Which statements describe a combination of hours James can work mowing lawns and at the clothing store? Select TWO correct responses.

Understand the Problem

The question is asking us to determine which combinations of hours for mowing lawns and working at a clothing store meet the specified constraints: working less than 10 hours in total and earning at least $56. We will analyze each option to see if it complies with these constraints.

Answer

The compliant combinations are: $(0, 5)$, $(1, 5)$, $(2, 4)$, $(3, 3)$, $(4, 2)$, $(5, 1)$.
Answer for screen readers

The combinations that meet the specified constraints are:

  • $m = 0$, $c = 5$ (5 hours at the clothing store)
  • $m = 1$, $c = 5$ (1 hour mowing and 5 hours at the store)
  • $m = 2$, $c = 4$ (2 hours mowing, 4 hours at the store)
  • $m = 3$, $c = 3$ (3 hours mowing, 3 hours at the store)
  • $m = 4$, $c = 2$ (4 hours mowing, 2 hours at the store)
  • $m = 5$, $c = 1$ (5 hours mowing, 1 hour at the store)

Steps to Solve

  1. Define Variables

Let:

  • $m$ = hours spent mowing lawns
  • $c$ = hours spent working at the clothing store
  1. Set Up Inequalities

From the problem, we know two things:

  • Total hours worked should be less than 10: $$ m + c < 10 $$

  • The total earnings should be at least $56. Assuming mowing lawns pays $8 per hour and working at the clothing store pays $12 per hour, we set up the equation: $$ 8m + 12c \geq 56 $$

  1. Analyze Combinations

Now, we will analyze different combinations of $m$ and $c$ to find which meet both inequalities. We can start by trying various combinations that keep the total hours (m + c) under 10:

  • For instance, if $m = 0$:

    • Then $c < 10$ and $12c \geq 56 \Rightarrow c \geq \frac{56}{12} \approx 4.67$. Hence, $c$ must be at least 5.
  • If $m = 1$:

    • Then $c < 9$, so we calculate $8(1) + 12c \geq 56 \Rightarrow 12c \geq 48 \Rightarrow c \geq 4$ (c must be 4 or more).

Continue testing reasonable combinations of $m$ and $c$ until we find suitable values.

  1. Final Verification

Ensure the chosen combination satisfies:

  • $m + c < 10$
  • $8m + 12c \geq 56$
  1. List Compliant Combinations

Once you find combinations that fulfill both constraints, record them.

The combinations that meet the specified constraints are:

  • $m = 0$, $c = 5$ (5 hours at the clothing store)
  • $m = 1$, $c = 5$ (1 hour mowing and 5 hours at the store)
  • $m = 2$, $c = 4$ (2 hours mowing, 4 hours at the store)
  • $m = 3$, $c = 3$ (3 hours mowing, 3 hours at the store)
  • $m = 4$, $c = 2$ (4 hours mowing, 2 hours at the store)
  • $m = 5$, $c = 1$ (5 hours mowing, 1 hour at the store)

More Information

The constraints representing total hours and earnings create a boundary in which you can efficiently manage time and income. This is a common real-life problem where optimization is key.

Tips

  • Miscalculating the inequalities. Ensure both constraints are maintained.
  • Ignoring possible combinations by not thoroughly testing each scenario.
  • Confusing total hours with total earnings. Keep them separate in your calculations.
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