Answer :
Answer: 17 minutes
Step-by-step explanation:
To calculate the duration of each routine, let's use a system of equations based on the information you provided. Let \(A\) represent the duration of an arm routine (in minutes) and \(B\) represent the duration of an abdominal routine (also in minutes).
From the information given:
Last week:
- You spent 104 minutes doing 1 arm routine and 3 abdominal routines, which can be expressed as: \(1A + 3B = 104\).
This morning:
- You completed 3 arm routines and 1 abdominal routine, spending 80 minutes, which can be expressed as: \(3A + 1B = 80\).
Now, you have a system of two equations:
1. \(1A + 3B = 104\)
2. \(3A + 1B = 80\)
You can solve this system of equations to find the duration of each routine (A and B).
Let's use the elimination method to solve this system:
First, multiply equation (1) by 3 and equation (2) by -1 to eliminate \(B\):
1. \(3A + 9B = 312\)
2. \(-3A - B = -80\)
Now, add equation (1) to equation (2) to eliminate \(A\):
\((3A - 3A) + (9B - B) = (312 - 80)\)
This simplifies to:
\(8B = 232\)
Now, divide both sides by 8 to solve for \(B\):
\(B = \frac{232}{8}\)
\(B = 29\)
So, an abdominal routine lasts 29 minutes.
Now, substitute this value back into either equation (1) or (2) to solve for \(A\). Let's use equation (1):
\(1A + 3(29) = 104\)
\(A + 87 = 104\)
Subtract 87 from both sides:
\(A = 104 - 87\)
\(A = 17\)
So, an arm routine lasts 17 minutes.
To summarize:
- An arm routine lasts 17 minutes.
- An abdominal routine lasts 29 minutes.
