}F-32)*(5/9)}$$ $$\ce{^{o}G = (^{o}C-29$.$78)*($0.4599$)}$$

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