Solving Ratio Problems with Increased Numbers

The question asks us to find the sum of two original numbers given their initial ratio and a new ratio formed after increasing each number by a fixed value.

Understanding the Initial Ratio

We are told that the ratio between two numbers is 2:3. This means we can represent the two numbers using a common multiplier. Let the common multiplier be $x$.

• The first number can be represented as $2x$.
• The second number can be represented as $3x$.

Here, $x$ must be a non-zero value. Since we are dealing with numbers that will be increased, $x$ is likely a positive value in this context.

Forming the New Ratio After Increasing the Numbers

Each original number is increased by 2.

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

The ratio of these new numbers is given as 3:4.

So, we can write the equation based on this new ratio:

\(\frac{\text{New first number}}{\text{New second number}} = \frac{3}{4}\)

\(\frac{2x + 2}{3x + 2} = \frac{3}{4}\)

Solving the Equation for the Multiplier $x$

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

\(4(2x + 2) = 3(3x + 2)\)

Distribute the numbers on both sides of the equation:

\(8x + 8 = 9x + 6\)

Now, we need to isolate the term with $x$. Subtract $8x$ from both sides:

\(8 = 9x - 8x + 6\)

\(8 = x + 6\)

Subtract 6 from both sides to find the value of $x$:

\(8 - 6 = x\)

\(2 = x\)

So, the common multiplier $x$ is 2.

Calculating the Original Numbers

Now that we have the value of $x$, we can find the original numbers:

• Original first number = $2x = 2(2) = 4$.
• Original second number = $3x = 3(2) = 6$.

Let's verify if these numbers satisfy the initial condition. The ratio of 4 to 6 is \(\frac{4}{6} = \frac{2}{3}\), which matches the given ratio 2:3.

Finding the Sum of the Original Numbers

The question asks for the sum of the original numbers. The original numbers are 4 and 6.

Sum = Original first number + Original second number

Sum = \(4 + 6 = 10\)

Step-by-Step Solution to the Ratio Problem

1. Represent the original numbers using the initial ratio 2:3 as $2x$ and $3x$.
2. Form expressions for the numbers after increasing each by 2: $2x + 2$ and $3x + 2$.
3. Set up an equation using the new ratio 3:4: \(\frac{2x + 2}{3x + 2} = \frac{3}{4}\).
4. Solve the equation for $x$ by cross-multiplying and simplifying.
   • \(4(2x + 2) = 3(3x + 2)\)
   • \(8x + 8 = 9x + 6\)
   • \(8 - 6 = 9x - 8x\)
   • \(2 = x\)
5. Substitute the value of $x=2$ back into the original expressions to find the original numbers: $2(2) = 4$ and $3(2) = 6$.
6. Calculate the sum of the original numbers: $4 + 6 = 10$.

Revision Table: Key Concepts for Ratio Problems

Additional Information on Ratios

A ratio is a way to show the relationship between two or more quantities. It tells us how much of one quantity there is compared to another. Ratios can be simplified like fractions by dividing all parts of the ratio by their greatest common divisor.

When solving problems involving ratios where quantities are increased or decreased by a fixed amount, it's often useful to represent the quantities using a common variable (like $x$ in this problem). This allows you to set up an algebraic equation based on the new ratio, which can then be solved to find the value of the variable and subsequently the original quantities.

Ratio problems often appear in various contexts, including mixing ingredients, scaling recipes, comparing speeds, and dividing amounts proportionally.