Explain how the procedure for changing $\frac{5}{9}$ to $\frac{10}{18}$ requires the use of the multiplicative identity element, 1.
Choose the correct choice below.
A. One must add $\frac{5}{9}$ to 1 to obtain $\frac{10}{18}$.
B. One must multiply the reciprocal of $\frac{5}{9}$ by 1 in the form of a fraction, $\frac{2}{2}$, to obtain $\frac{10}{18}$.
C. One must multiply $\frac{5}{9}$ by 1 in the form of a fraction, $\frac{2}{2}$, to obtain $\frac{10}{18}$.
D. One must divide $\frac{5}{9}$ by the reciprocal of $\frac{10}{18}$ to obtain 1.
E. One must subtract $\frac{5}{9}$ from 1 to obtain $\frac{10}{18}$.
Answer (1)
Recognize that changing 9 5 to 18 10 involves finding an equivalent fraction.
Multiply 9 5 by 2 2 , which is equal to 1, the multiplicative identity.
Perform the multiplication: 9 5 × 2 2 = 18 10 .
Conclude that multiplying by 1 (in the form of 2 2 ) maintains the fraction's value while changing its representation: C .
Explanation
Understanding the Problem The problem asks us to explain how changing the fraction 9 5 to the equivalent fraction 18 10 involves the multiplicative identity element, which is 1. The key idea here is that multiplying any number by 1 doesn't change its value. We need to identify how this principle is applied when converting 9 5 to 18 10 .
Applying the Multiplicative Identity To convert 9 5 to 18 10 , we need to multiply both the numerator and the denominator of 9 5 by the same number. This is equivalent to multiplying the entire fraction by 1, expressed as a fraction. In this case, we multiply by 2 2 , which is equal to 1.
Performing the Multiplication Let's perform the multiplication: 9 5 × 2 2 = 9 × 2 5 × 2 = 18 10 This shows that multiplying 9 5 by 2 2 (which is equal to 1) results in the equivalent fraction 18 10 .
Analyzing the Options Now, let's examine the given options to find the one that correctly describes this process.
A. One must add 9 5 to 1 to obtain 18 10 . (Incorrect, addition is not the operation used) B. One must multiply the reciprocal of 9 5 by 1 in the form of a fraction, 2 2 , to obtain 18 10 . (Incorrect, we multiply 9 5 , not its reciprocal) C. One must multiply 9 5 by 1 in the form of a fraction, 2 2 , to obtain 18 10 . (Correct) D. One must divide 9 5 by the reciprocal of 18 10 to obtain 1. (Incorrect, division is not the primary operation here) E. One must subtract 9 5 from 1 to obtain 18 10 . (Incorrect, subtraction is not the operation used)
Final Answer The correct choice is C. One must multiply 9 5 by 1 in the form of a fraction, 2 2 , to obtain 18 10 .
Examples
When baking, you might need to double a recipe. If a recipe calls for 2 1 cup of flour, doubling the recipe means multiplying 2 1 by 2. You can think of this as 2 1 × 2 2 , which equals 4 2 . So, 2 1 cup is the same as 4 2 cup when you double the recipe. This ensures the proportions remain the same, and your baked goods turn out perfectly!
