Solving a Ratio Problem: Number Division and Ratio Change

This problem involves a number that is initially divided into two parts according to a specific ratio. We are told how this ratio changes when a constant value is added to each part. We need to find the greater of the original two parts.

Setting up the Initial Ratio

Let the number be divided into two parts in the ratio $3:2$. This means the two parts can be represented as $3x$ and $2x$, where $x$ is a common multiplier. The original number itself would be the sum of these parts, $3x + 2x = 5x$.

The two original numbers (parts) are $3x$ and $2x$.

Applying the Change and the New Ratio

According to the problem, 8 was added to each of these two numbers.

• The first number becomes $3x + 8$.
• The second number becomes $2x + 8$.

After adding 8, the ratio of the new numbers changed to $7:5$. We can write this as an equation:

$$\frac{3x + 8}{2x + 8} = \frac{7}{5}$$

Solving for the Unknown Variable

To solve for $x$, we can cross-multiply the equation:

$$5 \times (3x + 8) = 7 \times (2x + 8)$$

Distribute the numbers on both sides of the equation:

$$15x + 40 = 14x + 56$$

Now, we need to isolate the term with $x$ on one side. Subtract $14x$ from both sides:

$$15x - 14x + 40 = 14x - 14x + 56$$

$$x + 40 = 56$$

Subtract 40 from both sides to find the value of $x$:

$$x + 40 - 40 = 56 - 40$$

$$x = 16$$

Finding the Original Numbers

Now that we have the value of $x$, we can find the original two numbers which were $3x$ and $2x$.

• First number = $3x = 3 \times 16 = 48$
• Second number = $2x = 2 \times 16 = 32$

The original number that was divided is $48 + 32 = 80$. We can check the original ratio: $48:32$. Dividing both by 16 gives $3:2$, which matches the problem statement.

Let's also check the new ratio after adding 8:

• First number + 8 = $48 + 8 = 56$
• Second number + 8 = $32 + 8 = 40$

The new ratio is $56:40$. Dividing both by 8 gives $7:5$, which also matches the problem statement.

Identifying the Greater Number

The question asks for the greater of the two original numbers. The two original numbers are 48 and 32.

Comparing them, $48 > 32$.

So, the greater of the two numbers is 48.

Revision Table: Ratio Problem Steps

Additional Information: Understanding Ratios

A ratio is a way to compare two or more quantities. It shows how much of one quantity there is compared to another. Ratios are often written using a colon (:) or as a fraction.

• A ratio $a:b$ means that for every unit of $a$, there are $b$ units of the other quantity.
• If a quantity is divided in the ratio $a:b$, the total number of parts is $a+b$. The first part is $\frac{a}{a+b}$ of the total, and the second part is $\frac{b}{a+b}$ of the total.
• When solving ratio problems involving unknown quantities, using a variable like $x$ to represent a common multiplier is a standard approach. For a ratio $a:b$, the quantities can be written as $ax$ and $bx$.
• Adding or subtracting a value from the quantities changes their values and consequently changes their ratio, unless the value added/subtracted is proportional to the original quantities in a specific way.
• Ratio problems often require setting up algebraic equations based on the given information and then solving for the unknown variable.

Understanding how to work with ratios and solve equations is fundamental for many quantitative problems.