Question

1. a mail carrier delivered mail on two different routes: the morning route and the afternoon route. last month, he delivered the morning route 16 times and the afternoon route 12 times for a total distance of 141 miles. this month, he delivered the morning route 10 times and the afternoon route 15 times for a total distance of 123.75 miles. what is the distance of the morning route?

Explanation:

Step1: Define variables

Let \( x \) be the distance of the morning route (in miles) and \( y \) be the distance of the afternoon route (in miles).
From the problem, we can set up the following system of equations:
Last month: \( 16x + 12y = 141 \)
This month: \( 10x + 15y = 123.75 \)

Step2: Simplify the equations

First, simplify the first equation by dividing all terms by 4:
\( 4x + 3y=\frac{141}{4} = 35.25 \) (Equation 1)
Simplify the second equation by dividing all terms by 5:
\( 2x + 3y=\frac{123.75}{5}=24.75 \) (Equation 2)

Step3: Subtract Equation 2 from Equation 1

\( (4x + 3y)-(2x + 3y)=35.25 - 24.75 \)
Simplify the left - hand side: \( 4x+3y - 2x - 3y = 2x \)
Simplify the right - hand side: \( 35.25-24.75 = 10.5 \)
So, \( 2x=10.5 \)

Step4: Solve for x

Divide both sides of the equation \( 2x = 10.5 \) by 2:
\( x=\frac{10.5}{2}=5.25 \)

Answer:

The distance of the morning route is \( 5.25 \) miles.