Understanding the Age Ratio Problem

This problem involves finding the future ratio of ages given their current ratio and the product of their current ages. We are told that the ratio of a father's age to his son's age is 3 ∶ 2. The product of the numbers representing their ages is 486. We need to find the ratio of their ages after 5 years.

Setting up the Equations for Ages

Let the current age of the father be represented by \(3x\) and the current age of the son be represented by \(2x\). This is because their current age ratio is given as 3 ∶ 2, and \(x\) is a common multiplying factor.

We are given that the product of their current ages is 486. So, we can write the equation:

\((3x) \times (2x) = 486\)

Solving for the Unknown Factor \(x\)

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

• Multiply the terms on the left side: \(6x^2 = 486\)
• Divide both sides by 6 to isolate \(x^2\): \(x^2 = \frac{486}{6}\)
• Perform the division: \(x^2 = 81\)
• Take the square root of both sides to find \(x\): \(x = \sqrt{81}\)
• Since age is positive, we take the positive square root: \(x = 9\)

Calculating Current Ages

Now that we have the value of \(x\), we can find the current ages of the father and the son:

• Father's current age = \(3x = 3 \times 9 = 27\) years
• Son's current age = \(2x = 2 \times 9 = 18\) years

Let's quickly check if the product of these ages is 486: \(27 \times 18 = 486\). This confirms our current ages are correct.

Calculating Ages After 5 Years

Next, we need to find their ages after 5 years. We add 5 years to their current ages:

• Father's age after 5 years = Current age + 5 = \(27 + 5 = 32\) years
• Son's age after 5 years = Current age + 5 = \(18 + 5 = 23\) years

Determining the Ratio of Ages After 5 Years

Finally, we need to find the ratio of the father's age to the son's age after 5 years. The ratio is:

Ratio = Father's age after 5 years ∶ Son's age after 5 years

Ratio = 32 ∶ 23

This ratio cannot be simplified further as 32 and 23 have no common factors other than 1.

Conclusion

The ratio of their ages after 5 years will be 32 ∶ 23.

Revision Table: Key Steps in Age Ratio Problems

Additional Information on Ratio and Age Problems

Ratio and age problems are common types of word problems in mathematics. They often involve using ratios to represent relationships between quantities (like ages) and setting up equations based on given conditions (like sums, products, or differences).

• Ratio Basics: A ratio \(a \ratio b\) means that for every \(a\) units of one quantity, there are \(b\) units of another quantity. When solving problems, we often represent these quantities as \(ax\) and \(bx\), where \(x\) is a common factor.
• Age Problem Variations: Problems can ask about ages in the past, present, or future. Pay close attention to whether you need to add or subtract years from the current age.
• Product vs. Sum: Be careful when setting up equations. If the product is given, you multiply the age expressions \((ax)(bx)\). If the sum is given, you add them \((ax + bx)\).

Solving these problems systematically by defining variables, setting up equations, and carefully calculating the required values helps ensure accuracy.