In Diana's stamp collection, 4/5 of the stamps are Canadian, and 3/7 of the Canadian stamps were issued before 1940....

1. Translate the problem requirements: Clarify what "\(\frac{4}{5}\) of stamps are Canadian" and "\(\frac{3}{7}\) of Canadian stamps were issued before 1940" means, and understand that we need to find non-Canadian stamps when given that 192 Canadian stamps were issued in 1940 or later.
2. Identify the fraction relationship for post-1940 Canadian stamps: If \(\frac{3}{7}\) of Canadian stamps were issued before 1940, then \(\frac{4}{7}\) of Canadian stamps were issued in 1940 or later.
3. Work backwards from the given quantity: Use the fact that 192 stamps represents \(\frac{4}{7}\) of all Canadian stamps to find the total number of Canadian stamps.
4. Calculate total stamps and non-Canadian stamps: Since Canadian stamps are \(\frac{4}{5}\) of the total collection, find the total collection size, then determine how many are not Canadian.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday language:

• Diana has a stamp collection where most stamps are Canadian
• Specifically, 4 out of every 5 stamps are Canadian
• Among the Canadian stamps, some are old (before 1940) and some are newer (1940 or later)
• We're told that 3 out of every 7 Canadian stamps were issued before 1940
• We know that exactly 192 Canadian stamps were issued in 1940 or later
• We need to find how many stamps are NOT Canadian

Process Skill: TRANSLATE - Converting the fraction relationships into clear categories

2. Identify the fraction relationship for post-1940 Canadian stamps

This is a key insight that students often miss. Let's think about it step by step:

If 3 out of every 7 Canadian stamps were issued BEFORE 1940, then what about the remaining Canadian stamps?

Well, if 3 out of 7 were issued before 1940, then the remaining 4 out of 7 must have been issued in 1940 or later.

So we can say: \(\frac{4}{7}\) of all Canadian stamps were issued in 1940 or later.

This means that the 192 stamps we're given represents \(\frac{4}{7}\) of all Canadian stamps.

Process Skill: INFER - Drawing the non-obvious conclusion about the complementary fraction

3. Work backwards from the given quantity

Now we can find the total number of Canadian stamps using simple reasoning:

If 192 stamps represent \(\frac{4}{7}\) of all Canadian stamps, then:

• \(\frac{1}{7}\) of Canadian stamps = \(192 ÷ 4 = 48\) stamps
• \(\frac{7}{7}\) of Canadian stamps (the total) = \(48 × 7 = 336\) stamps

So Diana has 336 Canadian stamps in total.

Let's verify this makes sense: \(\frac{4}{7} × 336 = 4 × 48 = 192\) ✓

4. Calculate total stamps and non-Canadian stamps

Now we use the fact that Canadian stamps make up \(\frac{4}{5}\) of Diana's entire collection:

If 336 Canadian stamps represent \(\frac{4}{5}\) of the total collection, then:

• \(\frac{1}{5}\) of the collection = \(336 ÷ 4 = 84\) stamps
• \(\frac{5}{5}\) of the collection (the total) = \(84 × 5 = 420\) stamps

Since the total collection is 420 stamps and 336 are Canadian:

Non-Canadian stamps = \(420 - 336 = 84\) stamps

Alternatively, we could note that non-Canadian stamps represent \(\frac{1}{5}\) of the collection:

\(\frac{1}{5} × 420 = 84\) stamps

4. Final Answer

Diana has 84 stamps that are not Canadian.

This matches answer choice A.

Verification:

• Total stamps: 420
• Canadian stamps: 336 (which is \(\frac{4}{5}\) of 420 ✓)
• Canadian stamps before 1940: \(\frac{3}{7} × 336 = 144\)
• Canadian stamps 1940 or later: \(\frac{4}{7} × 336 = 192\) ✓
• Non-Canadian stamps: \(420 - 336 = 84\) ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the fraction relationships
Students often struggle to clearly understand what each fraction represents. They might confuse "\(\frac{3}{7}\) of Canadian stamps were issued before 1940" with "\(\frac{3}{7}\) of ALL stamps were issued before 1940." The key insight is recognizing that the \(\frac{3}{7}\) fraction only applies to the subset of Canadian stamps, not the entire collection.

2. Missing the complementary fraction concept
Many students fail to realize that if \(\frac{3}{7}\) of Canadian stamps were issued before 1940, then \(\frac{4}{7}\) must have been issued in 1940 or later. They might try to work directly with the \(\frac{3}{7}\) fraction or attempt to find another relationship, missing this crucial complementary insight.

3. Not establishing the correct working backwards strategy
Students often get overwhelmed by multiple fraction relationships and don't recognize that they should work backwards from the given concrete number (192 stamps). They might try to set up complex equations instead of using the simpler approach of finding what fraction 192 represents.

Errors while executing the approach

1. Calculation errors when finding total Canadian stamps
When working backwards from "192 stamps represent \(\frac{4}{7}\) of Canadian stamps," students might incorrectly calculate the total. Common errors include: \(192 × \frac{7}{4} = 192 × 1.75 = 336\), but some students might mistakenly do \(192 ÷ 7 × 4\) or other incorrect operations.

2. Mixing up fraction relationships
Students might correctly find that there are 336 Canadian stamps but then incorrectly apply the \(\frac{4}{5}\) relationship. For example, they might think \(336 = \frac{4}{5} ×\) (total Canadian stamps) instead of recognizing that 336 Canadian stamps = \(\frac{4}{5} ×\) (total stamps in collection).

Errors while selecting the answer

1. Selecting an intermediate calculation as the final answer
Students might correctly calculate the total collection size (420 stamps) or total Canadian stamps (336 stamps) but then select one of these intermediate values instead of the final answer (84 non-Canadian stamps). This happens when they lose track of what the question is actually asking for.

2. Confusing Canadian vs non-Canadian stamps
Even after correctly calculating that there are 84 non-Canadian stamps and 336 Canadian stamps, some students might select 336 as their answer because they misread the question and think it's asking for Canadian stamps rather than non-Canadian stamps.