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A ratio is a way to compare two numbers or quantities. It shows how much of one thing there is compared to another. We often write ratios using a colon, like this: 14:21. This means for every 14 of something, there are 21 of something else. We can also write ratios as fractions, like \(\frac{14}{21}\). That way, we can simplify them easily using division.

A simple example of a ratio would be comparing the number of apples to oranges in a basket. If you have 3 apples and 2 oranges, the ratio of apples to oranges would be 3:2. This tells you that for every 3 apples, there are 2 oranges.

The greatest common divisor (GCD) is the largest number that can divide two numbers without leaving a remainder. It helps us to simplify fractions and ratios. For example, the GCD of 14 and 21 is 7 because 7 is the largest number that both 14 and 21 can be divided by.

To find the GCD of two numbers, you can use a few methods like:

• Listing all factors of each number and choosing the largest common one.
• Using the Euclidean algorithm, which involves repeated division.
• Prime factorization, which means breaking down numbers into their prime factors and identifying the common factors.

Simplifying ratios is easier when you understand GCD. For instance, for the ratio 48:72, we find that the GCD is 24. This helps us simplify the ratio to \(\frac{48}{72} = \frac{2}{3}\) by dividing both numbers by 24.

Simplifying fractions means reducing them to their simplest form. This happens when both the numerator (top number) and the denominator (bottom number) of the fraction are divided by their greatest common divisor (GCD).

For example, if we want to simplify \(\frac{14}{21}\), we find the GCD of 14 and 21, which is 7. Then we divide both the numerator and the denominator by 7, giving us \(\frac{2}{3}\). This process makes the fraction simpler and easier to understand.

To check if two ratios are in proportion, we simplify them and compare. For instance, simplifying both 14:21 and 48:72 gives us \(\frac{2}{3}\). Since both ratios simplify to the same fraction, they are in proportion. This means the two ratios are equivalent, conveying the same relationship between their numbers.