Sure, let's solve this step by step.
We are given two pieces of information:
1. The sum of two numbers is 72.
2. The difference between the two numbers is 19.
Let the larger number be [tex]\( x \)[/tex] and the smaller number be [tex]\( y \)[/tex]. Therefore, we can write the following equations based on the given information:
[tex]\[
x + y = 72 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
x - y = 19 \quad \text{(Equation 2)}
\][/tex]
To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can add these two equations together. By doing this, we eliminate [tex]\( y \)[/tex]:
[tex]\[
(x + y) + (x - y) = 72 + 19
\][/tex]
Simplifying the left-hand side, we get:
[tex]\[
2x = 91
\][/tex]
To solve for [tex]\( x \)[/tex], we divide both sides by 2:
[tex]\[
x = \frac{91}{2} = 45.5
\][/tex]
So, the larger number [tex]\( x \)[/tex] is [tex]\( 45.5 \)[/tex].
Now let's confirm the options:
A) [tex]\( 45 \frac{1}{2} \)[/tex]
B) [tex]\( 44 \frac{1}{2} \)[/tex]
C) 45
D) 46
E) [tex]\( 26 \frac{1}{2} \)[/tex]
The correct answer is [tex]\( 45 \frac{1}{2} \)[/tex], which is [tex]\( 45.5 \)[/tex].
Thus, the larger number is [tex]\( 45 \frac{1}{2} \)[/tex], and the correct option is:
A) [tex]\( 45 \frac{1}{2} \)[/tex].
