Explain how the procedure for changing $\frac{5}{9}$ to $\frac{10}{18}$ requires the use of the multiplicative identity element, 1.
Answer (1)
To change 9 5 to 18 10 , we multiply both the numerator and the denominator by 2.
This is equivalent to multiplying the fraction by 2 2 .
Since 2 2 = 1 , we are using the multiplicative identity.
Therefore, 9 5 = 9 5 × 1 = 9 5 × 2 2 = 18 10 .
Explanation
Understanding the Problem We need to explain how changing the fraction 9 5 to 18 10 involves using the multiplicative identity element, which is 1. The multiplicative identity property states that any number multiplied by 1 remains unchanged.
Transforming the Fraction To transform 9 5 into 18 10 , we multiply both the numerator and the denominator by 2: 9 × 2 5 × 2 = 18 10
Expressing as Multiplication by a Fraction We can express this multiplication by 2 in both the numerator and the denominator as multiplication by the fraction 2 2 . So, we have: 9 5 × 2 2 = 9 × 2 5 × 2 = 18 10
Using the Multiplicative Identity Since 2 2 is equal to 1, we are essentially multiplying 9 5 by 1: 2 2 = 1 Therefore, we can rewrite the transformation as: 9 5 = 9 5 × 1 = 9 5 × 2 2 = 18 10
Conclusion Multiplying by 2 2 is the same as multiplying by 1, which doesn't change the value of the fraction. This demonstrates the use of the multiplicative identity element in changing 9 5 to 18 10 .
Examples
Imagine you are baking a cake and the recipe calls for 2 1 cup of sugar. If you want to double the recipe, you need to multiply the amount of sugar by 2. Instead of directly changing 2 1 to 1, you can think of it as multiplying by 2 2 , which is equal to 1. So, 2 1 × 2 2 = 2 × 2 1 × 2 = 4 2 = 1 cup of sugar. This ensures you maintain the correct proportions while scaling the recipe.
