cross multiplication prove a proportion

Cross Multiplication in Proportions

• Definition of Proportion: A proportion is an equation that states that two ratios are equal.

  • Example: The proportion ( \frac{3}{4} = \frac{x}{8} ) states that the ratio of 3 to 4 is equal to the ratio of x to 8.

• Cross Multiplication Method: Cross multiplication is a technique used to show that two ratios are equal by multiplying the numerator of one ratio by the denominator of the other ratio and vice versa.

Solving for a Missing Value

• Given Proportion: ( \frac{3}{4} = \frac{x}{8} )

Step-by-Step Solution:

1. Multiply to Find Constant Product:

   • First Multiplication: Multiply the numerator of the first ratio by the denominator of the second ratio:

     • Calculation: ( 3 \times 8 = 24 )

2. Set Up the Equation for Missing Value:

   • Next, since ( 4 \times x ) must also equal the product obtained from step 1, write:

     • Equation: ( 4 \times x = 24 )

3. Solve for x:

   • To isolate x, divide both sides of the equation by 4:

     • Calculation: ( x = \frac{24}{4} = 6 )

Conclusion of the Calculation:

• We find that ( x = 6 ). Therefore, the complete proportion is now:

  • Result: ( \frac{3}{4} = \frac{6}{8} )

Verification of the Solution

• Checking Work Using Cross Multiplication: To confirm that the solution is correct, we can use cross multiplication to ensure both products are equal:

  1. First Product:

  • Calculation: ( 3 \times 8 = 24 )

  1. Second Product:

  • Calculation: ( 4 \times 6 = 24 )

• Since both products equal 24, we have verified the proportion:

  • Conclusion: The equation ( \frac{3}{4} = \frac{6}{8} ) is true, confirming our solution for x is correct.