In the expression 3.2% of 500 multiplied by 2.4% of an unknown number is equal to 312. Find the value of the unknown number.

Introduction / Context:
This question involves a product of two percentage expressions equated to a fixed number. Specifically, 3.2% of 500 multiplied by 2.4% of some unknown quantity results in 312. The task is to find that unknown quantity. This type of problem tests your ability to handle small decimal percentages and to set up a clean algebraic equation involving products.

Given Data / Assumptions:

First factor: 3.2% of 500.

Second factor: 2.4% of an unknown number, say N.

The product of these two factors equals 312.

We must find the value of N.

Concept / Approach:
We use the percentage formula value = (percentage / 100) * base for both percentages. So 3.2% of 500 is (3.2 / 100) * 500, and 2.4% of N is (2.4 / 100) * N. Their product is then set equal to 312. This gives us a single equation in the variable N, which we can solve by isolating N. Careful handling of decimal percentages is important to avoid computational errors.

Step-by-Step Solution:
Let the unknown number be N.3.2% of 500 = (3.2 / 100) * 500.Compute: (3.2 / 100) * 500 = 0.032 * 500 = 16.2.4% of N = (2.4 / 100) * N = 0.024N.We are told that (3.2% of 500) * (2.4% of N) = 312.So, 16 * 0.024N = 312.Compute the coefficient: 16 * 0.024 = 0.384.Therefore, 0.384N = 312.Solve for N: N = 312 / 0.384.Compute: 312 / 0.384 = 812.5.So the unknown number N is 812.5.

Verification / Alternative check:
Substitute N = 812.5 back into the original expression. First factor: 3.2% of 500 = 16 as before. Second factor: 2.4% of 812.5 = 0.024 * 812.5. Compute 0.024 * 812.5 = 19.5. Now the product is 16 * 19.5 = 312. This matches the given product exactly, so N = 812.5 is correct.

Why Other Options Are Wrong:

829.75, 749.85, 799.5 and 780 all fail when substituted back into the equation, because the product 16 * (2.4% of N) does not equal 312.

Only N = 812.5 gives exactly 312 as the product, satisfying the given condition.

Common Pitfalls:
Some students misinterpret the phrase 3.2% of 500 multiplied by 2.4% of N and add the percentages instead of multiplying the resulting values. Others mistakenly divide 312 by 3.2% or 2.4% directly without taking into account the factor of 500. Another common issue is handling decimal multiplication incorrectly. Writing each percentage as a decimal and carefully forming the equation avoids these mistakes.

Final Answer:
The value of the unknown number that satisfies the given percentage product equation is 812.5.