<Question>
If 4 daps = 7 yaps, and 5 yaps = 3 baps, how many daps equal 42 baps?
</Question>

<think>
<Plan_1>
Okay, let's see. I need to figure out how many daps equal 42 baps, given the conversion rates between daps, yaps, and baps. Hmm. Let me start by writing down the given relationships.
First, 4 daps equal 7 yaps. So, 4 daps = 7 yaps. Then, 5 yaps equal 3 baps. So, 5 yaps = 3 baps. The goal is to convert 42 baps into daps.

Maybe I can convert baps to yaps first, then yaps to daps. Let's break it down step by step.

Starting with 42 baps. Since 5 yaps equal 3 baps, I can find out how many yaps are equivalent to 42 baps. To do that, I need to set up a conversion factor.

<prompt_1.1> Convert 42 baps to yaps using the conversion factor 5 yaps = 3 baps. </prompt_1.1>
-----

Alternatively, maybe set up equations to convert baps to daps directly. Let's see.

We have:

4 daps = 7 yaps --> equation 1

5 yaps = 3 baps --> equation 2

We need to find x daps = 42 baps. Let's express 42 baps in terms of yaps first, then in terms of daps.

<prompt_1.2> Directly convert 42 baps to daps using the equations 4 daps = 7 yaps and 5 yaps = 3 baps. </prompt_1.2>

Alternatively, using unit conversion factors:

42 baps * (5 yaps / 3 baps) = 42 * (5/3) yaps = 70 yaps.

Then, 70 yaps * (4 daps / 7 yaps) = 70 * (4/7) daps = 40 daps.

Same result. So, seems solid.

But let me check with different numbers to ensure the conversion factors are correctly derived.

<prompt_1.3> Validate the conversion factors with different test cases. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
If 5 yaps = 3 baps, then 1 bap = 5/3 yaps. So, 42 baps would be 42 * (5/3) yaps. Let me calculate that. 42 divided by 3 is 14, multiplied by 5 is 70. So, 42 baps equal 70 yaps.
</execution_1.1>

<execution_1.2>
From equation 2: 5 yaps = 3 baps. So, 1 bap = 5/3 yaps. Therefore, 42 baps = 42 * (5/3) yaps. Let's compute that: 42 divided by 3 is 14, multiplied by 5 is 70. So, 42 baps = 70 yaps.

Now, convert 70 yaps to daps using equation 1. From equation 1: 4 daps = 7 yaps. So, 1 yap = 4/7 daps. Therefore, 70 yaps = 70 * (4/7) daps. Let's calculate that: 70 divided by 7 is 10, multiplied by 4 is 40. So, 70 yaps = 40 daps.
</execution_1.2>

<execution_1.3>
Suppose I take the first conversion: 4 daps = 7 yaps. So, 1 dap = 7/4 yaps, and 1 yap = 4/7 daps. Similarly, 5 yaps = 3 baps, so 1 yap = 3/5 baps, and 1 dap would be (3/5)*(4/7) = 12/35 baps? Wait, maybe not necessary. But at least the conversion factors seem reciprocal.

Alternatively, if I set up a chain:

42 baps * (5 yaps / 3 baps) * (4 daps / 7 yaps). Let's compute this:

42 * 5 / 3 * 4 /7. Let's compute step by step.

42 divided by 3 is 14. 14 multiplied by 5 is 70. 70 multiplied by 4 is 280. 280 divided by 7 is 40.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, convert those 70 yaps to daps. Since 4 daps = 7 yaps, we can find out how many daps are in 70 yaps. If 7 yaps equal 4 daps, then 70 yaps would be (4 daps / 7 yaps) * 70 yaps.

<prompt_2.1> Convert 70 yaps to daps using the conversion factor 4 daps = 7 yaps. </prompt_2.1>
-----

Based on execution_1.2:
Therefore, 42 baps equal 40 daps.
-----

Just to make sure I didn't mix up any steps, let me go through the unit conversion again.

<prompt_2.2> Recheck the unit conversion steps to ensure no mix-up. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Let me compute that. 70 divided by 7 is 10, multiplied by 4 is 40. So, 70 yaps equal 40 daps. Therefore, 42 baps equal 40 daps.
</execution_2.1>

<execution_2.2>
Starting with 42 baps, convert to yaps using the ratio from yaps to baps. The ratio is 5 yaps per 3 baps. So, the number of yaps is (5/3) times the number of baps. So, 42 * 5 / 3 = (42 / 3) * 5 = 14 * 5 = 70 yaps. Then, converting yaps to daps: since 4 daps equal 7 yaps, the number of daps is (4/7) times the number of yaps. So, 70 * 4 / 7 = (70 / 7) * 4 = 10 * 4 = 40 daps.
</execution_2.2>
</Execution_2>

<Final_answer>
Therefore, the number of daps equal to 42 baps is $\boxed{40}$.
</Final_answer>
</think>