<Question>
Write the expression \[\frac{4+6a}{5}-\frac{1+3a}{4}\] as a single fraction.
</Question>

<think>
<Plan_1>
Okay, so I need to combine these two fractions into a single expression. The original expression is (4 + 6a)/5 minus (1 + 3a)/4. Alright, let me think about how to do this step by step.

First, I remember that to subtract two fractions, they need to have the same denominator. So, the key here is to find a common denominator for both fractions. The denominators here are 5 and 4. The least common denominator (LCD) of 5 and 4 should be 20 because 20 is the smallest number that both 5 and 4 divide into evenly.

Let me verify that.
For 5: 20 divided by 5 is 4, and for 4: 20 divided by 4 is 5. So, the LCD is indeed 20.

Now, I need to convert each fraction to an equivalent one with the denominator of 20. Starting with the first fraction, (4 + 6a)/5. To change the denominator from 5 to 20, I multiply both the numerator and the denominator by 4. That would give me [(4 + 6a) * 4]/(5 * 4) which simplifies to (16 + 24a)/20.

Next, the second fraction is (1 + 3a)/4. To convert this to a denominator of 20, I multiply numerator and denominator by 5. So, [(1 + 3a) * 5]/(4 * 5) becomes (5 + 15a)/20.

Now, the original expression becomes (16 + 24a)/20 minus (5 + 15a)/20. Since the denominators are the same, I can combine the numerators: (16 + 24a) - (5 + 15a) all over 20.

Let me handle the numerator carefully. Distribute the negative sign to both terms in the second part: 16 + 24a - 5 - 15a. Now, combine like terms.

<prompt_1.1> Combine the terms in the numerator by following the above plan. </prompt_1.1>
-----

Alternatively, another way to check is to distribute the negative sign first: (16 +24a) -5 -15a. Then combine 16-5=11, 24a-15a=9a. So 11 +9a, which is the same as 11 +9a over 20. Yep, same result.
</Plan_1>

<Execution_1>
<execution_1.1>
The constants are 16 and -5. 16 minus 5 is 11. The terms with 'a' are 24a and -15a. 24a minus 15a is 9a. So, the combined numerator is 11 + 9a.

Therefore, putting that over the common denominator, the expression becomes (11 + 9a)/20.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, let me double-check my steps to make sure I didn't make a mistake. Starting with the original expression: (4 + 6a)/5 - (1 + 3a)/4.

<prompt_2.1> Verify the steps of converting the original expression to a single fraction by following the above plan. </prompt_2.1>
-----

Let me just make sure there are no arithmetic errors. Let's go through each step again.

<prompt_2.2> Recheck the arithmetic steps to ensure no errors were made by following the above plan. </prompt_2.2>

Alternatively, maybe I can use another method to verify. Let's expand the original expression without combining first. Let's distribute the subtraction sign.

<prompt_2.3> Verify the result using distribution and combining terms by following the above plan. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Convert to 20 denominators: (16 + 24a)/20 - (5 + 15a)/20. Subtract numerators: 16 -5 =11, 24a -15a=9a. So, yes, (11 +9a)/20.
</execution_2.1>

<execution_2.2>
1. Find LCD of 5 and 4: 20. Correct.

2. Convert first fraction: (4 + 6a)/5 * 4/4 = (16 + 24a)/20. Yes, 4*4=16, 6a*4=24a.

3. Convert second fraction: (1 + 3a)/4 *5/5= (5 +15a)/20. Correct, 1*5=5, 3a*5=15a.

4. Subtract numerators: (16 +24a) - (5 +15a) = 16 -5 +24a -15a = 11 +9a. Then over 20.
</execution_2.2>

<execution_2.3>
So, (4 + 6a)/5 - (1 + 3a)/4. Let's write it as (4/5 + 6a/5) - (1/4 + 3a/4). Then, distribute the subtraction: 4/5 + 6a/5 - 1/4 - 3a/4.

Now, group the constant terms and the 'a' terms: (4/5 - 1/4) + (6a/5 - 3a/4). Let's compute each part separately.

For the constants: 4/5 - 1/4. LCD is 20 again. 4/5 is 16/20, 1/4 is 5/20. So, 16/20 - 5/20 = 11/20.

For the 'a' terms: 6a/5 - 3a/4. Let's factor out the 'a': a*(6/5 - 3/4). Compute 6/5 - 3/4. LCD is 20. 6/5 is 24/20, 3/4 is 15/20. So, 24/20 - 15/20 = 9/20. Therefore, the 'a' terms combine to (9/20)a.

Putting it all together: 11/20 + (9/20)a, which is the same as (11 + 9a)/20.
</execution_2.3>
</Execution_2>

<Final_answer>
Therefore, the expression written as a single fraction is $\boxed{\frac{9a + 11}{20}}$.
</Final_answer>
</think>