Colin had some badges, postcards and magnets. He had 85 badges. 30% of his items were postcards. He had 51 less postcards than magnets.

1. How many postcards did he have?
2. He gave some postcards away. The total collection decreased by 20%. What was the new percentage of postcards he had in the end?

Let's solve the problem step by step.

### 1. How many postcards did Colin have?

Let's define:
- \( b = 85 \) (number of badges)
- \( p \) as the number of postcards
- \( m \) as the number of magnets

Given that 30% of Colin's items were postcards:
\[
p = 0.30 \times \text{(total number of items)}
\]

The total number of items is:
\[
\text{Total items} = b + p + m
\]

So:
\[
p = 0.30 \times (85 + p + m)
\]

Also, it is given that Colin had 51 less postcards than magnets:
\[
p = m - 51
\]

Now we can substitute \( m = p + 51 \) into the equation for \( p \):
\[
p = 0.30 \times (85 + p + (p + 51))
\]

Simplifying:
\[
p = 0.30 \times (85 + 2p + 51)
\]
\[
p = 0.30 \times (136 + 2p)
\]
\[
p = 40.8 + 0.6p
\]

To solve for \( p \), subtract \( 0.6p \) from both sides:
\[
0.4p = 40.8
\]

Divide by 0.4:
\[
p = \frac{40.8}{0.4} = 102
\]

So, Colin had **102 postcards**.

### 2. What was the new percentage of postcards after the total collection decreased by 20%?

First, calculate the original total number of items:
\[
\text{Original Total} = b + p + m = 85 + 102 + 153 = 340
\]

After a 20% decrease:
\[
\text{New Total} = 340 \times 0.80 = 272
\]

If Colin gave away some postcards, let’s denote the number of postcards he gave away as \( x \). The number of postcards he has after giving them away is \( 102 - x \).

The new percentage of postcards is:
\[
\text{Percentage of postcards} = \frac{102 - x}{272} \times 100
\]

We don't know exactly how many postcards were given away, so let's assume the worst-case scenario where he gave away all 102 postcards:
\[
\text{Percentage of postcards} = \frac{0}{272} \times 100 = 0\%
\]

If he gave away none:
\[
\text{Percentage of postcards} = \frac{102}{272} \times 100 \approx 37.5\%
\]

So the new percentage of postcards depends on the number he gave away. Without knowing \( x \), the percentage would range from **0% to 37.5%**.

Would you like more details on any part of the solution? Here are some related questions:

1. How would the calculation change if we knew the exact number of postcards given away?
2. What if the percentage of postcards was originally 40% instead of 30%?
3. How does the problem change if the badges were given away instead of the postcards?
4. What if the total collection decreased by 30% instead of 20%?
5. How can you express the relationship between badges, postcards, and magnets in a different mathematical form?

**Tip:** To solve problems involving percentages, setting up equations based on the information given is crucial for finding unknown quantities.

Discover how to solve Colin's problem involving badges, postcards, and magnets. Calculate the number of postcards Colin had and determine the new percentage after giving some away. Suitable for grades 6-8 mathematics.

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