Solving proportions is an essential skill in mathematics that allows us to compare relationships between different quantities. While solving proportions with only one unknown is relatively straightforward, what happens when there are two unknowns involved? In this step-by-step guide, we will explore how to solve proportions with two unknowns, ensuring a clear understanding of the process.

1. Identify the Proportion

The first step is to identify the proportion to be solved. Look for the presence of two ratios separated by an equal sign, such as:

  • 3/4 = x/6
  • 2a/5 = 6/15

2. Cross-Multiply

To solve proportions with two unknowns, we need to cross-multiply. Multiply the numerator of the first ratio with the denominator of the second ratio and vice versa. This step will help us eliminate the fractions and create an equation to solve. Using the previous examples, we get:

  • 3 * 6 = 4 * x, or 18 = 4x
  • 2a * 15 = 5 * 6, or 30a = 30

3. Solve for the Unknowns

After cross-multiplying, we have simplified the proportion to a simple algebraic equation. Now, the next step is to solve for the unknown(s). In our examples, we can solve as follows:

  • 18 = 4x ⟹ Divide both sides by 4 ⟹ x = 4.5
  • 30a = 30 ⟹ Divide both sides by 30 ⟹ a = 1

4. Check the Solution

Once we have obtained the values for the unknowns, it is important to validate our solution. Substitute the found values back into the original proportion to ensure both sides are equal:

  • 3/4 = 4.5/6 ⟹ Cross-multiply ⟹ 3 * 6 = 4 * 4.5, or 18 = 18 ⟹ The solution is correct.
  • 2a/5 = 6/15 ⟹ Cross-multiply ⟹ 2 * 15 = 5 * 6, or 30 = 30 ⟹ The solution is correct.

Solving proportions with two unknowns may appear challenging initially, but by following these step-by-step guidelines, we can tackle such problems with confidence. Remember to identify the proportion, cross-multiply to eliminate fractions, solve the resulting equation, and check the solution for accuracy. Mastering proportions with two unknowns is a valuable tool to understand the relationships between variables in mathematical problems.

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