The reciprocals of the squares of the numbers 1 1/2 and 1 1/3 are taken. In what ratio are these reciprocals related?

Introduction / Context:
This question combines fractions, squaring, and reciprocals, and then asks for the ratio between the results. It checks the ability to manipulate mixed numbers, convert them to improper fractions, square them correctly, and then take reciprocals, which is common in quantitative aptitude problems involving algebraic fractions.

Given Data / Assumptions:

• The two given numbers are 1 1/2 and 1 1/3.• We must first find the squares of these numbers.• Then we must take the reciprocals of those squares.• Finally, we must determine the ratio of these two reciprocal values.

Concept / Approach:
To handle mixed numbers, we first convert them into improper fractions. Then we square each fraction by squaring both numerator and denominator. After squaring, we take the reciprocal by inverting numerator and denominator. Finally, the ratio of two fractions is found by forming a fraction of one over the other and simplifying. Working step by step avoids confusion and ensures accurate handling of fractions and powers.

Step-by-Step Solution:
Step 1: Convert the mixed numbers to improper fractions.• 1 1/2 = (2 * 1 + 1) / 2 = 3 / 2.• 1 1/3 = (3 * 1 + 1) / 3 = 4 / 3.Step 2: Square each fraction.• Square of 3 / 2 is (3 / 2)^2 = 9 / 4.• Square of 4 / 3 is (4 / 3)^2 = 16 / 9.Step 3: Take the reciprocal of each squared value.• Reciprocal of 9 / 4 is 4 / 9.• Reciprocal of 16 / 9 is 9 / 16.Step 4: Now find the ratio of these reciprocals: (4 / 9) : (9 / 16).Step 5: Express this ratio as a single fraction and simplify: (4 / 9) / (9 / 16) = (4 / 9) * (16 / 9).Step 6: Multiply numerators and denominators: (4 * 16) / (9 * 9) = 64 / 81.Step 7: Therefore, the ratio of the reciprocals is 64 : 81.

Verification / Alternative check:
We can also think directly in terms of proportionality. Since the squared values are 9 / 4 and 16 / 9, their reciprocals 4 / 9 and 9 / 16 are natural inverses. Writing the ratio as 4 / 9 : 9 / 16 and simplifying through the same multiplication process confirms that no simplification has been overlooked. The final reduced form 64 : 81 is already in simplest form because 64 and 81 have no common factor other than 1. This confirms that the answer is consistent and fully simplified.

Why Other Options Are Wrong:
• 8 : 9: This might arise from confusing squaring with taking square roots or simplifying incorrectly.• 81 : 64: This is the inverse of the correct ratio and would correspond to reversing the order of the two reciprocals.• 9 : 8: This again reflects mixing up the fraction operations and not handling squares and reciprocals step by step.

Common Pitfalls:
Typical errors include failing to convert mixed numbers into improper fractions, squaring only the numerator but not the denominator, or misinterpreting the word reciprocal. Some students also directly square 1.5 and 1.333 without expressing them as fractions, which can introduce rounding errors. Another pitfall is to compute the ratio of the squares instead of the ratio of the reciprocals of the squares. Careful reading and systematic handling of fraction operations is essential.

Final Answer:
The reciprocals of the squares are in the ratio 64 : 81.