Solving a Ratio and Number Word Problem

This problem involves two numbers whose relationship is defined first by a ratio and then by conditions applied after modifying the numbers. We need to find the value of the larger number.

Defining the Numbers Based on the Ratio

The ratio of the two numbers is given as 9 : 5. This means that for some common factor, let's call it \(x\), the numbers can be represented as:

• Larger number = \(9x\)
• Smaller number = \(5x\)

Since the ratio is 9:5, \(9x\) is clearly the larger number and \(5x\) is the smaller number (assuming \(x\) is a positive value, which it must be for a ratio of actual numbers unless specified otherwise).

Setting Up the Equation from the Given Conditions

According to the problem, modifications are made to these numbers:

• 8 is added to the larger number: The new larger number is \(9x + 8\).
• 4 is subtracted from the smaller number: The new smaller number is \(5x - 4\).

The relationship between these new numbers is that the new greater number becomes twice the new smaller number. This can be written as an equation:

\( \text{New larger number} = 2 \times \text{New smaller number} \)

Substituting our expressions for the new numbers, we get:

\( 9x + 8 = 2(5x - 4) \)

Solving the Equation to Find the Value of x

Now, we solve the equation for \(x\):

\( 9x + 8 = 2(5x - 4) \)

First, distribute the 2 on the right side:

\( 9x + 8 = 10x - 8 \)

Next, we want to get all the \(x\) terms on one side and the constant terms on the other. Subtract \(9x\) from both sides:

\( 8 = 10x - 9x - 8 \)

\( 8 = x - 8 \)

Now, add 8 to both sides to isolate \(x\):

\( 8 + 8 = x \)

\( 16 = x \)

So, the common factor \(x\) is 16.

Finding the Larger Number

The larger number was defined as \(9x\). Now that we know \(x = 16\), we can calculate the larger number:

\( \text{Larger number} = 9 \times x \)

\( \text{Larger number} = 9 \times 16 \)

\( \text{Larger number} = 144 \)

Verification

Let's check if these numbers satisfy the conditions:

• Original numbers: 144 (larger) and 80 (smaller). Ratio is 144:80, which simplifies to 9:5. (144/16 = 9, 80/16 = 5)
• New larger number: 144 + 8 = 152
• New smaller number: 80 - 4 = 76
• Is the new larger number twice the new smaller number? \(152 = 2 \times 76\)? Yes, \(152 = 152\).

The conditions are satisfied, and the larger number is 144.

Revision Table: Key Steps in Ratio Problems

Additional Information: Understanding Ratios and Equations

A ratio is a comparison of two quantities. When we say the ratio of two numbers is 9:5, it means the first number is \(\frac{9}{5}\) times the second number. Using a variable \(x\) helps us represent the actual values of the numbers while maintaining their ratio. \(x\) acts as a common factor that has been scaled up or down.

Word problems often translate real-world situations into mathematical equations. The key is to carefully read the problem, identify the unknown quantities, represent them using variables, and then write an equation that describes the relationship given in the problem. Solving the equation gives the value of the variable, which can then be used to find the required quantities.

In this problem, the condition "the greater number becomes twice the smaller number" after the modifications is crucial for setting up the equation. It directly translates to an equality between the new larger number and two times the new smaller number.