6th grade Q2

A proportion states that two ratios are equal.

Example:

Let's see how you can verify that two ratios equal. Consider two rooms filled with cats and dogs. In one room, there are 3 cats and 4 dogs. The ratio of cats to dogs is  $3:4$ or $\frac{3}{4}$.

In the second room, there are 9 cats and 12 dogs. The ratio of cats to dogs here is $9:12$ or $\frac{9}{12}$.

The ratio $\frac{9}{12}$ can be simplified by dividing the numerator and the denominator by the common factor of 3, $\frac{9}{12}=\frac{9÷3}{12÷3}=\frac{3}{4}$, which matches the ratio in the first room.

This confirms that the ratio of cats to dogs is the same in both rooms, thereby establishing a proportion:

$\frac{3}{4}=\frac{9}{12}$

Cross products are equal:

An important property of proportions is that the cross products are equal.

To find the cross products, you multiply the numbers diagonally across from each other:

• Multiply the top number of the first ratio by the bottom number of the second ratio. So, you multiply 3 by 12.
• Then, multiply the bottom number of the first ratio by the top number of the second ratio. So, multiply 4 by 9.

For the given proportion $\frac{3}{4}=\frac{9}{12}$, the cross products are: $3\times 12=4\times 9$. Both calculations result in 36, verifying that the proportion holds.

Verifying proportions:

Verifying Proportions:

So there are two methods you can use to check if two ratios form a proportion:

1. Simplify the fractions: Reduce each fraction to its simplest form. If they simplify to the same ratio, then they are proportional.
2. Cross product check: Calculate the cross products as described. If they are equal, the ratios are proportional.