Ratio Problem Solving: Step-by-Step Explanation

This problem involves working with ratios and how they change when numbers are modified. We start with two numbers related by a given ratio, apply some changes, and then calculate a new ratio after further changes.

Representing the Numbers with a Ratio

We are told that two numbers are in the ratio 2 : 3. This means we can represent the numbers using a common multiplier, let's call it $x$.

• First number = $2x$
• Second number = $3x$

Setting Up the Equation from the Second Condition

The problem states that if 5 is subtracted from the first number and 6 is added to the second number, the new ratio becomes 5 : 12.

• New first number = $2x - 5$
• New second number = $3x + 6$

The ratio of these new numbers is $(2x - 5) : (3x + 6)$, which is equal to 5 : 12. We can write this as an equation:

$$\frac{2x - 5}{3x + 6} = \frac{5}{12}$$

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

$$12 \times (2x - 5) = 5 \times (3x + 6)$$

$$24x - 60 = 15x + 30$$

Solving for the Unknown Value

Now, we solve the equation for $x$:

Subtract $15x$ from both sides:

$$24x - 15x - 60 = 30$$

$$9x - 60 = 30$$

Add 60 to both sides:

$$9x = 30 + 60$$

$$9x = 90$$

Divide by 9:

$$x = \frac{90}{9}$$

$$x = 10$$

Finding the Original Numbers

Using the value of $x = 10$, we can find the original numbers:

• First number = $2x = 2 \times 10 = 20$
• Second number = $3x = 3 \times 10 = 30$

We can check this: the original ratio is 20 : 30, which simplifies to 2 : 3. If we subtract 5 from the first (20 - 5 = 15) and add 6 to the second (30 + 6 = 36), the new ratio is 15 : 36. Dividing both by 3 gives 5 : 12, which matches the condition.

Calculating the Final Ratio

The question asks what the ratio would become when eight is added to each number. The original numbers are 20 and 30.

• First number + 8 = $20 + 8 = 28$
• Second number + 8 = $30 + 8 = 38$

The new ratio is $28 : 38$.

To simplify this ratio, we need to find the greatest common divisor (GCD) of 28 and 38. Both numbers are divisible by 2.

• $28 \div 2 = 14$
• $38 \div 2 = 19$

So, the simplified ratio is 14 : 19.

Summary of Steps

Revision Table: Key Ratio Concepts

Understanding ratios is fundamental to solving problems like this. Here's a quick review:

• Ratio Definition: A ratio compares two or more quantities. It shows the relative sizes of the quantities.
• Writing Ratios: Ratios can be written as $a:b$, $a/b$, or "a to b".
• Simplifying Ratios: Ratios are simplified by dividing all terms by their greatest common divisor (GCD), similar to simplifying fractions. For example, 20:30 simplifies to 2:3.
• Using a Common Multiplier: If numbers are in the ratio $a:b$, they can be represented as $ax$ and $bx$, where $x$ is a common factor. This is often useful in algebraic problems.

Additional Information: Solving Ratio Word Problems

Ratio word problems often require setting up an equation. Here are some tips:

• Identify the Ratio: Clearly identify the ratios given in the problem.
• Use Variables: Represent the unknown quantities using variables, often by multiplying the ratio terms by a common variable (like $x$).
• Formulate Equations: Translate the word problem's conditions into algebraic equations. Pay close attention to what operations (addition, subtraction, multiplication, division) are performed on the numbers and how they affect the ratio.
• Solve the Equation: Use algebraic techniques to solve for the variable.
• Find the Actual Numbers: Substitute the value of the variable back into your initial representations to find the actual values of the numbers.
• Answer the Specific Question: Make sure you answer what the problem specifically asks for, which might be the original numbers, modified numbers, or a new ratio.
• Check Your Work: Plug your calculated numbers back into the original conditions to see if they hold true.

This systematic approach helps break down complex ratio problems into manageable steps.