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In mathematics, a ratio is a way to compare two quantities. You can think of it as describing how much of one thing there is compared to another thing. For example, if you have 7 apples and 9 oranges, the ratio of apples to oranges is 7 to 9, written as \( \frac{7}{9} \).

Ratios help us understand relationships. The order in which we write the quantities matters. In this case, 7 comes first because we're comparing apples to oranges, not the other way around.

A ratio can be simplified by dividing both quantities by their greatest common divisor. This helps in using ratios in further calculations. For instance, the ratio \( \frac{35}{45} \) can be simplified by dividing both by 5, resulting in \( \frac{7}{9} \) . Hence, 7:9 and 35:45 are equivalent ratios.

A proportion is an equation stating that two ratios are equal. For example, \( \frac{7}{9} = \frac{35}{45} \) is a proportion because the ratios on both sides of the equation are equivalent.

To check if a proportion is true, you can cross-multiply. Cross-multiplication involves multiplying the numerator of one ratio by the denominator of the other ratio. If both products are equal, the proportion is correct.

For our example, we would cross-multiply 7 and 45 and then 9 and 35. This means:

• 7 \( \times \) 45 = 315
• 9 \( \times \) 35 = 315

Since both products are equal (315), we have verified that \( \frac{7}{9} = \frac{35}{45} \) is a true proportion.

Cross-multiplication is a straightforward method often used to verify whether two ratios form a proportion. It involves multiplying across the 'equals' sign in a proportion to compare the products.

Here's a step-by-step guide for cross-multiplication:

• Write your proportion: \( \frac{a}{b} = \frac{c}{d} \)
• Multiply the numerator of the first ratio by the denominator of the second ratio: \( a \cdot d \)
• Multiply the numerator of the second ratio by the denominator of the first ratio: \( b \cdot c \)
• Compare the two products: if \( a \cdot d = b \cdot c \), then \( \frac{a}{b} = \frac{c}{d} \) is a true proportion.

Using cross-multiplication for the example \( \frac{7}{9} = \frac{35}{45} \):

• 7 \( \times \) 45 = 315
• 9 \( \times \) 35 = 315

Both products are 315, so the proportion is verified as correct.