Solving Ratio Problems: Finding the Amount Difference

This problem involves dividing an amount between two friends based on a specific ratio (4:7) and an additional condition. We need to determine the exact difference in the amounts they receive.

Understanding the Ratio

The ratio 4:7 tells us how the total amount is split. Let the amounts received by the two friends be \(4x\) and \(7x\), respectively. Here, '\(x\)' is a common factor representing one unit of the ratio. This means:

• Amount for the first friend = \(4x\)
• Amount for the second friend = \(7x\)

Applying the Condition

The question states that the second friend's share is ₹6 more than the first friend's share. We can express this relationship as an equation:

Second friend's amount = First friend's amount + ₹6

Substituting the ratio representations:

\(7x = 4x + 6\)

Calculating the Value of the Common Factor 'x'

To find the specific amounts, we first need to solve the equation for '\(x\)'.

Subtract \(4x\) from both sides of the equation:

\(7x - 4x = 6\)

\(3x = 6\)

Now, divide both sides by 3 to find the value of '\(x\)':

\(x = \frac{6}{3}\)

\(x = 2\)

This means each unit of the ratio is equivalent to ₹2.

Determining the Difference in Amounts

The question asks for the difference in the amount between the two friends. We can find this using the value of '\(x\)' we just calculated.

Method 1: Calculating Individual Shares

Calculate the exact amount each friend gets:

• First friend's share = \(4x = 4 \times 2 = ₹8\)
• Second friend's share = \(7x = 7 \times 2 = ₹14\)

The difference is:

Difference = \(₹14 - ₹8 = ₹6\)

Method 2: Using the Ratio Difference Directly

The difference in the ratio parts is \(7 - 4 = 3\) parts.

Since each part ('\(x\)') is equal to ₹2, the total difference in amount is:

Difference = (Difference in ratio parts) \(\times x\)

Difference = \(3 \times ₹2 = ₹6\)

Final Answer

Both calculation methods confirm that the difference in the amount between the two friends is ₹6.