Understanding the Age Ratio Problem

This problem asks us to find the ratio of the ages of two individuals, M and N, after 6 years, given their current age ratio and the difference in their current ages.

Setting up the Present Ages

The problem states that the ratio of the present ages of M and N is 5 ∶ 8.

We can represent their present ages using a common multiplier, say \(x\).

• Present age of M = \(5x\) years
• Present age of N = \(8x\) years

Using the Age Difference to Find the Multiplier

We are told that N is 6 years older than M. This means the difference between N's age and M's age is 6 years.

We can write this as an equation:

\( \text{Present age of N} - \text{Present age of M} = 6 \)

\( 8x - 5x = 6 \)

Solving for \(x\)

Now, let's solve the equation to find the value of \(x\):

\( 3x = 6 \)

Divide both sides by 3:

\( x = \frac{6}{3} \)

\( x = 2 \)

Calculating the Present Ages

Now that we have the value of \(x\), we can find the present ages of M and N.

• Present age of M = \(5x = 5 \times 2 = 10\) years
• Present age of N = \(8x = 8 \times 2 = 16\) years

Let's quickly check if N is 6 years older than M: \(16 - 10 = 6\). Yes, this is correct.

Calculating Ages After 6 Years

We need to find the ratio of their ages after 6 years. To do this, we add 6 years to their present ages.

• Age of M after 6 years = Present age of M + 6 = \(10 + 6 = 16\) years
• Age of N after 6 years = Present age of N + 6 = \(16 + 6 = 22\) years

Finding the Ratio After 6 Years

Finally, we find the ratio of the ages of M and N after 6 years.

\( \text{Ratio after 6 years} = \frac{\text{Age of M after 6 years}}{\text{Age of N after 6 years}} \)

\( \text{Ratio} = \frac{16}{22} \)

We can simplify this ratio by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

\( \frac{16 \div 2}{22 \div 2} = \frac{8}{11} \)

So, the ratio of the ages of M and N after 6 years will be 8 ∶ 11.

Revision Table: Solving Age Ratio Problems

Additional Information: Understanding Ratios and Age Word Problems

Age word problems often involve setting up equations based on relationships between ages at different points in time (present, future, or past).

• Ratio Basics: A ratio like \(a:b\) means that for every \(a\) units of the first quantity, there are \(b\) units of the second quantity. We often use a variable (\(x\)) to represent the common part of the ratio, so the quantities become \(ax\) and \(bx\).
• Difference/Sum: Information about the difference or sum of ages at a specific time is crucial for creating an equation to solve for the variable \(x\).
• Future/Past Ages: To find an age in the future, add the number of years to the present age. To find an age in the past, subtract the number of years from the present age.
• Simplifying Ratios: Always simplify the final ratio to its lowest terms by dividing both parts by their greatest common divisor.

Solving these problems systematically by representing ages with variables, setting up equations, and calculating ages at the required time makes them easier to solve.