Problems on Ages
Problems on ages form one of the friendliest chapters of arithmetic, because the situation is always the same: the ages of a few people at one point of time, and how those ages change as the years pass. Every person's age grows at the same rate, one year for every year, and this single fact is the whole key to the chapter. The method followed in the Master Book: Maths Masti (D. Hurina) is to take the present age of the youngest or the first person as x, to write every other age in terms of x, and to translate the sentence of the question into a single equation. Once the equation is set, the rest is ordinary algebra. Because the questions are framed in words, the real work lies in reading the statement slowly and fixing the time at which each age is quoted.
Basic Terms
- Present age: the age of a person at the current time.
- Past age: the age some years ago, found by subtracting those years.
- Future age: the age after some years, found by adding those years.
- Age ratio: one age divided by another at a stated time.
- Sum of ages: the total of the ages of all the persons.
- Difference of ages: the gap between two ages, which never changes with time.
Two ideas must be held firmly. Every age changes by the same number of years, and the difference between two ages remains constant for ever. These two facts solve most of the questions in the chapter.
The Core Idea of Ages
If a person is x years old now, then that person was x minus n years old n years ago, and will be x plus n years old n years later. When two people are compared, both ages move together, so the difference between them does not change. Writing all the ages of a question in terms of a single unknown is the standard first step, and it turns a word problem into an equation.
Ages in the Past and Future
Statements such as five years ago or ten years hence tell the candidate which way to move. If the present age is x, then the past age is x minus the given years and the future age is x plus the given years. A common mistake is to add the years to one person and not to the other; in a question about two people, the same number of years must be applied to both.
Age Ratios and Their Change
Age ratios are usually given at a past, present or future time. When a ratio is given, one may write the ages as multiples of a common factor, for example three to four as three k and four k, and then use the second condition to find k. The ratio of ages changes with time even though the difference does not, and questions often give two ratios at two different times to build two equations.
Ages of a Family or a Group
Some questions give the sum of the ages of a family and then add or remove a member, or state the relation between the parents and the children. Here the candidate must keep the total and the number of persons separate, and remember that when a member leaves or joins, both the total and the count change. The same method of a single unknown works, but the equation is formed from the sum rather than from a ratio.
Average Age and Age Sums
Average age is the total of the ages divided by the number of persons. If the average age of a group rises or falls after a new member joins, the change in the total age reveals the age of the new member. Questions that mix average with ages are solved by finding the two totals, before and after, and taking their difference.
How This Chapter Appears in the Examination
In ADRE Grade IV, APSC and Assam Police papers the age questions usually give a relation between the ages of two or three persons, sometimes at present and sometimes with a past or future condition, and ask for one person's age. Other forms give a ratio that changes over time, or the sum of the ages of a family. The questions are always solvable with one unknown and one equation.
Shortcut Methods for Rapid Calculation
Start from the person whose age is asked, unless the question clearly points to another. Use the constant difference whenever two ratios are given at two times. Check the answer by substituting it back into the original sentence. For questions involving a family, add or subtract the years for every member together, not one by one.
Common Mistakes to Avoid
- Applying the number of years to only one person instead of all.
- Subtracting instead of adding when the question speaks of a future age.
- Treating a past age as a present age.
- Forgetting that the difference of two ages never changes.
- Using the wrong person for the unknown and making the equation complicated.
How to Prepare This Topic
- Fix the habit of taking the present age of one person as x.
- Write every other age in terms of x before forming the equation.
- Practise past, present and future conditions in the same question.
- Work on two-ratio questions using the constant difference.
- Finish with family and average-age problems, and always verify by substitution.
Frequently Asked Questions
What is the key fact in age problems?
Every age increases by the same number of years, and the difference between two ages never changes. Both facts together turn the word problem into a simple equation.
How do I handle a ratio of ages at two different times?
Write the ages at the first time as multiples of a common factor, move them forward or backward by the given years, and equate the result to the second ratio. The constant difference is often the quickest route.
Which age should I choose as x?
Choose the age that the question asks for, or the youngest person, because the other ages then become simple expressions in x and the equation stays short.
Can the difference between two ages change?
No. As long as both persons are alive, both grow by one year each year, so the difference between their ages remains exactly the same at every time.
How is average age used in these questions?
The average age is the total age divided by the number of persons. When a member joins or leaves, compare the total age before and after; the change gives the new or the departed member's age.