Finding Numbers from Ratio Changes

This problem involves finding two unknown numbers based on how their ratio changes when a constant value is added to each. We start by representing the numbers using a variable based on their initial ratio.

Setting up the Ratio Problem

The two numbers are initially in the ratio 5 ∶ 7. This means we can represent the numbers as $5x$ and $7x$, where $x$ is a common multiplier.

According to the problem, if 6 is added to each number, the new ratio becomes 3 ∶ 4.

The new numbers will be:

• First number: $5x + 6$
• Second number: $7x + 6$

The ratio of these new numbers is $\frac{5x + 6}{7x + 6}$. We are told this new ratio is 3 ∶ 4, which can be written as $\frac{3}{4}$.

Formulating and Solving the Equation

We can set up an equation based on the new ratio:

\(\frac{5x + 6}{7x + 6} = \frac{3}{4}\)

To solve for $x$, we use cross-multiplication:

\(4 \times (5x + 6) = 3 \times (7x + 6)\)

Distribute the numbers on both sides of the equation:

\(20x + 24 = 21x + 18\)

Now, we need to isolate $x$. Subtract $20x$ from both sides:

\(24 = 21x - 20x + 18\)

\(24 = x + 18\)

Subtract 18 from both sides to find the value of $x$:

\(24 - 18 = x\)

\(6 = x\)

So, the common multiplier $x$ is 6.

Calculating the Original Numbers

The original numbers were represented as $5x$ and $7x$. Now that we know $x = 6$, we can find the numbers:

• First number: $5x = 5 \times 6 = 30$
• Second number: $7x = 7 \times 6 = 42$

The original numbers are 30 and 42.

Verifying the Solution

Let's check if these numbers satisfy the second condition. If we add 6 to each number:

• New first number: $30 + 6 = 36$
• New second number: $42 + 6 = 48$

The new ratio is $\frac{36}{48}$. We can simplify this fraction:

\(\frac{36}{48} = \frac{12 \times 3}{12 \times 4} = \frac{3}{4}\)

The new ratio is indeed 3 ∶ 4, which matches the problem statement. Therefore, the numbers 30 and 42 are the correct solution.

Reviewing the Options

Let's quickly look at the given options:

Only the numbers 30 and 42 satisfy both conditions stated in the problem.

Revision Table: Ratio Problems

Additional Information: Working with Ratios and Equations

Ratio and proportion problems are common in mathematics. They often require setting up an algebraic equation based on the given information.

• When a quantity is added or subtracted from numbers in a ratio, the ratio changes.
• Representing the numbers as $ax$ and $bx$ is a key step in solving these types of problems, as it maintains the initial ratio while allowing for changes.
• Cross-multiplication is a fundamental technique for solving equations where a fraction is equal to another fraction. Ensure you distribute any multiplied terms correctly.
• Always verify your final answer by plugging the found numbers back into the original problem statement to ensure all conditions are met.

Understanding how to translate word problems about ratios into algebraic equations is crucial for success in this topic.