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 is 24 gm and weight of a 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 the three fruits in terms of weight, that an orphan gets :

Options

A. 90:75:42
B. 180:150:82
C. 75:42:90
D. none of these

Answer

To find the ratio of the weights of bananas, oranges, and grapes that an orphan receives, we can follow these steps:

### 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: Calculate the weight of each fruit
1. **Weight of grapes**: The weight of one grape is given as 24 grams.
2. **Weight of oranges**: The weight of one orange is given as 150 grams.
3. **Weight of bananas**: The weight of bananas and oranges is in the ratio of 4:5. Since the weight of an orange is 150 grams, we can find the weight of a banana.

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

### Step 3: Calculate the total weight of each type of fruit
Now we need to calculate the total weight for each fruit based on the ratio of dozens:
- **Bananas**:
\[ \text{Weight of bananas} = 3 \text{ dozens} \times 12 \text{ bananas/dozen} \times 120 \text{ grams/banana} = 3 \times 12 \times 120 = 4320 \text{ grams} \]

- **Oranges**:
\[ \text{Weight of oranges} = 2 \text{ dozens} \times 12 \text{ oranges/dozen} \times 150 \text{ grams/orange} = 2 \times 12 \times 150 = 3600 \text{ grams} \]

- **Grapes**:
\[ \text{Weight of grapes} = 7 \text{ dozens} \times 12 \text{ grapes/dozen} \times 24 \text{ grams/grape} = 7 \times 12 \times 24 = 2016 \text{ grams} \]

### Step 4: Find the ratio of the weights
Now we have the total weights:
- Bananas: 4320 grams
- Oranges: 3600 grams
- Grapes: 2016 grams

To find the ratio, we can express it as:
\[ \text{Ratio} = 4320 : 3600 : 2016 \]

### Step 5: Simplify the ratio
To simplify the ratio, we can divide each term by the greatest common divisor (GCD).

Calculating the GCD of 4320, 3600, and 2016, we find that:
- GCD = 24

Now we divide each term by 24:
\[ \frac{4320}{24} : \frac{3600}{24} : \frac{2016}{24} = 180 : 150 : 84 \]

### Step 6: Further simplify the ratio
Now, we can simplify this ratio further:
- Divide by 6:
\[ \frac{180}{6} : \frac{150}{6} : \frac{84}{6} = 30 : 25 : 14 \]

Thus, the final ratio of the weights of bananas, oranges, and grapes is:
\[ \text{Final Ratio} = 30 : 25 : 14 \]