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: We need to find how many stamps are NOT Canadian. We know: \(\frac{4}{5}\) of all stamps are Canadian, \(\frac{3}{7}\) of Canadian stamps were issued before 1940, and 192 Canadian stamps were issued in 1940 or later.
2. Work backwards from the concrete number: Use the 192 Canadian stamps issued in 1940 or later to find the total number of Canadian stamps in the collection.
3. Scale up to find the total collection size: Once we know the number of Canadian stamps, use the fact that they represent \(\frac{4}{5}\) of the total collection to find the total number of stamps.
4. Calculate the non-Canadian stamps: Subtract the number of Canadian stamps from the total to find how many stamps are not Canadian.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English first. Diana has a stamp collection where most stamps are Canadian, and we want to find how many stamps are NOT Canadian.

Here's what the problem tells us:

• 4 out of every 5 stamps in her collection are Canadian
• Of those Canadian stamps, 3 out of every 7 were issued before 1940
• This means 4 out of every 7 Canadian stamps were issued in 1940 or later
• We know that exactly 192 Canadian stamps were issued in 1940 or later

Our goal is to find how many stamps are not Canadian.

Process Skill: TRANSLATE - Converting the fraction relationships into a clear understanding of the collection structure

2. Work backwards from the concrete number

Since we know that 192 Canadian stamps were issued in 1940 or later, let's use this to find the total number of Canadian stamps.

We established that 4 out of every 7 Canadian stamps were issued in 1940 or later. In fraction terms, this means \(\frac{4}{7}\) of all Canadian stamps equals 192 stamps.

Let's call the total number of Canadian stamps 'C'. Then:
\(\frac{4}{7} \times \mathrm{C} = 192\)

To find C, we multiply both sides by \(\frac{7}{4}\):
\(\mathrm{C} = 192 \times \frac{7}{4}\)
\(\mathrm{C} = 192 \times 7 ÷ 4\)
\(\mathrm{C} = 1344 ÷ 4\)
\(\mathrm{C} = 336\)

So there are 336 Canadian stamps in total.

3. Scale up to find the total collection size

Now we know there are 336 Canadian stamps, and these represent \(\frac{4}{5}\) of Diana's entire collection.

Let's call the total number of stamps in the collection 'T'. Then:
\(\frac{4}{5} \times \mathrm{T} = 336\)

To find T, we multiply both sides by \(\frac{5}{4}\):
\(\mathrm{T} = 336 \times \frac{5}{4}\)
\(\mathrm{T} = 336 \times 5 ÷ 4\)
\(\mathrm{T} = 1680 ÷ 4\)
\(\mathrm{T} = 420\)

So Diana has 420 stamps in total.

4. Calculate the non-Canadian stamps

Now we can find how many stamps are not Canadian by subtracting the Canadian stamps from the total collection:

Non-Canadian stamps = Total stamps - Canadian stamps
Non-Canadian stamps = 420 - 336
Non-Canadian stamps = 84

Let's verify this makes sense: If \(\frac{4}{5}\) of the collection is Canadian, then \(\frac{1}{5}\) should be non-Canadian.
\(\frac{1}{5} \times 420 = 84\) ✓

Final Answer

The number of stamps in Diana's collection that are not Canadian is 84.

This matches answer choice A.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting the fraction relationships
Students often confuse what the fractions represent. They might think that \(\frac{3}{7}\) of Canadian stamps issued before 1940 means \(\frac{3}{7}\) of ALL stamps, rather than \(\frac{3}{7}\) of just the Canadian stamps. This leads to setting up incorrect equations from the start.

Faltering Point 2: Not recognizing the complement relationship
If \(\frac{3}{7}\) of Canadian stamps were issued before 1940, then \(\frac{4}{7}\) were issued in 1940 or later. Students frequently miss this complement calculation and try to work directly with the \(\frac{3}{7}\) fraction, making the problem much more complicated than necessary.

Faltering Point 3: Working forward instead of backward
Many students try to assume a total collection size and work forward, rather than starting with the concrete number (192 stamps) and working backward. This approach often leads to complex algebraic setups that are prone to errors.

Errors while executing the approach

Faltering Point 1: Fraction multiplication and division errors
When solving equations like \(\left(\frac{4}{7}\right) \times \mathrm{C} = 192\), students often make mistakes when multiplying or dividing by fractions. Common errors include forgetting to flip the fraction when dividing, or making arithmetic mistakes like calculating \(192 \times 7 ÷ 4\) incorrectly.

Faltering Point 2: Losing track of what each variable represents
As students move through multiple steps (finding Canadian stamps, then total stamps), they may confuse what each calculated number represents. For example, mixing up the total Canadian stamps (336) with the total collection size (420).

Errors while selecting the answer

Faltering Point 1: Selecting an intermediate calculation instead of the final answer
Students correctly calculate the total Canadian stamps (336) or total collection size (420) but then select one of these intermediate values as their final answer, forgetting that the question specifically asks for non-Canadian stamps (84).