In Maa Yatri Temple each orphan receives bananas, oranges and grapes in the ratio 3 : 2 : 7 in terms of dozens. The weight of one grape is 24 g, the weights of a banana and an orange are in the ratio 4 : 5, and the weight of one orange is 150 g. What is the ratio of the total weight of bananas, oranges and grapes an orphan receives?

Introduction / Context:
This problem combines ratio, unit conversion and basic arithmetic in a context of fruit distribution to orphans. Although the initial distribution is given in dozens, the question finally asks for a ratio in terms of total weight of each fruit type. Converting counts to weights using the given mass of each fruit and then simplifying the resulting ratio is the central task.

Given Data / Assumptions:

• The number of dozens of bananas, oranges and grapes per orphan is 3 : 2 : 7.
• One dozen equals 12 individual fruits.
• The weight of one grape is 24 g.
• The weight of one banana and one orange are in the ratio 4 : 5.
• The weight of one orange is 150 g.
• We assume all fruits are of uniform size and weight.

Concept / Approach:
The basic approach is to convert the ratios in dozens into actual counts of fruits, then multiply by the weight of each single fruit to obtain total mass. Banana weight is found using the given ratio with the orange. Once total weights for bananas, oranges and grapes are computed in grams, we form a three term ratio and simplify it by dividing through by a common factor to obtain the simplest integer ratio.

Step-by-Step Solution:
Bananas: 3 dozens means 3 * 12 = 36 bananas. Oranges: 2 dozens means 2 * 12 = 24 oranges. Grapes: 7 dozens means 7 * 12 = 84 grapes. Weight of one orange is 150 g. Banana to orange weight ratio is 4 : 5, so banana weight = (4 / 5) * 150 = 120 g. Weight of one grape is 24 g. Total banana weight = 36 * 120 = 4320 g. Total orange weight = 24 * 150 = 3600 g. Total grape weight = 84 * 24 = 2016 g. Weight ratio = 4320 : 3600 : 2016. Divide all three numbers by 48 to simplify: 4320 / 48 = 90, 3600 / 48 = 75, 2016 / 48 = 42. Therefore, the simple ratio is 90 : 75 : 42.

Verification / Alternative check:
We can verify by trying another common factor, such as 24. Dividing 4320, 3600 and 2016 by 24 gives 180, 150 and 84. Dividing these again by 2 returns 90, 75 and 42, confirming that the simplified ratio is consistent. There is no larger common factor that can reduce 90, 75 and 42 further while keeping all terms integral, so this is the simplest ratio.

Why Other Options Are Wrong:
The option 180 : 150 : 82 retains a wrong grape weight and does not correspond to correct multiplication. The ratio 75 : 42 : 90 mixes the order of fruits. Options 45 : 30 : 21 and 60 : 50 : 28 are other arbitrary reductions or misinterpretations of dozens or weights and do not match the given numeric data when calculated carefully.

Common Pitfalls:
Common mistakes include forgetting that each dozen corresponds to 12 fruits, misapplying the ratio of weights between bananas and oranges, or mixing up which numbers correspond to bananas, oranges and grapes when forming the final ratio. Another frequent slip is failing to fully simplify the ratio to its lowest terms.

Final Answer:
The ratio of the total weight of bananas, oranges and grapes that each orphan receives is 90 : 75 : 42.