Question

lesson 9: when are they the same?
let’s use equations to think about situations.
warm-up
1 if you were babysitting, which would you rather? explain your reasoning.

• charge $5 for the first hour and $8 for each additional hour?

or

• charge $15 for the first hour and $6 for each additional hour?

Explanation:

Step1: Define variables

Let \( h \) be the number of additional hours (so total hours is \( h + 1 \)). Let \( C_1 \) be the cost for the first option and \( C_2 \) be the cost for the second option.

For the first option: \( C_1=5 + 8h \) (since $5 for the first hour and $8 for each additional hour).

For the second option: \( C_2 = 15+6h \) (since $15 for the first hour and $6 for each additional hour).

Step2: Analyze for different \( h \)

• Case 1: \( h = 0 \) (only 1 hour)

\( C_1 = 5+8(0)=5 \), \( C_2=15 + 6(0)=15 \). So first option is better.

• Case 2: \( h = 1 \) (2 hours total)

\( C_1=5 + 8(1)=13 \), \( C_2=15+6(1)=21 \). First option better.

• Case 3: \( h = 2 \) (3 hours total)

\( C_1=5 + 8(2)=21 \), \( C_2=15+6(2)=27 \). First option better.

• Case 4: \( h = 3 \) (4 hours total)

\( C_1=5 + 8(3)=29 \), \( C_2=15+6(3)=33 \). First option better.

• Case 5: \( h = 4 \) (5 hours total)

\( C_1=5 + 8(4)=37 \), \( C_2=15+6(4)=39 \). First option better.

• Case 6: \( h = 5 \) (6 hours total)

\( C_1=5 + 8(5)=45 \), \( C_2=15+6(5)=45 \). Equal here.

• Case 7: \( h>5 \) (e.g., \( h = 6 \), 7 hours total)

\( C_1=5 + 8(6)=53 \), \( C_2=15+6(6)=51 \). Now second option is better.

But since babysitting is usually for a short time (likely \( h\leq5 \), i.e., total hours \( \leq6 \)), for short durations (most common babysitting times, like 1 - 5 additional hours, or 2 - 6 total hours), the first option (\( 5 + 8h \)) is cheaper. If babysitting for a long time (more than 6 hours total), the second option becomes cheaper. But typically, babysitting is for a few hours, so the first option is better for typical babysitting durations.

Answer:

If babysitting for a short time (e.g., less than 6 hours total), I would rather charge $5 for the first hour and $8 for each additional hour. If babysitting for a long time (more than 6 hours total), I would rather charge $15 for the first hour and $6 for each additional hour. For typical babysitting durations (a few hours), the first option is better as it is cheaper.