Certainly! Let's solve the problem step-by-step, and I'll guide you through the process.
Problem:
If the result of the sum of a certain number and 415, multiplied by 3, is the same as multiplying the number by 9. Determine the number.
Step-by-Step Solution:
1. Define the variable:
Let the unknown number be [tex]\( x \)[/tex].
2. Set up the equation:
According to the problem, when we add 415 to the number [tex]\( x \)[/tex] and then multiply the result by 3, it is the same as multiplying the number [tex]\( x \)[/tex] by 9.
In equation form:
[tex]\[
3 \cdot (x + 415) = 9 \cdot x
\][/tex]
3. Expand and simplify the equation:
First, we apply the distributive property on the left side of the equation:
[tex]\[
3x + 3 \cdot 415 = 9x
\][/tex]
Simplify the multiplication:
[tex]\[
3x + 1245 = 9x
\][/tex]
4. Isolate the variable [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], we need to get all terms involving [tex]\( x \)[/tex] on one side of the equation and the constants on the other side.
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[
1245 = 9x - 3x
\][/tex]
Simplify the right side:
[tex]\[
1245 = 6x
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Divide both sides of the equation by 6 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{1245}{6}
\][/tex]
6. Simplify the fraction:
Simplifying the fraction on the right-hand side:
[tex]\[
x = \frac{1245}{6}
\][/tex]
It reduces to:
[tex]\[
x = \frac{415}{2}
\][/tex]
Thus, the number [tex]\( x \)[/tex] is [tex]\( \frac{415}{2} \)[/tex], or if you prefer a decimal form, approximately [tex]\( 207.5 \)[/tex].
Therefore, the number that satisfies the given condition is [tex]\( \boxed{\frac{415}{2}} \)[/tex].
