If 175 billion French francs is equivalent to 35 billion United States dollars, which of the following expressions represents the...

1. Translate the problem requirements: We're given that 175 billion French francs equals 35 billion US dollars, and we need to find an expression that converts any amount f French francs into US dollars
2. Establish the conversion rate: Determine how many French francs equal one US dollar by simplifying the given ratio
3. Set up the conversion formula: Use the rate to create an expression that converts f French francs to US dollars
4. Verify against answer choices: Check which answer choice matches our derived expression

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're given and what we need to find.

What we know: 175 billion French francs equals 35 billion US dollars

What we need: An expression that tells us how many US dollars we get for any amount f French francs

Think of this like any exchange situation - if someone told you that 175 apples cost 35 dollars, you'd want to figure out how much 1 apple costs, and then you could find the cost of any number of apples.

2. Establish the conversion rate

Now let's find out how many French francs equal one US dollar.

If 175 billion French francs = 35 billion US dollars, then we can find how many French francs equal 1 US dollar by dividing both sides by 35 billion:

\(175 \text{ billion} ÷ 35 \text{ billion} = 5\)

\(35 \text{ billion} ÷ 35 \text{ billion} = 1\)

So 5 French francs = 1 US dollar

This means that French francs are worth less than US dollars - it takes 5 French francs to equal just 1 US dollar.

3. Set up the conversion formula

Now we know the rate: 5 French francs = 1 US dollar

If we have f French francs, how many US dollars is that worth?

Since it takes 5 French francs to make 1 US dollar, we need to divide the number of French francs by 5:

f French francs = \(\mathrm{f} ÷ 5\) US dollars = \(\frac{\mathrm{f}}{5}\) US dollars

Let's check this makes sense: If f = 10 French francs, then \(\frac{\mathrm{f}}{5} = \frac{10}{5} = 2\) US dollars. Since 5 French francs = 1 US dollar, then 10 French francs should indeed equal 2 US dollars. ✓

4. Verify against answer choices

Our expression is \(\frac{\mathrm{f}}{5}\), which matches answer choice (D).

Let's double-check with the original information:

If f = 175 billion French francs, then \(\frac{\mathrm{f}}{5} = \frac{175 \text{ billion}}{5} = 35 \text{ billion}\) US dollars ✓

This matches exactly what the problem told us!

Final Answer

Answer: (D) \(\frac{\mathrm{f}}{5}\)

The expression \(\frac{\mathrm{f}}{5}\) correctly converts any amount f of French francs into the equivalent number of US dollars, since 5 French francs equal 1 US dollar.

Common Faltering Points

Errors while devising the approach

1. Setting up the conversion in the wrong direction: Students often confuse which currency should be in the numerator versus denominator. They might think "175 francs = 35 dollars" means 1 franc = \(\frac{35}{175}\) dollars, leading them toward answer choice (E) \(\frac{\mathrm{f}}{7}\) instead of recognizing they need to find how many francs equal 1 dollar first.

2. Misunderstanding what the variable 'f' represents: Some students might think 'f' represents dollars instead of francs, completely reversing their setup and leading them to look for expressions like \(5\mathrm{f}\) (choice B) or \(7\mathrm{f}\) (choice C).

3. Attempting to use subtraction instead of division: Students might notice that 175 - 35 = 140 and incorrectly think this difference is relevant to the conversion, leading them toward answer choice (A) \(\mathrm{f} - 140\), not recognizing this is a ratio/proportion problem.

Errors while executing the approach

1. Arithmetic errors when finding the conversion rate: When dividing \(175 ÷ 35\), students might incorrectly calculate this as 7 instead of 5, leading them to think 7 francs = 1 dollar, which would give the wrong expression \(\frac{\mathrm{f}}{7}\).

2. Inverting the final conversion formula: Even after correctly finding that 5 francs = 1 dollar, students might incorrectly conclude that f francs = \(5\mathrm{f}\) dollars instead of \(\frac{\mathrm{f}}{5}\) dollars, leading them to answer choice (B) \(5\mathrm{f}\).

Errors while selecting the answer

1. Failing to verify the answer with the original data: Students might arrive at \(\frac{\mathrm{f}}{5}\) but then second-guess themselves and switch to \(\frac{\mathrm{f}}{7}\) or another choice without checking that when f = 175 billion, their expression should yield 35 billion dollars as stated in the problem.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart numbers that satisfy the given relationship

We know that 175 billion French francs = 35 billion US dollars. Let's use these exact numbers as our smart numbers since they're already given in the problem.

Step 2: Test with a convenient value for f

Let's choose f = 175 (representing 175 French francs, without the 'billion' for simplicity). According to our conversion rate, this should equal 35 US dollars.

Step 3: Calculate the conversion rate

From the given information: 175 French francs = 35 US dollars
This means 1 French franc = \(\frac{35}{175} = \frac{1}{5}\) US dollars

Step 4: Apply this rate to our test value

For f = 175 French francs:
US dollars = \(175 × \frac{1}{5} = \frac{175}{5} = 35\) US dollars ✓

Step 5: Check which answer choice gives us this result

Testing each choice with f = 175:

1. \(\mathrm{f} - 140 = 175 - 140 = 35\) ✓
2. \(5\mathrm{f} = 5(175) = 875\) ✗
3. \(7\mathrm{f} = 7(175) = 1,225\) ✗
4. \(\frac{\mathrm{f}}{5} = \frac{175}{5} = 35\) ✓
5. \(\frac{\mathrm{f}}{7} = \frac{175}{7} = 25\) ✗

Step 6: Test with another smart number to eliminate the coincidence

Let's try f = 35 French francs. This should equal \(\frac{35}{5} = 7\) US dollars.

1. \(\mathrm{f} - 140 = 35 - 140 = -105\) ✗
2. \(\frac{\mathrm{f}}{5} = \frac{35}{5} = 7\) ✓

Only choice (D) \(\frac{\mathrm{f}}{5}\) works consistently for different values of f.