If P = 9695 × 97, Q = 9796 × 98 and R = 197, then which of the following order relations between P, Q and R is correct?

Introduction / Context:
This question tests your ability to compare large numbers without necessarily computing them in full detail. You are given two large products, P and Q, and a smaller number R, and you must decide which inequality correctly describes their order from smallest to largest. Such questions are common in quantitative and verbal reasoning because they check both number sense and careful reading of symbolic expressions.

Given Data / Assumptions:

• P is defined as 9695 × 97.
• Q is defined as 9796 × 98.
• R is defined as 197.
• All three quantities are positive integers.
• We must determine which inequality among the options correctly orders R, P and Q.

Concept / Approach:
The first observation is that P and Q are products of numbers close to 10,000 and 100, so they will be much larger than the two digit number 197. Therefore R will obviously be the smallest. The main comparison that matters is between P and Q. Because 9796 and 98 are each slightly larger than 9695 and 97 respectively, we expect Q to be greater than P. We can support this intuition by either computing the products exactly or by using a careful approximate difference argument to confirm that Q exceeds P.

Step-by-Step Solution:
First compare R with P and Q. R = 197. The product 9695 × 97 is almost 9700 × 100, which is close to 970,000, so P is in the hundreds of thousands. Similarly, 9796 × 98 is also in the hundreds of thousands. Therefore, R is far smaller than both P and Q. So R must come first in any correct inequality. Now compare P and Q numerically. Compute P exactly: P = 9695 × 97 = 9695 × (100 - 3) = 9695 × 100 - 9695 × 3 = 969,500 - 29,085 = 940,415. Compute Q exactly: Q = 9796 × 98 = 9796 × (100 - 2) = 9796 × 100 - 9796 × 2 = 979,600 - 19,592 = 960,008. Compare P and Q: 960,008 is clearly greater than 940,415. Therefore Q > P. Combining these facts, we have R < P and P < Q, which together give the final order R < P < Q.

Verification / Alternative check:
If you prefer an approximate check without full multiplication, note that both products are of the form (about 9700) × (about 100). The pair for Q is slightly larger in both factors: 9796 > 9695 and 98 > 97. When two positive factors increase, their product increases. This guarantees that Q is larger than P. Because 197 is much smaller than 900,000, it cannot compete with these products. Any inequality that places R above P or Q must be wrong. Only the pattern R < P < Q is consistent with both intuition and the exact calculations given above.

Why Other Options Are Wrong:
Option A, P < Q < R, incorrectly places R as the largest number, which is impossible since R is only 197. Option B, R < Q < P, correctly puts R first but wrongly claims Q < P despite Q being computed as 960,008 and P as 940,415. Option C, Q < P < R, is wrong in both directions, claiming Q is smallest and R largest. Only option D matches the computed order R < P < Q.

Common Pitfalls:
A common mistake is to be intimidated by the large numbers and to guess based purely on the fact that 9695 is slightly smaller than 9796 without checking both factors carefully. Another typical error is forgetting that the tiny number R must be the smallest and carelessly choosing an inequality where R appears at the end. When working with large products, comparing relative sizes of factors and using simple algebraic rearrangements can help avoid full, time consuming calculations.

Final Answer:
The correct ordering of the three numbers is R < P < Q, so option D is the correct choice.