Solve the proportional equation with decimals: 3420 ÷ 19 = ( ? ÷ 0.01 ) × 7. Find the exact value of ?. Show each simplification step clearly.

Introduction / Context:
This item assesses your ability to manipulate equations involving decimals and fractions. Understanding how division by a small decimal (such as 0.01) scales a number is key to solving for the unknown quickly and accurately.

Given Data / Assumptions:

• 3420 ÷ 19 = ( ? ÷ 0.01 ) × 7
• All arithmetic is exact; avoid premature rounding.
• We must determine the exact rational value for ?.

Concept / Approach:
Note that dividing by 0.01 is the same as multiplying by 100. Therefore ( ? ÷ 0.01 ) × 7 equals 700? . Reduce the equation to a simple linear form and solve for ?. Then compare with the options provided.

Step-by-Step Solution:
( ? ÷ 0.01 ) × 7 = 700 ?Set 3420 ÷ 19 = 700 ?.Then ? = (3420 ÷ 19) ÷ 700 = 3420 / (19 × 700).Compute 3420/700 = 171/35.Hence ? = (171/35) ÷ 19 = 171 / (35 × 19) = 9/35.

Verification / Alternative check:
Evaluate RHS with ? = 9/35: ( (9/35) ÷ 0.01 ) × 7 = ( (9/35) × 100 ) × 7 = (900/35) × 7 = 900/5 = 180. LHS 3420/19 = 180. The equality is satisfied.

Why Other Options Are Wrong:

• 35/9, 18/7, 63/5, 7/18: None equals 9/35, so they do not balance the equation.
• None of these: Correct because the required value is 9/35, which is not listed explicitly.

Common Pitfalls:
Forgetting that ÷0.01 multiplies by 100; arithmetic slips when simplifying 3420/700; or not fully reducing the fraction before comparing with options.

Final Answer:
None of these