Q: Must two quantities have the same dimensions if you are subtracting them?

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Yes they must be in the same units of measurements.

No, it is not.

If they are not on the same dimension then you cannot do this. They will be different numbers and will not go together.

Same.

When adding or subtracting fractions their denominators must be the same

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Yes they must be in the same units of measurements.

No

If you intend 'dimensions' to mean units then whenever the two quantities are to be operated on each other then they must have the 'dimensions', refer to dimensional analysis

No, it is not true.

No, it is not.

If they are not on the same dimension then you cannot do this. They will be different numbers and will not go together.

Efficiency is a ratio of the same quantities. Usually, output power / input power. As the numerator and denominator have the same quantities, the dimensions cancel each other out.

It is called algebra. When two quantities are related as long as you do the same operation to both (add, subtract, multiply, or divide) you do not change the relation

if the signs are the same you must add its opposite.

if they have same units they must have same dimensions . but thy can have different units even if they have same dimensions i hope it helps :

They must have the same dimensions.

The denominators must be the same, the numerators can be different.