Final answer:
The smallest number to multiply both sides of the equation 5/16x - 9/4 = 7/12 to clear fractions is 16, not listed in the options. The least common multiple of the denominators is 48, but the common factor of 16 suffices to clear all the fractions.
Step-by-step explanation:
To find the smallest number to clear the equation 5/16x - 9/4 = 7/12 of fractions, we need to determine the least common multiple (LCM) of the denominators (16, 4, and 12). The prime factorization of the denominators yields 16 = 24, 4 = 22, and 12 = 22 × 3. We take the highest powers of the primes which gives us 24 × 3 = 48. However, because the equation involves the denominator 16, which is already a multiple of 4, and 12, which is a multiple of 4 as well, we can actually use a smaller number which is common to all which is 16 itself. The number 16 is a multiple of 4 and also evenly divides 12 (16 is 4 times 4, and 12 is 3 times 4), hence it would clear all the fractions when multiplied with each term of the equation.
Therefore, option (C) 32 is not the smallest multiple, and option (A) 4 is too small to clear the fraction with a denominator of 16. Option (B) 12 is not enough to clear the fraction with a denominator of 16, so the correct answer is, indeed, (D) none of these, because the actual smallest number to use is 16. Thus, when you are simplifying equations, it is important to look for the LCM, but also consider the specific numbers in the equation to find the smallest multiple that works for all terms.