Understanding Multiplication Value Equivalence

The question asks us to find which option is equivalent in value to the product $6893.5 \times 603.3 \times 0.3019$. This requires understanding how shifting decimal points in the factors affects the overall product and how to balance these shifts to maintain the original value.

Analyzing the Original Expression

Let the original expression be represented by E:

E = 6893.5 \times 603.3 \times 0.3019

To find an equivalent expression, any increase in the value of a factor (by moving the decimal point to the right) must be offset by a corresponding decrease in the value of another factor (by moving the decimal point to the left), and vice versa, such that the overall product remains the same.

Evaluating Each Option Against the Original Expression

Option 1: $6.8935 \times 6.033 \times 301.9$

• Comparing $6893.5$ to $6.8935$: The decimal point moved 3 places to the left, which is equivalent to dividing by $10^3$.
• Comparing $603.3$ to $6.033$: The decimal point moved 2 places to the left, equivalent to dividing by $10^2$.
• Comparing $0.3019$ to $301.9$: The decimal point moved 3 places to the right, equivalent to multiplying by $10^3$.
• The combined effect is a change factor of $\frac{1}{10^3} \times \frac{1}{10^2} \times 10^3 = \frac{1}{10^2} = 10^{-2}$.
• Therefore, Option 1's value is $E \times 10^{-2}$, which is not equal to E.

Option 2: $68.935 \times 6033 \times 3.019$

• Comparing $6893.5$ to $68.935$: The decimal point moved 2 places to the left, equivalent to dividing by $10^2$.
• Comparing $603.3$ to $6033$: The decimal point moved 1 place to the right, equivalent to multiplying by $10^1$.
• Comparing $0.3019$ to $3.019$: The decimal point moved 1 place to the right, equivalent to multiplying by $10^1$.
• The combined effect is a change factor of $\frac{1}{10^2} \times 10^1 \times 10^1 = 10^{-2 + 1 + 1} = 10^0 = 1$.
• Therefore, Option 2's value is $E \times 1$, which is equal to E.

Option 3: $6.8935 \times 60.33 \times 30.19$

• Comparing $6893.5$ to $6.8935$: The decimal point moved 3 places to the left, equivalent to dividing by $10^3$.
• Comparing $603.3$ to $60.33$: The decimal point moved 1 place to the left, equivalent to dividing by $10^1$.
• Comparing $0.3019$ to $30.19$: The decimal point moved 2 places to the right, equivalent to multiplying by $10^2$.
• The combined effect is a change factor of $\frac{1}{10^3} \times \frac{1}{10^1} \times 10^2 = 10^{-3 - 1 + 2} = 10^{-2}$.
• Therefore, Option 3's value is $E \times 10^{-2}$, which is not equal to E.

Option 4: $689.35 \times 603.3 \times 301.9$

• Comparing $6893.5$ to $689.35$: The decimal point moved 1 place to the left, equivalent to dividing by $10^1$.
• Comparing $603.3$ to $603.3$: No change, equivalent to multiplying by $10^0$.
• Comparing $0.3019$ to $301.9$: The decimal point moved 3 places to the right, equivalent to multiplying by $10^3$.
• The combined effect is a change factor of $\frac{1}{10^1} \times 10^0 \times 10^3 = 10^{-1 + 0 + 3} = 10^2$.
• Therefore, Option 4's value is $E \times 10^2$, which is not equal to E.

Conclusion on Value Equivalence

After analyzing the shifts in decimal places for each factor in all the options, only Option 2 resulted in a net change factor of $10^0 = 1$. This means Option 2 is the only expression that is mathematically equivalent to the original expression $6893.5 \times 603.3 \times 0.3019$.