Understanding the Money Gifted Problem

This problem involves calculating the total money Joseph gifted to his family. The gift was divided into two parts: one for his wife and one for his three children. The money given to the children was distributed based on their ages, which is a classic ratio distribution problem.

Here's a breakdown of the information given:

• Money gifted to wife: ₹20000
• Children's ages: 12 years, 14 years, and 16 years.
• Money to children was distributed in the ratio of their ages.
• Youngest child (age 12) received: ₹3000.
• We need to find the total money gifted to the family.

Calculating Money Gifted to Children by Age Ratio

The money gifted to the children is in the ratio of their ages: 12 : 14 : 16.

We can simplify this ratio by dividing each number by the greatest common divisor, which is 2.

Simplified ratio = \( \frac{12}{2} \) : \( \frac{14}{2} \) : \( \frac{16}{2} \) = 6 : 7 : 8.

Let the amount of money received by the three children be \( 6x \), \( 7x \), and \( 8x \) respectively, where \( x \) is a constant value representing the common multiple in the ratio.

We are told that the youngest child, who is 12 years old (corresponding to the ratio part 6), received ₹3000.

So, we have the equation:

\( 6x = 3000 \)

Now, we can solve for \( x \):

\( x = \frac{3000}{6} \)

\( x = 500 \)

Now that we have the value of \( x \), we can calculate the amount received by each of the other children:

• Middle child (age 14, ratio part 7): Amount = \( 7x = 7 \times 500 = 3500 \)
• Eldest child (age 16, ratio part 8): Amount = \( 8x = 8 \times 500 = 4000 \)

Let's verify the amount received by the youngest child using this \( x \) value:

• Youngest child (age 12, ratio part 6): Amount = \( 6x = 6 \times 500 = 3000 \)

This matches the information given in the problem (₹3000), confirming our value for \( x \) is correct.

Total Money Received by Children

The total money gifted to the three children is the sum of the amounts each child received:

Total money to children = Amount to youngest + Amount to middle + Amount to eldest

Total money to children = ₹3000 + ₹3500 + ₹4000

Total money to children = ₹10500

Calculating Total Money Gifted to the Family

The total money gifted to the family is the sum of the money gifted to the wife and the total money gifted to the children.

Total money gifted to family = Money to wife + Total money to children

Total money gifted to family = ₹20000 + ₹10500

Total money gifted to family = ₹30500

So, Joseph gifted a total of ₹30500 to his family.

Revision Table: Key Concepts

Additional Information: Ratio and Proportion in Distribution

Problems involving dividing a quantity in a given ratio are common in mathematics. If a total quantity \( Q \) is to be divided among individuals in the ratio \( a:b:c \), the total number of ratio parts is \( a+b+c \). The share of the first individual is \( \frac{a}{a+b+c} \times Q \), the share of the second is \( \frac{b}{a+b+c} \times Q \), and so on.

In this specific problem, we didn't start with the total money for children. Instead, we were given the share of one child and the ratio. This allowed us to find the value of one 'ratio unit' (\( x \)), and then calculate the shares of the others and the total for the children.

Let the simplified ratio be \( a:b:c \). If the amount corresponding to ratio part \( a \) is \( A \), then \( ax = A \), which gives \( x = A/a \). The total amount distributed according to the ratio is \( (a+b+c)x = (a+b+c) \times (A/a) \). In our case, \( a=6 \), \( A=3000 \), and the simplified ratio parts are 6, 7, 8. The total ratio parts are \( 6+7+8=21 \). The total money for children is \( (6+7+8) \times x = 21x = 21 \times 500 = 10500 \).

Understanding how to work forwards (from total to shares) and backwards (from a share to the total or other shares) in ratio problems is crucial.