Question

Last week William worked out for a total of 110 minutes by doing 2 arm routines and 3 abdominal routines. This week he has completed 4 arm routines and 4 abdominal routines and spent a total of 160 minutes exercising. How long does each routine last
An arm routines takes_______minutes to complete and an abdominal routine takes_____minutes to complete

Answer 1

Let \( x \) represent the time in minutes for each arm routine, and \( y \) represent the time in minutes for each abdominal routine.

From the information provided, we can set up the following equations based on the total time spent on each type of routine for the two weeks.

**Week 1:**
- 2 arm routines and 3 abdominal routines for a total of 110 minutes:
\[
2x + 3y = 110 \quad \text{(1)}
\]

**Week 2:**
- 4 arm routines and 4 abdominal routines for a total of 160 minutes:
\[
4x + 4y = 160 \quad \text{(2)}
\]

We can simplify equation (2) by dividing everything by 4:
\[
x + y = 40 \quad \text{(3)}
\]

Now we have a system of equations:
1. \( 2x + 3y = 110 \)
2. \( x + y = 40 \)

Next, we can use equation (3) to express \( y \) in terms of \( x \):
\[
y = 40 - x
\]

Now substitute \( y \) from equation (3) into equation (1):
\[
2x + 3(40 - x) = 110
\]
\[
2x + 120 - 3x = 110
\]
Combine like terms:
\[
-1x + 120 = 110
\]
Subtract 120 from both sides:
\[
-x = -10
\]
Multiply both sides by -1:
\[
x = 10
\]

Now substitute \( x = 10 \) back into equation (3) to find \( y \):
\[
10 + y = 40
\]
Subtract 10 from both sides:
\[
y = 30
\]

Thus, we find that:
- An arm routine takes **10 minutes**.
- An abdominal routine takes **30 minutes**.

So, the final answer is:
- An arm routine takes **10** minutes to complete, and an abdominal routine takes **30** minutes to complete.