A rectangular piece of cardboard is 45 cm long and 26 cm wide (a) Find the area of the piece of cardboard. (b) cardboard is cut into squares each of which measures 4 cm by 4 Find the largest number of whole squares that can be cut.

Let's solve each part of the problem step by step. ### **Problem Statement:** A rectangular piece of cardboard is **45 cm** long and **26 cm** wide. **(a) Find the area of the piece of cardboard.** **(b) The cardboard is cut into squares, each measuring **4 cm by 4 cm**. Find the largest number of whole squares that can be cut. --- ### **Solution:** #### **Part (a): Finding the Area** The **area** of a rectangle is calculated using the formula: \[ \text{Area} = \text{Length} \times \text{Width} \] Given: - Length (\( L \)) = **45 cm** - Width (\( W \)) = **26 cm** Plugging the values into the formula: \[ \text{Area} = 45\, \text{cm} \times 26\, \text{cm} = 1170\, \text{cm}^2 \] **Answer to Part (a):** The area of the cardboard is **1170 cm²**. --- #### **Part (b): Maximizing the Number of 4 cm × 4 cm Squares** To determine the maximum number of whole **4 cm × 4 cm** squares that can be cut from the cardboard, follow these steps: 1. **Determine How Many Squares Fit Along Each Dimension:** - **Along the Length (45 cm):** \[ \text{Number of squares} = \left\lfloor \frac{45\, \text{cm}}{4\, \text{cm}} \right\rfloor = \left\lfloor 11.25 \right\rfloor = 11\, \text{squares} \] *(Here, \(\left\lfloor x \right\rfloor\) denotes the floor function, which rounds down to the nearest whole number.)* - **Along the Width (26 cm):** \[ \text{Number of squares} = \left\lfloor \frac{26\, \text{cm}}{4\, \text{cm}} \right\rfloor = \left\lfloor 6.5 \right\rfloor = 6\, \text{squares} \] 2. **Calculate the Total Number of Squares:** \[ \text{Total squares} = \text{Number along length} \times \text{Number along width} = 11 \times 6 = 66\, \text{squares} \] 3. **Verify Remaining Material:** - **Unused Length:** \[ 45\, \text{cm} - (11 \times 4\, \text{cm}) = 45\, \text{cm} - 44\, \text{cm} = 1\, \text{cm} \] - **Unused Width:** \[ 26\, \text{cm} - (6 \times 4\, \text{cm}) = 26\, \text{cm} - 24\, \text{cm} = 2\, \text{cm} \] The leftover pieces (1 cm by 26 cm and 2 cm by 45 cm) are insufficient to form additional **4 cm × 4 cm** squares. **Answer to Part (b):** The largest number of whole squares that can be cut is **66**. --- ### **Final Answer:** **a)** The area of the cardboard is \( 1170\ \text{cm}^2 \). **b)** The largest number of whole \( 4\ \text{cm} \times 4\ \text{cm} \) squares that can be cut is **66**.