Full solution

Q. Gabriel is designing a rectangular planter box for their garden. It needs to cover an area of

15 \frac{1}{2} \text{ m}^2

. Gabriel wants it to be

7 \frac{3}{4}

long. How wide does the planter box need to be?

Understand Relationship and Formula: Understand the relationship between area, length, and width for a rectangle. $\newline$ The area $A$ of a rectangle is given by the formula $A = \text{length} \times \text{width}$ .
Convert Mixed Numbers to Fractions: Convert the mixed numbers to improper fractions to make calculations easier. $\newline$ The area is $15 \frac{1}{2} \text{ m}^2$ , which is $(15\times2 + 1)/2 = \frac{31}{2} \text{ m}^2$ . $\newline$ The length is $7 \frac{3}{4} \text{ m}$ , which is $(7\times4 + 3)/4 = \frac{31}{4} \text{ m}$ .
Find Width Using Area Formula: Use the area formula to find the width. $\newline$ Let $w$ be the width of the rectangle. $\newline$ $A = \text{length} \times \text{width}$ $\newline$ $\frac{31}{2} \, \text{m}^2 = \frac{31}{4} \, \text{m} \times w$
Solve for Width: Solve for the width $w$ . $w = \left(\frac{31}{2}\right) m^2 \div \left(\frac{31}{4}\right) m$ $w = \left(\frac{31}{2}\right) \times \left(\frac{4}{31}\right) m$ $w = \left(\frac{31 \times 4}{2 \times 31}\right) m$ $w = \frac{4}{2} m$ $w = 2 m$
Check Calculation: Check the calculation for any mathematical errors. $\newline$ $(\frac{31}{4}) m \times 2 m = (\frac{31}{2}) m^2$ $\newline$ $(\frac{31}{4}) \times 2 = \frac{31}{2}$ $\newline$ $(\frac{31}{2}) = (\frac{31}{2})$ $\newline$ The calculation is correct.