<Question>
Simplify $\frac{1}{5}\cdot \frac{8}{7}\div \frac{12}{20}$.
</Question>

<think>
<Plan_1>
Okay, let's see. I need to simplify the expression (1/5) multiplied by (8/7) divided by (12/20). Hmm, fractions can sometimes be tricky, but I think I remember that dividing by a fraction is the same as multiplying by its reciprocal. Let me start by writing down the problem again to make sure I have it right: (1/5) * (8/7) ÷ (12/20). Yep, that's what it says.

First, maybe I should handle the division part. Since dividing by a fraction is the same as multiplying by its reciprocal, I can rewrite the last part of the expression. So, instead of dividing by (12/20), I'll multiply by (20/12).
Let me check that: ÷ (12/20) is indeed * (20/12).

So now the expression becomes (1/5) * (8/7) * (20/12). Now, I need to multiply these three fractions together. To multiply fractions, I just multiply the numerators and the denominators.

Let me compute the numerators first: 1 * 8 * 20. Let's do that step by step. 1 times 8 is 8, then 8 times 20 is 160. So the numerator is 160.

Now the denominators: 5 * 7 * 12. Again, step by step. 5 times 7 is 35, then 35 times 12.
Let me calculate that. 35 times 12: 35*10 is 350, and 35*2 is 70, so 350 + 70 = 420. So the denominator is 420.

So now the fraction is 160/420. The next step is to simplify this fraction. To simplify, I need to find the greatest common divisor (GCD) of 160 and 420 and then divide both numerator and denominator by that GCD.

Let me find the GCD of 160 and 420. One way to do this is by listing the prime factors of both numbers.

Starting with 160. Let's break it down:

<prompt_1.1> Find the prime factors of 160. </prompt_1.1>

Now 420. Let's factorize that:

<prompt_1.2> Find the prime factors of 420. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
160 ÷ 2 = 80
80 ÷ 2 = 40
40 ÷ 2 = 20
20 ÷ 2 = 10
10 ÷ 2 = 5
5 ÷ 5 = 1
So the prime factors of 160 are 2^5 * 5^1.

Now 420:

420 ÷ 2 = 210
210 ÷ 2 = 105
105 ÷ 3 = 35
35 ÷ 5 = 7
7 ÷ 7 = 1
So the prime factors of 420 are 2^2 * 3^1 * 5^1 * 7^1.

The GCD is the product of the lowest powers of all common prime factors. The common prime factors between 160 and 420 are 2 and 5.

For prime factor 2: the lowest power is 2^2 (since 160 has 2^5 and 420 has 2^2).
For prime factor 5: both have 5^1.

So GCD = 2^2 * 5^1 = 4 * 5 = 20.
</execution_1.1>

<execution_1.2>
420 ÷ 2 = 210  
210 ÷ 2 = 105  
105 ÷ 3 = 35  
35 ÷ 5 = 7  
7 is prime. 

So the prime factors of 420 are 2^2 * 3 * 5 * 7.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Therefore, we can divide both numerator and denominator by 20. Let's do that:

<prompt_2.1> Simplify the fraction 160/420 using the GCD. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
160 ÷ 20 = 8  
420 ÷ 20 = 21  

So the simplified fraction is 8/21.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait a second, let me check if 8 and 21 have any common factors.
The factors of 8 are 1, 2, 4, 8. The factors of 21 are 1, 3, 7, 21. The only common factor is 1, so 8/21 is indeed in its simplest form.

But just to make sure I didn't make any mistakes earlier, let me go through the steps again.

<prompt_3.1> Re-evaluate the steps of the original expression to verify the simplification. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Original expression: (1/5) * (8/7) ÷ (12/20). 

First, handle the division: ÷ (12/20) becomes * (20/12). So the expression is (1/5)*(8/7)*(20/12). 

Multiply numerators: 1*8*20 = 160. Multiply denominators: 5*7*12 = 420. So 160/420. Simplify by dividing numerator and denominator by GCD 20, resulting in 8/21.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Alternatively, maybe there's a way to simplify before multiplying all the numerators and denominators. Let me try that approach to cross-check.

<prompt_4.1> Simplify the expression by canceling common factors before multiplying. </prompt_4.1>

Let me verify once again. If I compute the original expression step by step:

<prompt_4.2> Verify the result by computing the original expression step by step. </prompt_4.2>

Alternatively, converting the divisions into decimals to check:

<prompt_4.3> Verify the result by converting the expression into decimal form. </prompt_4.3>
</Plan_4>

<Execution_4>
<execution_4.1>
When multiplying fractions, we can cancel out common factors between numerators and denominators before multiplying. Let's see if that's possible here.

We have (1/5)*(8/7)*(20/12). Let's write all numerators and denominators out:

Numerators: 1, 8, 20  
Denominators: 5, 7, 12

Looking for common factors between numerators and denominators. Let's start with the 20 in the numerator and the 12 in the denominator. 20 and 12 have a common factor of 4. So 20 ÷ 4 = 5, and 12 ÷ 4 = 3. So we can simplify 20/12 to 5/3.

Wait, but we have to adjust the other numbers accordingly. Let me see. Alternatively, maybe break it down step by step.

Alternatively, let's look at each numerator and denominator:

First fraction: 1/5. Second: 8/7. Third: 20/12.

Looking at 1 and 5: no common factors. 8 and 7: no common factors. 20 and 12: can simplify as mentioned before. So perhaps simplifying 20/12 with something else.

But actually, since 20 is in the numerator and 12 is in the denominator, we can divide both by 4, as I did before. So 20 ÷ 4 = 5, 12 ÷ 4 = 3. So the third fraction becomes 5/3.

So now the expression becomes (1/5)*(8/7)*(5/3). Now, notice that there's a 5 in the numerator of the third fraction and a 5 in the denominator of the first fraction. So we can cancel those out. 5 ÷ 5 = 1, so the first fraction becomes 1/1, and the third fraction becomes 1/3.

So now we have (1/1)*(8/7)*(1/3). Multiplying these together: numerators are 1*8*1=8, denominators are 1*7*3=21. So 8/21. Same result as before. So that confirms that 8/21 is the simplified form.
</execution_4.1>

<execution_4.2>
First, compute (1/5) * (8/7). That would be (1*8)/(5*7) = 8/35. 

Then, divide that result by (12/20). So 8/35 ÷ 12/20. Which is the same as 8/35 * 20/12. 

Simplify 8/35 * 20/12. Let's see if we can cancel out any factors before multiplying. 

The 8 in the numerator and the 12 in the denominator: 8 and 12 have a common factor of 4. So 8 ÷ 4 = 2, and 12 ÷ 4 = 3. So that becomes 2/35 * 20/3. 

Now, 20 in the numerator and 35 in the denominator: 20 and 35 have a common factor of 5. 20 ÷ 5 = 4, 35 ÷ 5 = 7. So now it's 2/7 * 4/3. 

Multiply those: 2*4=8, 7*3=21. So 8/21. 

Same result. So that's another way to simplify by canceling factors before multiplying everything out.
</execution_4.2>

<execution_4.3>
First, compute (1/5) = 0.2.

Then, (0.2) * (8/7) ≈ 0.2 * 1.142857 ≈ 0.2285714.

Then, divide by (12/20). Since 12/20 = 0.6, so 0.2285714 ÷ 0.6 ≈ 0.3809523.

Now, 8/21 ≈ 0.3809523, which matches. So the decimal check also confirms that 8/21 is the correct simplified fraction.
</execution_4.3>
</Execution_4>

<Final_answer>
Therefore, the simplified expression is $\boxed{\frac{8}{21}}$.
</Final_answer>
</think>