Okay, let me try to figure out how to simplify this expression: (1/5) multiplied by (8/7) divided by (12/20). Hmm, division of fractions can be a bit tricky, but I remember that dividing by a fraction is the same as multiplying by its reciprocal. So maybe I should start by converting that division into multiplication.

First, let me write down the original expression again to make sure I have it right: (1/5) * (8/7) ÷ (12/20). Right. So the division part is ÷ (12/20). To handle that, I need to take the reciprocal of 12/20, which would be 20/12, and then multiply it by the existing fractions. So the expression becomes (1/5) * (8/7) * (20/12). 

Wait, let me confirm that. Dividing by a fraction is the same as multiplying by its reciprocal, yes. So if I have a ÷ (b/c), it's the same as a * (c/b). So applying that here, the division by 12/20 becomes multiplication by 20/12. That makes sense.

Now, I need to multiply these three fractions together: 1/5, 8/7, and 20/12. Let me recall how to multiply fractions. Multiply the numerators together and the denominators together. So the numerator would be 1 * 8 * 20, and the denominator would be 5 * 7 * 12. Let me compute that.

First, the numerator: 1 * 8 is 8, and 8 * 20 is 160. So the numerator is 160. The denominator: 5 * 7 is 35, and 35 * 12. Let me calculate 35*12. 35*10 is 350, and 35*2 is 70, so 350+70=420. So the denominator is 420. Therefore, the product is 160/420.

But now I need to simplify this fraction. Let's see. Both 160 and 420 are divisible by 20. Let me check. 160 ÷ 20 is 8, and 420 ÷ 20 is 21. So that simplifies to 8/21. Wait, is 8 and 21 divisible by any common factors? 8 is 2^3, and 21 is 3*7. They don't have any common factors other than 1, so 8/21 is the simplified form.

Wait, let me verify my steps again to make sure I didn't make a mistake. Starting with 1/5 * 8/7 ÷ 12/20. Convert division to multiplication by reciprocal: 1/5 * 8/7 * 20/12. Then multiply numerators: 1*8*20 = 160. Denominators: 5*7*12 = 420. 160/420 simplifies by dividing numerator and denominator by 20: 8/21. Yep, that seems right.

Alternatively, maybe I can simplify before multiplying. Let's try that approach to cross-cancel common factors. That might make the calculation easier. Let's see:

Original expression after converting division to multiplication: (1/5) * (8/7) * (20/12). Let's write all the numerators and denominators:

Numerators: 1, 8, 20

Denominators: 5, 7, 12

Now, let's look for common factors between numerators and denominators.

First, 8 and 12: 8 is 2^3, 12 is 2^2*3. So we can cancel a 4 from 8 and 12. Wait, 8 divided by 4 is 2, and 12 divided by 4 is 3. So that would leave us with 2 in the numerator and 3 in the denominator.

Similarly, 20 and 5: 20 divided by 5 is 4, and 5 divided by 5 is 1. So 20/5 simplifies to 4/1, which is 4.

Then, 1 is just 1, and 7 remains as is. So after simplifying, the numerators would be 1, 2, 4, and denominators would be 1, 7, 3. Wait, let me think again.

Wait, maybe I should approach it step by step. Let's take each fraction and see if we can cancel with another fraction's numerator or denominator.

Looking at the first fraction: 1/5. The second fraction is 8/7. The third is 20/12.

Looking at 8/7 and 20/12. The 8 in the numerator of the second fraction can be simplified with the 12 in the denominator of the third fraction. 8 and 12 have a common factor of 4. So 8 ÷4=2, 12 ÷4=3. So that simplifies 8/12 to 2/3. Wait, but in the original expression, the 8 is in the numerator of the second fraction and the 12 is in the denominator of the third fraction. So actually, it's 8/7 * 20/12. So the 8 in the numerator and the 12 in the denominator can be simplified. So 8 divided by 4 is 2, 12 divided by 4 is 3. So that gives 2/3. Then the 20 in the numerator of the third fraction and the 5 in the denominator of the first fraction can be simplified. 20 divided by 5 is 4, and 5 divided by 5 is 1. So 20/5 is 4. Then the 7 in the denominator of the second fraction remains as is.

So after simplifying, the numerators are 1 (from the first fraction), 2 (from the second fraction's 8/4), 4 (from the third fraction's 20/5), and the denominators are 1 (from the first fraction's 5/5), 7 (from the second fraction's denominator), and 3 (from the third fraction's 12/4). Wait, maybe I need to represent this differently.

Alternatively, when cross-canceling, you can cancel across the entire multiplication. So:

1/5 * 8/7 * 20/12

We can write this as (1 * 8 * 20) / (5 * 7 * 12)

Now, let's factor each number:

1 is 1

8 is 2^3

20 is 2^2 * 5

5 is 5

7 is 7

12 is 2^2 * 3

So the numerator factors: 2^3 (from 8) * 2^2 * 5 (from 20) = 2^(3+2) * 5^1 = 2^5 * 5^1

Denominator factors: 5^1 (from 5) * 7^1 (from 7) * 2^2 * 3^1 (from 12) = 2^2 * 3^1 * 5^1 * 7^1

So numerator: 2^5 * 5

Denominator: 2^2 * 3 * 5 * 7

Now, cancel common factors. Both have 2^2 and 5^1.

So numerator becomes 2^(5-2) = 2^3 = 8

Denominator becomes 3 * 7 = 21

So 8/21, same as before. So that confirms the answer is 8/21.

Alternatively, maybe there's a simpler way. Let me try calculating the original expression step by step without combining all the fractions first.

First, compute 1/5 multiplied by 8/7. That would be (1*8)/(5*7) = 8/35.

Then divide that result by 12/20. Dividing by 12/20 is the same as multiplying by 20/12. So 8/35 * 20/12.

Again, multiply numerators: 8*20=160

Multiply denominators: 35*12=420

160/420. Simplify by dividing numerator and denominator by 20: 8/21. Same result. So regardless of the method, the answer is 8/21.

I think that's solid. Let me check once more with a different approach. Maybe decimal conversion.

First, convert each fraction to decimal:

1/5 is 0.2

8/7 is approximately 1.142857...

12/20 is 0.6

So the original expression is 0.2 * 1.142857... ÷ 0.6

First, multiply 0.2 and 1.142857: 0.2 * 1.142857 ≈ 0.2285714

Then divide by 0.6: 0.2285714 / 0.6 ≈ 0.38095238...

Now, 8/21 as a decimal is approximately 0.38095238... So that matches. Therefore, 8/21 is the correct simplified fraction.

Therefore, after all these steps and checks, the simplified form of the given expression is 8/21.
Therefore, the simplified form of the expression is $\boxed{\frac{8}{21}}$.
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