A total of 30 percent of the geese included in a certain migration study were male. If some of the...

1. Translate the problem requirements: We need to understand that migration rate means the fraction of each gender that migrated, and we're looking for the ratio of these two rates (male rate ÷ female rate).
2. Set up concrete numbers using the given percentages: Use a convenient total number to make calculations simple and track the male/female breakdown in the original population.
3. Determine the migrating population composition: Use the constraint that 20% of migrating geese were male to find how many males and females actually migrated.
4. Calculate individual migration rates: Find what fraction of all males migrated and what fraction of all females migrated using the definition provided.
5. Form the required ratio: Divide the male migration rate by the female migration rate to get the final answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is really asking in plain English. We have a group of geese being studied, and we want to compare how likely males are to migrate versus how likely females are to migrate.

The key term here is "migration rate" - this means: out of all the males in the study, what fraction migrated? And out of all the females in the study, what fraction migrated?

We're not just counting how many males vs females migrated - we're comparing their rates (percentages) of migration. Then we want the ratio of these two rates: male migration rate ÷ female migration rate.

Process Skill: TRANSLATE - Converting the problem's language about "migration rates" into clear mathematical understanding

2. Set up concrete numbers using the given percentages

Let's use nice round numbers to make our arithmetic simple. Since we're dealing with 30% males, let's say we started with 100 total geese.

Original population:
• Total geese: 100
• Males: 30% of 100 = 30 male geese
• Females: 70% of 100 = 70 female geese

This concrete setup will help us track exactly what happens during migration without getting lost in variables and formulas.

3. Determine the migrating population composition

Now we need to figure out how many geese actually migrated. The problem tells us that 20% of the migrating geese were male.

Let's say M geese total migrated. Then:
• Males who migrated: \(20\% \text{ of } \mathrm{M} = 0.2\mathrm{M}\)
• Females who migrated: \(80\% \text{ of } \mathrm{M} = 0.8\mathrm{M}\)

But we know there were only 30 males total in our population. So \(0.2\mathrm{M}\) males migrated out of 30 total males.
And \(0.8\mathrm{M}\) females migrated out of 70 total females.

We can work with this M for now - it will cancel out when we form our ratio!

4. Calculate individual migration rates

Now let's find the migration rate for each gender using the definition given:

Male migration rate = (males who migrated) ÷ (total males)
= \(0.2\mathrm{M} \div 30 = \frac{0.2\mathrm{M}}{30} = \frac{\mathrm{M}}{150}\)

Female migration rate = (females who migrated) ÷ (total females)
= \(0.8\mathrm{M} \div 70 = \frac{0.8\mathrm{M}}{70} = \frac{8\mathrm{M}}{700} = \frac{2\mathrm{M}}{175}\)

Notice how we're calculating what fraction of each gender actually migrated.

5. Form the required ratio

Now we can find the ratio we want:

Ratio = (male migration rate) ÷ (female migration rate)
= \(\left(\frac{\mathrm{M}}{150}\right) \div \left(\frac{2\mathrm{M}}{175}\right)\)
= \(\left(\frac{\mathrm{M}}{150}\right) \times \left(\frac{175}{2\mathrm{M}}\right)\)
= \(\frac{175}{150 \times 2}\)
= \(\frac{175}{300}\)
= \(\frac{7}{12}\)

Notice how the M's cancelled out perfectly - this confirms our approach was correct!

Process Skill: SIMPLIFY - Using the fact that M cancels out, avoiding unnecessary complexity

Final Answer

The ratio of the migration rate for male geese to the migration rate for female geese is \(\frac{7}{12}\).

This means that males migrate at a rate that is \(\frac{7}{12}\) (about 58%) of the female migration rate. In other words, females are more likely to migrate than males in this study.

The answer is B: \(\frac{7}{12}\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "migration rate" means: Students often confuse migration rate with simple counts. They might think the question is asking for the ratio of "number of males who migrated" to "number of females who migrated" rather than understanding that migration rate means "what fraction of each gender migrated." This leads to setting up the wrong calculation entirely.

2. Confusion about "20% of migrating geese were male": Students may misinterpret this constraint. Some think it means 20% of all males migrated, rather than understanding it describes the gender composition of the migrating group. This fundamental misreading changes the entire problem setup.

3. Getting overwhelmed by the complex definition: The formal definition "Migration rate = (number of geese of that sex migrating) / (total number of geese of that sex)" can cause students to panic and miss that this is simply asking "what fraction of males migrated vs. what fraction of females migrated."

Errors while executing the approach

1. Arithmetic errors when simplifying fractions: Even with the correct setup, students often make mistakes when simplifying complex fractions like \(\left(\frac{\mathrm{M}}{150}\right) \div \left(\frac{2\mathrm{M}}{175}\right)\). They may incorrectly multiply by the reciprocal or make errors when canceling terms, especially when dealing with the variable M.

2. Not recognizing that M cancels out: Students may get stuck thinking they need to find the actual value of M (total migrating geese) and spend time trying to solve for it, not realizing that M will cancel out completely in the final ratio calculation.

3. Percentage calculation errors: When working with the 30% males, 70% females, and 20% male migrants, students may make basic percentage errors, such as calculating 80% of M incorrectly for female migrants or mixing up which percentages apply to which groups.

Errors while selecting the answer

1. Inverting the final ratio: Students may correctly calculate \(\frac{7}{12}\) but then select answer choice E \(\left(\frac{8}{7}\right)\) because they confused which migration rate should be in the numerator versus denominator of the final ratio. The question asks for "male rate to female rate" but they might flip it.

2. Not fully simplifying the fraction: Students might arrive at an equivalent fraction like \(\frac{175}{300}\) and not recognize they need to reduce it to \(\frac{7}{12}\), potentially leading them to think their answer doesn't match any of the choices and second-guessing their work.

Alternate Solutions

Smart Numbers Approach

This problem is well-suited for smart numbers because we can choose a convenient total number of geese that makes all percentage calculations clean and straightforward.

Step 1: Choose a smart total number of geese

Let's use 100 total geese to make percentage calculations simple.

• Male geese: 30% of 100 = 30 males
• Female geese: 70% of 100 = 70 females

Step 2: Determine how many geese migrated

We don't know the exact number, so let's call it M migrating geese.

From the constraint that 20% of migrating geese were male:

• Migrating males: \(0.20\mathrm{M}\)
• Migrating females: \(0.80\mathrm{M}\)

Step 3: Calculate migration rates using the definition

Male migration rate = (migrating males) ÷ (total males) = \(0.20\mathrm{M} \div 30 = \frac{\mathrm{M}}{150}\)

Female migration rate = (migrating females) ÷ (total females) = \(0.80\mathrm{M} \div 70 = \frac{0.80\mathrm{M}}{70} = \frac{8\mathrm{M}}{700} = \frac{2\mathrm{M}}{175}\)

Step 4: Find the ratio of migration rates

Ratio = (Male rate) ÷ (Female rate) = \(\left(\frac{\mathrm{M}}{150}\right) \div \left(\frac{2\mathrm{M}}{175}\right)\)

= \(\left(\frac{\mathrm{M}}{150}\right) \times \left(\frac{175}{2\mathrm{M}}\right)\)

= \(\frac{175}{150 \times 2}\)

= \(\frac{175}{300}\)

= \(\frac{7}{12}\)

Verification with specific numbers

Let's verify with M = 60 migrating geese:

• Migrating males: 20% of 60 = 12
• Migrating females: 80% of 60 = 48
• Male migration rate: \(\frac{12}{30} = \frac{2}{5} = 0.4\)
• Female migration rate: \(\frac{48}{70} = \frac{24}{35}\)
• Ratio: \(\left(\frac{2}{5}\right) \div \left(\frac{24}{35}\right) = \left(\frac{2}{5}\right) \times \left(\frac{35}{24}\right) = \frac{70}{120} = \frac{7}{12}\) ✓

Answer: B. \(\frac{7}{12}\)