Five years ago the age of the son was one third of that of his mother at that time. If the sum of their present ages is 70 years, then find the present age of the mother.
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Solving Age Word Problems: Finding Present Ages
This problem involves finding the present ages of a mother and her son based on given conditions about their ages five years ago and the sum of their present ages.
Understanding the Given Conditions
We are provided with two key pieces of information:
- Five years ago, the son's age was one third of the mother's age at that time.
- The sum of their present ages is 70 years.
Our goal is to determine the present age of the mother.
Setting Up Equations for Ages
Let's use variables to represent their present ages:
- Let the present age of the mother be \$M$ years.
- Let the present age of the son be \$S$ years.
Now, let's translate the given conditions into mathematical equations:
-
Condition 1: Ages five years ago
Five years ago, the mother's age was \$M - 5$ years.
Five years ago, the son's age was \$S - 5$ years.
According to the condition, the son's age five years ago was one third of the mother's age five years ago.
Equation 1: \$S - 5 = \frac{1}{3}(M - 5)$ -
Condition 2: Sum of present ages
The sum of their present ages is 70 years.
Equation 2: \$M + S = 70$
Solving the System of Equations to Find Mother's Age
We now have a system of two linear equations with two variables:
1. \$S - 5 = \frac{1}{3}(M - 5)$
2. \$M + S = 70$
We can solve this system using substitution. From Equation 2, we can express \$S$ in terms of \$M$:
\$S = 70 - M$
Now, substitute this expression for \$S$ into Equation 1:
\$(70 - M) - 5 = \frac{1}{3}(M - 5)$
Simplify and solve for \$M$:
\$65 - M = \frac{1}{3}(M - 5)$
Multiply both sides by 3 to eliminate the fraction:
\$3(65 - M) = 3 \times \frac{1}{3}(M - 5)$
\$195 - 3M = M - 5$
Gather the terms with \$M$ on one side and the constants on the other:
\$195 + 5 = M + 3M$
\$200 = 4M$
Divide by 4 to find the value of \$M$:
\$M = \frac{200}{4}$
\$M = 50$
So, the present age of the mother is 50 years.
Finding Son's Present Age (Optional Check)
We can also find the son's present age using Equation 2:
\$S = 70 - M$
\$S = 70 - 50$
\$S = 20$
The son's present age is 20 years.
Verification
Let's check if these ages satisfy the first condition (ages five years ago):
- Five years ago, mother's age was \$50 - 5 = 45$ years.
- Five years ago, son's age was \$20 - 5 = 15$ years.
Is the son's age (15) one third of the mother's age (45) at that time?
\$\frac{1}{3} \times 45 = 15$
Yes, it is. The values satisfy both conditions.
The present age of the mother is 50 years.
| Person | Present Age | Age Five Years Ago |
|---|---|---|
| Mother | \$M = 50$ | \$M - 5 = 45$ |
| Son | \$S = 20$ | \$S - 5 = 15$ |
Revision Table: Key Steps in Age Problems
| Step | Description | Action |
|---|---|---|
| 1 | Identify unknowns | Assign variables (e.g., \$M$, \$S$) to present ages. |
| 2 | Express ages at different times | Write expressions for ages in the past or future based on variables (e.g., \$M-5$, \$S+10$). |
| 3 | Formulate equations | Translate problem conditions into algebraic equations. |
| 4 | Solve the system | Use substitution or elimination to find the value(s) of the variable(s). |
| 5 | Verify the solution | Check if the calculated ages satisfy all original conditions. |
Additional Information: Solving Age Problems
Age problems are a common type of word problem in algebra. They typically involve relationships between the ages of different people at different points in time (past, present, or future).
- Always define variables for the present ages, as this is usually the most straightforward starting point.
- Carefully read the conditions related to past or future ages. An age 'x years ago' is the present age minus x, and an age 'y years from now' is the present age plus y.
- Set up as many equations as there are unknown variables using the given information.
- Systematically solve the equations. Substitution is often useful when one variable can be easily expressed in terms of another.
- Ratio conditions (like "one third of") translate into multiplication or division in equations.
- Sum or difference conditions translate into addition or subtraction equations.
- Always perform a final check to ensure your calculated ages fit all the original statements in the problem. This helps catch arithmetic errors.
Important Questions from Problem on Age
The ratio of present ages of Neelam and Rajni is 6 : 7. After four years, their ages will be in the ratio of 8 : 9. What is the present age of Rajni?
Jaya is 36 years old and her son Bharat is 11 years old. In how many years will Jaya be twice as Bharat's age?
Five years ago, father's age was 5 times that of his son and after 3 years he will be 3 times as old as his son. Find the present age of son.
The ratio of a father's age to his son's age is 3 ∶ 2 The product of the numbers representing their age is 486. The ratio of their ages after 5 years will be:
X said to Y, "At the time of your birth I was twice as old as you are at present." If the present age of X is 42 years, then consider the following statements:
1. 8 years ago, the age of X was five times the age of Y.
2. After 14 years, the age of X would be two times the age of Y.
Which of the above statements is/are correct?
