Question

mai earns $7 per hour mowing her neighbors lawns. she also earned $14 for hauling away bags of recyclables for some neighbors. priya babysits her neighbors children. the table shows the amount of money, m, that priya earns in h hours.

number of hours (h)	amount of money (m)
2	$16.80
4	$33.60

priya and mai have agreed to go to the movies the weekend after they have earned the same amount of money for the same number of work hours.
a. how many hours do they each have to work before they go to the movies?
b. how much will each of them have earned?
c. explain where the solution can be seen in tables of values, graphs, and equations that represent priyas and mais hourly earnings.

Explanation:

Part (a)

Step1: Find Priya's hourly rate

To find Priya's hourly rate, we can use the table. For \( h = 1 \), \( m=\$8.40 \), so her hourly rate \( r_{Priya}=\frac{8.40}{1} = 8.40 \) dollars per hour.

Step2: Write Mai's earnings equation

Mai earns \( \$7 \) per hour mowing lawns and a flat \( \$14 \) for hauling recyclables. So her earnings equation is \( m_{Mai}=7h + 14 \), where \( h \) is the number of hours she works.

Step3: Write Priya's earnings equation

Priya's earnings are based on her hourly rate, so her earnings equation is \( m_{Priya}=8.4h \), where \( h \) is the number of hours she babysits.

Step4: Set the earnings equal and solve for \( h \)

We want to find when \( m_{Mai}=m_{Priya} \), so:
\[
7h + 14=8.4h
\]
Subtract \( 7h \) from both sides:
\[
14 = 8.4h-7h
\]
\[
14 = 1.4h
\]
Divide both sides by \( 1.4 \):
\[
h=\frac{14}{1.4}=10
\]

Part (b)

Step1: Use Priya's equation to find earnings

We can use Priya's earnings equation \( m = 8.4h \). We know \( h = 10 \) from part (a), so:
\[
m=8.4\times10 = 84
\]
(We could also use Mai's equation: \( m = 7\times10+14=70 + 14=84 \))

Part (c)

• Tables of Values: We would create a table for Mai's earnings (\( m = 7h + 14 \)) and Priya's earnings (\( m = 8.4h \)) with different \( h \) values. We would look for the row where the \( m \)-values are the same; this row would have \( h = 10 \) and \( m = 84 \).
• Graphs: We would graph both linear equations. Mai's graph has a y - intercept of 14 and a slope of 7, Priya's graph has a y - intercept of 0 and a slope of 8.4. The solution is the point of intersection of the two lines, which has coordinates \( (10, 84) \), so \( h = 10 \) and \( m = 84 \).
• Equations: As we did in part (a), we set the two earnings equations equal to each other (\( 7h + 14=8.4h \)) and solved for \( h \). The solution to this equation (\( h = 10 \)) gives the number of hours, and substituting \( h = 10 \) back into either equation gives the amount of money earned (\( m = 84 \)).

Answer:

s:
a. They each have to work \(\boldsymbol{10}\) hours.
b. Each of them will have earned \(\boldsymbol{\$84}\).
c. (Explanation as above)