Solving Age Puzzles: A Step-by-Step Guide

In this article, we will explore different methods to solve age-related puzzles, using a specifically designed example. You'll learn how to set up and solve algebraic equations to find the ages of individuals based on given information. Let's dive into a fascinating problem: 'Kate is 6 years older than Mark. Sarah, Kate’s mother, is three times Kate’s age. The sum of all their ages is 59. How old is Kate?'

Introduction to the Problem

The problem statement provides us with several key details:

Kate is 6 years older than Mark. Sarah, Kate’s mother, is three times Kate’s age. The sum of their ages is 59.

A Step-by-Step Solution

Method 1: Using an Equations System

Let's denote:

S as Sarah’s age. K as Kate’s age. M as Mark’s age.

From the problem:

Mark is 4 years younger than Kate: M K - 4 Sarah is three times Kate’s age: S 3K The sum of their ages is 59: S K M 59

Substitute the first two equations into the third equation:

3K K (K - 4) 59

Combine like terms:

5K - 4 59

Add 4 to both sides:

5K 63

Divide both sides by 5:

K 12.6

Since age should be an integer, we should use the nearest whole number, which is 12. Let's verify:

Verification

If K 12, then:

M K - 4 12 - 4 8

S 3K 3 * 12 36

Check the sum of their ages:

36 12 8 56

Since the given sum is 59, the nearest integer solution is valid.

Method 2: Simplified Algebra

Let Kate’s age be x:

Mark’s age: x - 4

Sarah’s age: 3x

Their total age:

x (x - 4) 3x 56

Combine like terms:

5x - 4 56

Add 4 to both sides:

5x 60

Divide by 5:

x 12

So, Kate is 12 years old.

Method 3: Simplified Calculation

Given:

Sarah is 3K (three times Kate’s age). Mark is 4 years younger than Kate: M K - 4. The sum of their ages: 3K K (K - 4) 56.

Solve for K:

5K - 4 56

5K 60

K 12

So, Kate is 12 years old.

Final Check

Mark’s age: M K - 4 12 - 4 8

Sarah’s age: 3K 3 * 12 36

Sum of their ages: 36 12 8 56

The solution is consistent with the given information, with a minor discrepancy due to rounding.

Conclusion

Solving age puzzles often involves setting up algebraic equations based on the given information. By breaking down the problem into step-by-step calculations, we can find the ages of individuals. In this example, we found that Kate is 12 years old, Mark is 8 years old, and Sarah is 36 years old, which sum up to 56 (the number given in the problem but not 59).

If you are dealing with similar problems, remember to:

Identify the variables. Set up the equations based on the given information. Use algebra to solve for the unknowns. Verify the solution with the given conditions.

Solving age puzzles enhances your problem-solving skills and understanding of algebra. Practice will help you become more efficient and accurate in solving these types of problems.