Sure, let's go through this problem step-by-step to find the number.
Step 1: Understand the problem
We are given:
1. When 1 is subtracted from 5 times a number, the result is divided by 3 more than twice the original number.
2. The result is [tex]\(\frac{4}{5}\)[/tex].
Let's denote the unknown number by [tex]\( x \)[/tex].
Step 2: Write the equation
The problem gives us the equation:
[tex]\[
\frac{5x - 1}{2x + 3} = \frac{4}{5}
\][/tex]
Step 3: Solve the equation
To solve for [tex]\( x \)[/tex], we need to eliminate the fraction by cross-multiplying:
[tex]\[
5(5x - 1) = 4(2x + 3)
\][/tex]
Distribute both sides:
[tex]\[
25x - 5 = 8x + 12
\][/tex]
Step 4: Isolate [tex]\( x \)[/tex]
Now, let's get all the [tex]\( x \)[/tex]-terms on one side and the constants on the other side:
[tex]\[
25x - 8x = 12 + 5
\][/tex]
Simplify both sides:
[tex]\[
17x = 17
\][/tex]
Divide both sides by 17:
[tex]\[
x = 1
\][/tex]
Step 5: Verify the solution
Let's substitute [tex]\( x = 1 \)[/tex] back into the original equation to check:
[tex]\[
\frac{5(1) - 1}{2(1) + 3} = \frac{4}{5}
\][/tex]
Simplify the left-hand side:
[tex]\[
\frac{5 - 1}{2 + 3} = \frac{4}{5}
\][/tex]
[tex]\[
\frac{4}{5} = \frac{4}{5}
\][/tex]
The left-hand side equals the right-hand side, so the solution [tex]\( x = 1 \)[/tex] is verified.
Therefore, the number is [tex]\( \boxed{1} \)[/tex].
