In Maa Yatri Temple every devotee offers fruits to the orphans. Thus every orphan receives bananas, oranges and grapes in the ratio of `3:2:7` in terms of dozen. But the weight of a grape in 24 gm and weight ofa banana and an orange are in the ratio of 4:5 while the weight of an orange is 150 gm. Find the ratio of all three fruits in terms of weight, that an orphan gets:

Options

A. a. `90:75:42`
B. b. `180:150:82`
C. c. `75:42:90`
D. d. None of these

Answer

To solve the problem, we need to find the ratio of the weights of bananas, oranges, and grapes that an orphan receives, given the ratios of the fruits in dozens and their respective weights. Here’s how we can approach this step-by-step:

### Step 1: Understand the given ratios
The ratio of bananas, oranges, and grapes is given as \(3:2:7\) in terms of dozens. This means for every 3 dozens of bananas, there are 2 dozens of oranges and 7 dozens of grapes.

### Step 2: Convert dozens to actual quantities
Let's denote the number of dozens of each fruit:
- Bananas = \(3x\)
- Oranges = \(2x\)
- Grapes = \(7x\)

### Step 3: Determine the weight of each fruit
We know:
- The weight of a grape = 24 gm
- The weight of an orange = 150 gm
- The weight of a banana and an orange are in the ratio \(4:5\).

Since the weight of the orange is 150 gm, we can find the weight of the banana.

### Step 4: Calculate the weight of the banana
Let the weight of the banana be \(b\). According to the ratio:
\[ \frac{b}{150} = \frac{4}{5} \]
Cross-multiplying gives:
\[ 5b = 4 \times 150 \]
\[ 5b = 600 \]
\[ b = \frac{600}{5} = 120 \text{ gm} \]

### Step 5: Calculate the total weight of each fruit
Now we can calculate the total weight of each fruit based on the number of dozens and their respective weights:
- Total weight of bananas = \(3x \times 120\) gm
- Total weight of oranges = \(2x \times 150\) gm
- Total weight of grapes = \(7x \times 24\) gm

### Step 6: Express total weights
Calculating each:
- Total weight of bananas = \(360x\) gm
- Total weight of oranges = \(300x\) gm
- Total weight of grapes = \(168x\) gm

### Step 7: Find the ratio of the weights
Now we can express the weights in ratio form:
\[ \text{Weight ratio} = 360x : 300x : 168x \]
We can simplify this by dividing each term by \(x\):
\[ 360 : 300 : 168 \]

### Step 8: Simplify the ratio
To simplify, we can divide each term by the greatest common divisor (GCD). The GCD of 360, 300, and 168 is 12.
\[ \frac{360}{12} : \frac{300}{12} : \frac{168}{12} = 30 : 25 : 14 \]

### Final Answer
The ratio of the weights of bananas, oranges, and grapes that an orphan receives is:
\[ \text{Ratio} = 30 : 25 : 14 \]