Solving a Ratio Problem: How Many Stamps Did John Have Initially?

In this article, we will walk through the solution to a ratio problem involving two individuals, Mary and John, and their collections of stamps. This problem not only requires a solid understanding of ratios but also algebraic manipulation to derive the answer. The steps we will follow will be clear and detailed, ensuring that even those with a moderate background in mathematics can follow along.

Problem Statement

The problem states that the ratio of the number of stamps Mary had to the number of stamps John had was 3:7. After John gave 6 stamps to Mary, the new ratio became 4:6. The task is to determine how many stamps John had initially.

Step-by-Step Solution

Let's denote the number of stamps Mary had as M and the number of stamps John had as J.

Initial Condition

The initial ratio of stamps is:

Initial ratio: M:J 3:7

This can be expressed as:

M 3/7 J

After the Transaction

After John gives 6 stamps to Mary, the new number of stamps Mary has is M - 6 and the new number of stamps John has is J - 6. The new ratio of their stamps is:

Ratio after transaction: (M - 6) : (J - 6) 4:6 2:3

This can be expressed as:

3(M - 6) 2(J - 6)

Solving for the Variables

Substituting the expression for M from the initial condition into the equation for the new ratio:

3(3/7 J - 6) 2(J - 6)

Expanding and combining like terms:

9/7 J - 18 2J - 12

Multiplying the entire equation by 7 to eliminate the fraction:

9J - 126 14J - 84

Rearranging the equation:

126 - 84 14J - 9J

Further simplifying:

210 5J

Solving for J:

J 210 / 5 42

Now, substituting J 42 back into the initial ratio equation to find M:

M 3/7 * 42 18

Conclusion

John initially had 42 stamps.

Alternative Solution Method

Another approach to solving this problem is to use the total number of stamps and the changing ratio:

The total number of stamps is:

0.7J 6 0.6J 60

Solving for J:

0.1J 60

J 600 / 10 60

So, John originally had 42 stamps, and Mary had 18.

When John gives 6 stamps to Mary:

John: 60 - 6 54 stamps, Mary: 18 6 24 stamps

Verifying the ratios:

54 / 24 27 / 12 9 / 4 2.25 (not accurate, simplifying ratios to 4:6)

Keywords

- Ratio problem

- Initial number of stamps

- Mathematical problem solving