Let the two original numbers be represented in terms of a variable based on their given ratio.

Understanding the Original Ratio

The problem states that the two numbers are in the ratio 3 : 5. This means we can represent the numbers as \(3x\) and \(5x\), where \(x\) is a common multiplier.

• First number: \(3x\)
• Second number: \(5x\)

Since \(3x < 5x\) for any positive \(x\), the smaller of the two original numbers is \(3x\).

Applying the Subtraction

According to the question, 4 is subtracted from each of these two numbers. The new numbers become:

• New first number: \(3x - 4\)
• New second number: \(5x - 4\)

Setting Up the Equation from the New Ratio

The problem also states that these new numbers are then in the ratio 4 : 7. We can write this relationship as an equation:

\[ \frac{3x - 4}{5x - 4} = \frac{4}{7} \]

Solving the Equation for \(x\)

To solve for \(x\), we can cross-multiply:

\[ 7(3x - 4) = 4(5x - 4) \] Distribute the numbers on both sides of the equation:

\[ 21x - 28 = 20x - 16 \] Now, collect the \(x\) terms on one side and the constant terms on the other side:

\[ 21x - 20x = -16 + 28 \] Simplify both sides:

\[ x = 12 \]

Finding the Original Numbers

Now that we have the value of \(x\), we can find the original numbers:

• First number: \(3x = 3 \times 12 = 36\)
• Second number: \(5x = 5 \times 12 = 60\)

Identifying the Smaller Original Number

The two original numbers are 36 and 60. The smaller of these two numbers is 36.

Verification

Let's check if these numbers satisfy the conditions given in the problem:

• Original numbers: 36 and 60. Ratio \(36 : 60\). Dividing both by 12, we get \(3 : 5\). This matches the first condition.
• Subtract 4 from each: \(36 - 4 = 32\) and \(60 - 4 = 56\).
• New numbers: 32 and 56. Ratio \(32 : 56\). Dividing both by 8, we get \(4 : 7\). This matches the second condition.

Both conditions are met, so our calculated numbers are correct.

The smaller of the two original numbers is 36.

Revision Table: Ratio Problem Steps

Additional Information: Working with Ratios

A ratio is a comparison of two or more quantities. It shows how much of one quantity there is compared to another. Ratios can be written in several ways, such as \(3:5\), 3 to 5, or \(\frac{3}{5}\).

When solving problems involving ratios of unknown numbers, it's common practice to represent the numbers using a common multiplier, like \(x\). If the ratio is \(a:b\), the numbers can be represented as \(ax\) and \(bx\). This allows you to set up algebraic equations when the relationship between the numbers changes.

Changing the numbers (like adding or subtracting) will change the actual values, but not necessarily the ratio, unless the change is proportional to the original numbers. In this problem, subtracting a constant value (4) from each number changes their relationship, resulting in a new ratio.

Solving ratio problems often involves setting up a proportion (an equation stating that two ratios are equal) and then solving for the unknown variable using cross-multiplication.