If Shubham's father bought 3 oranges, 4 apples, and 5 bananas for Rs. 62, and his sister Pinki bought 4 oranges, 5 apples, and 4 bananas for Rs. 72,then what amount (in Rs.) did Bill pay for 1 orange, 2 apples, and 7 bananas?

Let's denote the price of one orange, one apple, and one banana by \( x \), \( y \), and \( z \) respectively.

From the problem, we have the following two equations based on the purchases made by Shubham's father and his sister Pinki:

1. \( 3x + 4y + 5z = 62 \)
2. \( 4x + 5y + 4z = 72 \)

We need to find the amount Bill paid for 1 orange, 2 apples, and 7 bananas, i.e., \( x + 2y + 7z \).

### Step 1: Solve the system of equations

First, we'll multiply the first equation by 4 and the second equation by 3 to make the coefficients of \( x \) in both equations the same.

\[ 4(3x + 4y + 5z) = 4 \times 62 \]
\[ 12x + 16y + 20z = 248 \]

\[ 3(4x + 5y + 4z) = 3 \times 72 \]
\[ 12x + 15y + 12z = 216 \]

Next, we subtract the second new equation from the first new equation to eliminate \( x \):

\[ (12x + 16y + 20z) - (12x + 15y + 12z) = 248 - 216 \]
\[ 12x + 16y + 20z - 12x - 15y - 12z = 32 \]
\[ y + 8z = 32 \quad \text{(Equation 3)} \]

### Step 2: Express \( y \) in terms of \( z \) from Equation 3

\[ y = 32 - 8z \]

### Step 3: Substitute \( y = 32 - 8z \) into one of the original equations

Substituting into \( 3x + 4y + 5z = 62 \):

\[ 3x + 4(32 - 8z) + 5z = 62 \]
\[ 3x + 128 - 32z + 5z = 62 \]
\[ 3x - 27z = 62 - 128 \]
\[ 3x - 27z = -66 \]
\[ x - 9z = -22 \]
\[ x = 9z - 22 \quad \text{(Equation 4)} \]

### Step 4: Substitute \( y = 32 - 8z \) and \( x = 9z - 22 \) into \( x + 2y + 7z \)

\[ x + 2y + 7z = (9z - 22) + 2(32 - 8z) + 7z \]
\[ = 9z - 22 + 64 - 16z + 7z \]
\[ = 9z + 7z - 16z - 22 + 64 \]
\[ = 0z + 42 \]
\[ = 42 \]

Therefore, Bill paid \( Rs. 42 \) for 1 orange, 2 apples, and 7 bananas.

Do you want more details or have any questions?

Here are 8 related questions for further exploration:
1. How much does one apple cost?
2. What is the cost of 10 oranges?
3. How much did Pinki pay for her 4 bananas?
4. What would be the total cost if Shubham's father bought 10 bananas instead of 5?
5. What is the cost of one banana?
6. How many apples can be bought for Rs. 24?
7. If the price of apples doubled, how much would Bill pay for 1 orange, 2 apples, and 7 bananas?
8. What would be the new total if Bill bought 3 oranges, 2 apples, and 1 banana?

**Tip:** To solve systems of linear equations, methods such as substitution, elimination, and matrix operations (like Gaussian elimination) are very useful.

Learn how to calculate the amount Bill paid for 1 orange, 2 apples, and 7 bananas using a system of linear equations. This problem involves solving equations based on the purchases made by Shubham's father and his sister Pinki. Suitable for students in Grades 8-10.

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