Solving Age Proportional Problems in Proportions and Ages

Understanding and solving age-related problems that are presented in proportions is a valuable skill in both mathematics and real-life scenarios. This article explains how to solve a classic age problem using algebra and proportions. We will apply the steps to find the present ages of three individuals whose ages are in a 4:7:9 ratio, given specific conditions from the past.

Problem Statement

The present ages of three persons are in proportions 4:7:9. Six years ago, the sum of their ages was 56. The task is to find their current ages.

Setting Up the Equation

Let the present ages of the three persons be represented as:

4x 7x 9x

where x is a common multiplier.

Ages Six Years Ago

Six years ago, their ages would have been:

First person: 4x - 6 Second person: 7x - 6 Third person: 9x - 6

Solving the Equation

The sum of their ages six years ago can be expressed as:

[ (4x - 6) (7x - 6) (9x - 6) 56 ]

This simplifies to:

[4x 7x 9x - 18 56]

Solve the equation step by step:

[2 - 18 56][2 74][x frac{74}{20} 3.7]

Finding Their Present Ages

Using the value of x, we can find the present ages:

First person: 4x 4 times 3.7 14.8 approx; 15 years Second person: 7x 7 times 3.7 25.9 approx; 26 years Third person: 9x 9 times 3.7 33.3 approx; 33 years

Conclusion

The present ages of the three persons are approximately:

First person: 15 years Second person: 26 years Third person: 33 years

Summary

Person Presentation Age (Years) First Person 15 Second Person 26 Third Person 33

Note: For whole numbers, round these to 15, 26, and 33 respectively.

Alternative Methods

Here are a couple of alternative methods, although the primary solution is more straightforward:

Method 1: Let the present ages be a, b, c. The total is s abc. Eight years ago the total was 56 years. Present total must be s 56 3 times 8 80. a:b:c:s ~ 4:7:9:20. c/s 9/20. c 9/20s 8/20 times 80 36 years. Method 2: Let present ages are 4x 7x 9x. As per the question: 4x-8 7x-8 9x-8 56. 2x 80. x 4. So, present ages are 16, 28, 36. Method 3: Let ages be 4x, 7x, and 9x. 8 years ago sum of age was 56 then 4x-8 7x-8 9x-856. 4x 7x 9x-2456. 2x56-2432, then x32/201.6. Eldest person age 9x9times1.614.4 years. Method 4: Eight years age the sum of theirs ages 51. At present sum of their ages 51 3 times 8 75. 3p 5p 7p75. 15p75. p5. Ans - present age 3p15, 5p25, 7p35. Method 5: Let the current age be 3x, 5x, 7x. Sum of age was 51 8 years ago. So present age is 51 - 8 times 3 75. 3x 5x 7x 75. X5. Present age is 15, 25, 35.

Note in these methods: Similar to the initial solution, we can find the values for x and then determine their present ages.