<Question>
It's a well-known physics formula that force equals mass times acceleration.  Jen wants to throw a softball with the same force as Jack throws a baseball.  If the softball has a mass of $200$ g and the baseball has a mass of $150$ g, what is the ratio of acceleration of Jen's ball to Jack's?  Answer as a fraction in lowest terms.
</Question>

<think>
<Plan_1>
Okay, so Jen wants to throw a softball with the same force as Jack throws a baseball. The question is asking for the ratio of the acceleration of Jen's ball to Jack's. Hmm, let's see. I remember from physics that force equals mass times acceleration, which is F = ma. So, if both Jen and Jack are using the same force, then their masses and accelerations must be related in some way.

<prompt_1.1> Set up the equations for the forces exerted by Jen and Jack using the formula F = ma. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Let me write down what I know. The mass of the softball is 200 grams, and the baseball is 150 grams. Wait, but in the formula F = ma, mass is in kilograms, right? So I need to convert grams to kilograms. That means the softball is 0.2 kg and the baseball is 0.15 kg.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let's denote the force exerted by Jen as F_jen and Jack as F_jack. According to the problem, these forces are equal.
So, F_jen = F_jack. Using the formula F = ma, we can write this as m_jen * a_jen = m_jack * a_jack.

Since the forces are the same, the product of mass and acceleration for each ball must be equal. So, 0.2 kg * a_jen = 0.15 kg * a_jack.

But the question is asking for the ratio of Jen's acceleration to Jack's acceleration, which is a_jen / a_jack. Let me solve for that ratio.

<prompt_2.1> Solve for the ratio a_jen / a_jack using the equation 0.2 kg * a_jen = 0.15 kg * a_jack. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Dividing both sides of the equation by a_jack and then by 0.2 kg, we get a_jen / a_jack = 0.15 / 0.2. Let me compute that. 0.15 divided by 0.2 is the same as 15/20, which simplifies to 3/4. So the ratio is 3/4.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, but let me double-check. If Jen is using a ball that's heavier (200g vs 150g), then for the same force, her acceleration must be less, right?
Because acceleration is inversely proportional to mass when force is constant. So if Jen's ball is heavier, her acceleration should be smaller. So the ratio of Jen's acceleration to Jack's would be less than 1, which 3/4 is.

But let me confirm the steps again. Starting from F = ma.

<prompt_3.1> Reconfirm the solution by starting from the formula F = ma and ensuring the steps are correct. </prompt_3.1>

But let me check if I converted grams to kilograms correctly.

<prompt_3.2> Verify the conversion of grams to kilograms and confirm the solution. </prompt_3.2>

Wait, but maybe I made a mistake in the ratio. Let me think again. If F is the same, then m1*a1 = m2*a2. So a1/a2 = m2/m1.

<prompt_3.3> Re-evaluate the solution using the relationship m1*a1 = m2*a2 and confirm the ratio. </prompt_3.3>

Alternatively, maybe using variables without plugging in numbers first. Let's see.

<prompt_3.4> Solve the problem using variables without plugging in numbers first and confirm the solution. </prompt_3.4>

So I think that's the answer. Let me just confirm once more with actual numbers.

<prompt_3.5> Confirm the solution with actual numbers and ensure the ratio is correct. </prompt_3.5>
</Plan_3>

<Execution_3>
<execution_3.1>
For Jen: F = 0.2 * a_jen. For Jack: F = 0.15 * a_jack. Since F is the same, 0.2a_jen = 0.15a_jack. Therefore, a_jen / a_jack = 0.15 / 0.2. 0.15 divided by 0.2 is indeed 3/4.
</execution_3.1>

<execution_3.2>
The problem states the masses in grams, but the formula requires kilograms. However, when we take the ratio, the units (grams or kilograms) would cancel out. Let me see. If I use grams instead of kilograms, the ratio would be the same. Because if we have m_jen = 200g and m_jack = 150g, then the equation F = ma for Jen is 200g * a_jen = 150g * a_jack. Then the ratio a_jen / a_jack = 150 / 200 = 3/4. Same result. So whether we use grams or kilograms, the ratio remains the same because the units cancel out in the ratio.
</execution_3.2>

<execution_3.3>
So the ratio of accelerations is inverse of the ratio of masses. So if Jen's mass is 200g and Jack's is 150g, then a_jen / a_jack = m_jack / m_jen = 150/200 = 3/4. Yes, that's the same result. So 3/4 is the ratio.
</execution_3.3>

<execution_3.4>
Let F be the force. Then for Jen: F = m_jen * a_jen => a_jen = F / m_jen. For Jack: F = m_jack * a_jack => a_jack = F / m_jack. Therefore, the ratio a_jen / a_jack is (F / m_jen) / (F / m_jack) = (F / m_jen) * (m_jack / F) = m_jack / m_jen. So the ratio is m_jack / m_jen. But m_jack is 150g and m_jen is 200g, so 150/200 = 3/4. So that's the same result. Therefore, the ratio is 3/4.
</execution_3.4>

<execution_3.5>
Suppose Jack's acceleration is 4 m/s². Then Jen's acceleration would be (0.15/0.2)*4 = 3 m/s². So 3/4 is indeed the ratio. That makes sense. So the ratio is 3/4.
</execution_3.5>
</Execution_3>

<Final_answer>
Therefore, the ratio of acceleration of Jen's ball to Jack's is $\boxed{\frac{3}{4}}$.
</Final_answer>
</think>