Yes.
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Yes, it is true that the first numbers in two equivalent ratios will always have a common factor. This common factor is known as the scale factor, which is used to create equivalent ratios by multiplying or dividing both parts of the ratio by the same number. This ensures that the ratios maintain the same proportionality.
The ratio is simply 24/15. That fraction (or ratio) can be simplified by dividing both numbers by their highest common factor.
It is simply 6:8. You could, if you wanted to, simplify the ratio by dividing both the numbers by their greatest common factor.
No; depends on the signs of the rational numbers.
The ratio of 60 to 300 can be simplified by dividing both numbers by their greatest common factor, which is 60. This simplifies to a ratio of 1:5. This means that for every 1 unit of 60, there are 5 units of 300.