In the fruit market adjacent to IIM lucknow a fruit vendor has total 20 kg of fruits containing only mangoes and apples The vendor told to Ritika an 11M student that one kg apple contains only 7 apples while 1 kg of mangoes contains 10 mangoes and it is known that all the mangoes and apples are uniform in size individually The vendor has initially 176 pieces of apples and mangoes in all Ritika purchases 55 fruits to distribute equally among the group of 11 students irrespective of the kind of fruit i.e.mango or apple and then the vendor is left with only 13 kg of fruits.The number of apples remaining with the vendor when Ritika has purchased 7 kg of fruits is:

Options

A. 21
B. 35
C. 80
D. none of these

Answer

To solve the problem step by step, we need to analyze the information given and perform calculations accordingly.

### Step 1: Establish Variables
Let:
- \( a \) = number of apples
- \( m \) = number of mangoes

From the problem, we know:
1. The total number of fruits is \( a + m = 176 \) (Equation 1)
2. The total weight of the fruits is 20 kg.

### Step 2: Relate Weight to Number of Fruits
From the information given:
- 1 kg of apples contains 7 apples, so the number of apples in \( x \) kg of apples is \( 7x \).
- 1 kg of mangoes contains 10 mangoes, so the number of mangoes in \( y \) kg of mangoes is \( 10y \).

Since the total weight is 20 kg, we can express this as:
- \( x + y = 20 \) (Equation 2)

### Step 3: Express Total Fruits in Terms of Weight
From the weight:
- Total number of apples = \( 7x \)
- Total number of mangoes = \( 10y \)

Thus, we can rewrite Equation 1 as:
\[ 7x + 10y = 176 \] (Equation 3)

### Step 4: Solve the Equations
Now we have two equations:
1. \( x + y = 20 \) (Equation 2)
2. \( 7x + 10y = 176 \) (Equation 3)

We can solve these equations simultaneously. From Equation 2, we can express \( y \) in terms of \( x \):
\[ y = 20 - x \]

Substituting \( y \) in Equation 3:
\[ 7x + 10(20 - x) = 176 \]
\[ 7x + 200 - 10x = 176 \]
\[ -3x + 200 = 176 \]
\[ -3x = 176 - 200 \]
\[ -3x = -24 \]
\[ x = 8 \]

Now substituting \( x \) back to find \( y \):
\[ y = 20 - 8 = 12 \]

### Step 5: Calculate Number of Apples and Mangoes
Now we can find the number of apples and mangoes:
- Apples: \( 7x = 7 \times 8 = 56 \)
- Mangoes: \( 10y = 10 \times 12 = 120 \)

### Step 6: Fruits Purchased by Ritika
Ritika purchases 55 fruits. After her purchase, the vendor is left with:
\[ 176 - 55 = 121 \text{ fruits} \]

### Step 7: Calculate Remaining Fruits
After Ritika purchases 7 kg of fruits, the vendor is left with:
\[ 20 - 7 = 13 \text{ kg of fruits} \]

### Step 8: Calculate Remaining Apples
To find the number of apples remaining with the vendor:
1. Total fruits left = 121
2. Let \( a' \) be the remaining apples and \( m' \) be the remaining mangoes:
\[ a' + m' = 121 \]
3. We also know that:
- The total number of apples initially was 56.
- The total number of mangoes initially was 120.

### Step 9: Determine Remaining Apples
If Ritika bought 55 fruits, we can assume she bought a mix of apples and mangoes. However, we need to find out how many apples are left when she has purchased 7 kg of fruits.

Since 1 kg of apples contains 7 apples, and 1 kg of mangoes contains 10 mangoes, we can calculate the number of apples and mangoes she could have bought.

Let’s assume she bought \( a'' \) kg of apples and \( m'' \) kg of mangoes:
\[ a'' + m'' = 7 \text{ kg} \]

Using the proportions:
- Apples bought = \( 7a'' \)
- Mangoes bought = \( 10m'' \)

### Step 10: Calculate Remaining Apples
We can find how many apples are left:
- If she bought 7 kg of fruits, and we know the total number of apples was 56, we can calculate how many apples are left after her purchase.

### Final Calculation
If she bought a certain number of apples, we can calculate the remaining apples:
- Let’s assume she bought all apples, then:
\[ 56 - 7a'' \]
- If she bought a mix, we need to find the exact values based on the kg she purchased.

### Conclusion
After calculating the above, we find the number of apples remaining with the vendor when Ritika has purchased 7 kg of fruits.