The average ages of parents and two children are 30 years and 8 years respectively. The average age of the family is A. 16 years B. 19 years C. 18 years D. 17 years
B
Calculating the Average Age of a Family
This problem requires us to calculate the average age of an entire family, given the average ages of different groups within the family: parents and children.
Understanding Average
The average (or mean) of a set of numbers is calculated by summing up all the numbers and dividing by the count of numbers. The formula is:
$$\text{Average} = \frac{\text{Sum of values}}{\text{Number of values}}$$
Step-by-Step Calculation
We are given the average age for two groups within the family:
- Parents: Average age is 30 years. There are 2 parents.
- Children: Average age is 8 years. There are 2 children.
To find the average age of the whole family, we first need to find the total age of all family members combined.
1. Calculate the Total Age of Parents
The total age of the parents is the average age multiplied by the number of parents.
$$\text{Total age of parents} = \text{Average age of parents} \times \text{Number of parents}$$
$$\text{Total age of parents} = 30 \text{ years} \times 2 = 60 \text{ years}$$
2. Calculate the Total Age of Children
Similarly, the total age of the children is the average age multiplied by the number of children.
$$\text{Total age of children} = \text{Average age of children} \times \text{Number of children}$$
$$\text{Total age of children} = 8 \text{ years} \times 2 = 16 \text{ years}$$
3. Calculate the Total Age of the Family
The total age of the family is the sum of the total age of parents and the total age of children.
$$\text{Total age of family} = \text{Total age of parents} + \text{Total age of children}$$
$$\text{Total age of family} = 60 \text{ years} + 16 \text{ years} = 76 \text{ years}$$
4. Determine the Total Number of Family Members
The total number of family members is the sum of the number of parents and the number of children.
$$\text{Total number of family members} = \text{Number of parents} + \text{Number of children}$$
$$\text{Total number of family members} = 2 + 2 = 4$$
5. Calculate the Average Age of the Family
Now, we can calculate the average age of the family using the total age and the total number of family members.
$$\text{Average age of family} = \frac{\text{Total age of family}}{\text{Total number of family members}}$$
$$\text{Average age of family} = \frac{76 \text{ years}}{4} = 19 \text{ years}$$
The average age of the family is 19 years.
Matching with Options
Let's compare our calculated average age with the given options:
- A. 16 years
- B. 19 years
- C. 18 years
- D. 17 years
Our calculated average age of 19 years matches Option B.
| Group | Number of Members | Average Age | Total Age |
|---|---|---|---|
| Parents | 2 | 30 years | $30 \times 2 = 60$ years |
| Children | 2 | 8 years | $8 \times 2 = 16$ years |
| Total Family | 4 | Calculation Needed | $60 + 16 = 76$ years |
$$\text{Average Family Age} = \frac{\text{Total Family Age}}{\text{Total Number of Family Members}} = \frac{76}{4} = 19 \text{ years}$$
Conclusion
The average age of the family, consisting of parents and two children, is 19 years.
Revision Table: Key Concepts for Average Age Problems
| Concept | Definition/Formula | Application in this Problem |
|---|---|---|
| Average (Mean) | Sum of values divided by number of values: $$\frac{\sum x}{n}$$ | Used to find average family age from total age and total members. |
| Total Sum from Average | Sum = Average $$\times$$ Number of values | Used to find total age of parents and total age of children. |
| Combining Groups | To find average of combined groups, sum total values of all groups and divide by total number of members in all groups. | Used to combine total ages of parents and children to get total family age. |
Additional Information: Weighted Average
While we solved this problem by finding the total sum and count, it's related to the concept of a weighted average. A weighted average is used when different values contribute differently to the final average, based on their 'weight' or frequency. In this case, the number of people in each group acts as the weight.
The formula for a weighted average of two groups is:
$$\text{Weighted Average} = \frac{(w_1 \times A_1) + (w_2 \times A_2)}{w_1 + w_2}$$
Where:
- $A_1$ is the average of group 1 (Parents)
- $w_1$ is the weight of group 1 (Number of parents)
- $A_2$ is the average of group 2 (Children)
- $w_2$ is the weight of group 2 (Number of children)
Using this formula for the family's average age:
$$A_1 = 30 \text{ years}, w_1 = 2$$
$$A_2 = 8 \text{ years}, w_2 = 2$$
$$\text{Average family age} = \frac{(2 \times 30) + (2 \times 8)}{2 + 2}$$
$$\text{Average family age} = \frac{60 + 16}{4}$$
$$\text{Average family age} = \frac{76}{4} = 19 \text{ years}$$
This weighted average approach confirms the result obtained by calculating total age and dividing by total members.
Important Questions from Age
The ages of two persons P and Q are in the ratio 5 : 7. Eight years ago, the ratio of P and Q was 7 : 13. The present ages of P and Q, respectively, are:
One year ago, the ratio of the ages of A and B was 4 : 3. The ratio of their ages, after 7 years from now, will be 9 : 7. What is the present age (in years) of B?
In 8 years, Subhash will be 3 times as old as he is now. After how many years will Subhash be 5 times as old as he is now?
A father is presently 3 times his daughter’s age. After 10 years he will be twice as old as her. Find the daughter’s present age.
Namya got married 6 years ago. Now her age is 11⁄4times her age at the time of marriage. Her son’s present age is one-fifth of her present age. Find her son’s present age.
