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Relate area to multiplication

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5403523
Use the labeled dimensions in the figure to find the area of the rectangular strip.
Figure for problem 540352

Hints

- Read the width and height from the figure. - Think of the rectangle as equal rows of unit squares. - Multiply the two side lengths to find the area.

Solution

1. The figure shows a rectangle \(9\) units wide and \(3\) units tall. 2. Multiply the side lengths: \(9 \times 3 = 27\).

Answer

The area is \(27\) square units.
5402423
A game board has an area of \(24\) square units. It will be covered with identical rectangular tiles, and each tile covers exactly \(2\) square units. How many tiles are needed if there are no gaps or overlaps?

Hints

- Relate the area of one tile to the board's total area. - Find how many equal \(2\)-square-unit groups fit in \(24\) square units.

Solution

1. Each tile accounts for \(2\) of the \(24\) square units. 2. The number of tiles is \(24 \div 2 = 12\).

Answer

The board needs \(12\) tiles.
5402983
A rectangular fabric patch is \(6\) inches by \(4\) inches. Eli correctly calculates \(6 \times 4 = 24\), but writes the answer as \(24\) inches. What should the answer and unit be? Explain.

Hints

- Decide what kind of measurement the product of the side lengths represents. - Match the unit to a surface covered by squares.

Solution

1. Multiplying the side lengths counts \(24\) one-square-inch units covering the rectangle. 2. Area uses square units, not linear units.

Answer

The area is \(24\) square inches. “Inches” alone would describe a length, not an area.
5403173
Rectangle A is \(4\) units by \(9\) units. Square B has side length \(6\) units. Do the two figures have the same area? Show the products.

Hints

- Find each figure's area from its side lengths. - Compare the products rather than the shapes' appearances.

Solution

1. Rectangle A's area is \(4 \times 9 = 36\) square units. 2. Square B's area is \(6 \times 6 = 36\) square units. 3. The products are equal.

Answer

Yes. Both figures have an area of \(36\) square units.
5403623
The figure shows a rectangular mural with a rectangular section left unpainted. Use the dimensions in the figure to find the painted area.
Figure for problem 540362

Hints

- Read the dimensions of the whole rectangle and the unpainted rectangle from the figure. - Find both areas before subtracting. - The painted area is the whole area minus the unpainted area.

Solution

1. The whole mural is \(6\) units by \(3\) units, so its area is \(6 \times 3 = 18\) square units. 2. The unpainted section is \(2\) units by \(2\) units, so its area is \(2 \times 2 = 4\) square units. 3. Subtract the unpainted area: \(18 - 4 = 14\) square units.

Answer

The painted area is \(14\) square units.
5404133
Each rectangle has \(6\) unit squares in every row. Complete the table. <table><tr><th>Number of rows</th><th>Area</th></tr><tr><td>\(2\)</td><td>\(12\) square units</td></tr><tr><td>\(3\)</td><td>\(18\) square units</td></tr><tr><td>\(4\)</td><td>?</td></tr></table>

Hints

- Use the table to identify the number of unit squares in each row. - Multiply the number of rows by the number of squares in each row. - Express area in square units.

Solution

1. Each row contains \(6\) unit squares. 2. Four rows contain \(4 \times 6 = 24\) unit squares. 3. Therefore, the area is \(24\) square units.

Answer

The missing table entry is \(24\) square units.
5165253
A tile installer makes squares from rectangular tiles. Each rectangular tile is \(20\,\text{cm}\) long and \(10\,\text{cm}\) wide. The installer places two tiles together along their long sides to make one square. a) What is the side length of the square? b) The installer wants to cover a square area with side length \(60\,\text{cm}\) using the squares from part a). How many of the squares are needed? c) How many original rectangular tiles are needed altogether?

Hints

- Think about how two rectangles can make a square. - Find how many small-square side lengths fit along one side of the large square. - Remember that the squares form both rows and columns. - Each small square is made from two rectangular tiles.

Solution

1. Placing two \(20\,\text{cm} \times 10\,\text{cm}\) rectangles together along their \(20\,\text{cm}\) sides makes a \(20\,\text{cm} \times 20\,\text{cm}\) square. 2. Along each side of the larger square, \(60 \div 20 = 3\) small squares fit. Therefore, \(3 \times 3 = 9\) small squares are needed. 3. Each small square uses \(2\) rectangular tiles, so \(9 \times 2 = 18\) rectangular tiles are needed.

Answer

a) \(20\,\text{cm}\) b) \(9\) squares c) \(18\) rectangular tiles
5317153
Two figures, A and B, are shown on a geoboard. The small gray square in the lower-right corner represents one unit square. a) For each figure, write one multiplication-and-addition expression that splits the figure into two non-overlapping rectangles, and use it to find the area. b) Which figure has the greater area, and by how many square units?
Figure for problem 531715

Hints

- Find a way to view each figure as two rectangles that do not overlap. - For each rectangular part, use length times width rather than counting individual unit squares. - Add the two rectangular areas, then compare the two totals.

Solution

1. Figure A can be split into a \(4\)-by-\(2\) rectangle and a \(2\)-by-\(2\) rectangle: \((4 \times 2) + (2 \times 2) = 8 + 4 = 12\) square units. 2. Figure B can be split into a \(4\)-by-\(2\) rectangle and a \(2\)-by-\(3\) rectangle: \((4 \times 2) + (2 \times 3) = 8 + 6 = 14\) square units. 3. Compare the areas: \(14 - 12 = 2\) square units.

Answer

a) One valid pair of expressions is: A: \((4 \times 2) + (2 \times 2) = 12\) square units B: \((4 \times 2) + (2 \times 3) = 14\) square units b) Figure B has the greater area by \(2\) square units.
5402333
A rectangular display has \(7\) rows of \(9\) unit squares. Nia splits each row into a group of \(5\) squares and a group of \(4\) squares. Write the multiplication-and-addition equation that represents Nia's split, including both partial products, and use it to find the area.

Hints

- Keep the common \(7\) rows in both rectangular parts. - The two widths must match Nia's \(5 + 4\) split. - Your final equation should show both partial products before their sum.

Solution

1. Nia's split makes a \(7\)-by-\(5\) rectangle and a \(7\)-by-\(4\) rectangle. 2. Their areas are \(7 \times 5 = 35\) and \(7 \times 4 = 28\) square units. 3. The split is represented by \((7 \times 5) + (7 \times 4) = 35 + 28 = 63\).

Answer

\((7 \times 5) + (7 \times 4) = 35 + 28 = 63\) square units
5402383
A rectangular stage mat has an area of \(48\) square feet. It is \(6\) feet wide. How long is the mat?

Hints

- Think of the rectangle as equal rows of square feet. - Use the known area and one side length to find the missing factor.

Solution

1. The area equals width times length. 2. Find the missing factor: \(48 \div 6 = 8\).

Answer

The mat is \(8\,\text{ft}\) long.
5402433
The figure shown is made from two non-overlapping rectangles. Use the dimensions in the figure to find the area of the whole figure.
Figure for problem 540243

Hints

- Read the side lengths from the figure. - Find the area of each rectangle. - Add the two areas because the rectangles do not overlap.

Solution

1. Find the area of the larger rectangle: \(6 \times 8 = 48\) square units. 2. Find the area of the smaller rectangle: \(2 \times 5 = 10\) square units. 3. Add the non-overlapping areas: \(48 + 10 = 58\) square units.

Answer

The area of the figure is \(58\) square units.
5402543
The unit-square grid is split into a shaded rectangle and an unshaded rectangle. Which expression matches the **two-part split shown**, not merely the total area? Explain what the factors in your choice represent, then find the whole area. A. \(7 \times 8\) B. \((7 \times 5) + (7 \times 3)\) C. \((8 \times 5) + (8 \times 2)\) D. \(7 + 8\)
Figure for problem 540254

Hints

- Do not decide only by evaluating the expressions; more than one choice can equal the total area. - Read the number of rows in both parts and the width of each part from the grid. - Match those visible dimensions to the factors in one expression.

Solution

1. The grid has \(7\) rows. The shaded part is \(5\) columns wide, and the unshaded part is \(3\) columns wide. 2. Therefore, the shown split is represented by \((7 \times 5) + (7 \times 3)\), which is choice B. 3. The partial areas are \(35\) and \(21\) square units. 4. Add them: \(35 + 21 = 56\) square units. 5. Choices A and C also evaluate to \(56\), but they do not describe the two rectangular parts shown in the grid.

Answer

B. \((7 \times 5) + (7 \times 3) = 56\) square units
5402593
One rectangular photo mat is \(7\) inches by \(6\) inches. Another is \(8\) inches by \(5\) inches. Which mat has the greater area, and by how many square inches?

Hints

- Find each rectangle's area before comparing them. - The side lengths are not enough by themselves; compare the products.

Solution

1. Find the first area: \(7 \times 6 = 42\) square inches. 2. Find the second area: \(8 \times 5 = 40\) square inches. 3. Compare: \(42 - 40 = 2\) square inches.

Answer

The \(7\)-inch by \(6\)-inch mat has the greater area by \(2\) square inches.
5402693
Three non-overlapping rectangular panels are each \(4\) feet by \(5\) feet. What total area do the three panels cover?

Hints

- Find the area of one rectangle first. - The panels do not overlap, so their areas can be combined.

Solution

1. Find one panel's area: \(4 \times 5 = 20\) square feet. 2. Multiply by the number of panels: \(3 \times 20 = 60\) square feet.

Answer

The panels cover \(60\) square feet altogether.
5402743
A rectangular array has \(5\) rows and an area of \(35\) square units. One unit square is added to the end of every row. What is the new area?

Hints

- Use the original area and number of rows to find the row length. - Apply the same change to every row before finding the new area.

Solution

1. Find the original number of squares in each row: \(35 \div 5 = 7\). 2. Adding one square to every row makes \(8\) squares per row. 3. Find the new area: \(5 \times 8 = 40\) square units.

Answer

The new area is \(40\) square units.
5402793
A \(6\)-by-\(9\) rectangle is split into two smaller rectangles. One part is \(6\) by \(4\). The other part has the same height of \(6\). What is the missing width, and what is the area of that part?

Hints

- The two smaller widths must combine to make the whole rectangle’s width. - After finding the missing width, use the shared height to find that part’s area.

Solution

1. The whole width is \(9\), and one part uses width \(4\). 2. The missing width is \(9 - 4 = 5\). 3. The missing part’s area is \(6 \times 5 = 30\) square units. 4. Check the whole area: \(6 \times 4 + 6 \times 5 = 24 + 30 = 54\), and \(6 \times 9 = 54\).

Answer

The missing width is \(5\) units, and the missing area is \(30\) square units.
5402873
The figure shows a square courtyard with a square planter inside it. Rowan subtracts the planter's side length from the courtyard's side length and then squares the result, getting \(16\) square yards for the area outside the planter. Explain Rowan's error and find the correct area outside the planter.
Figure for problem 540287

Hints

- Read the side lengths of both squares from the figure. - Compare the region represented by Rowan's square with the entire region outside the planter. - Find the area of the whole courtyard and the area of the planter before subtracting.

Solution

1. From the figure, the courtyard is \(7\) yards by \(7\) yards and the planter is \(3\) yards by \(3\) yards. 2. Rowan's subtraction creates only one \(4\)-yard by \(4\)-yard square; it does not represent all the space around the planter. 3. The courtyard area is \(7 \times 7 = 49\) square yards. 4. The planter area is \(3 \times 3 = 9\) square yards. 5. Subtract the areas: \(49 - 9 = 40\) square yards.

Answer

Rowan subtracted side lengths instead of subtracting areas. The area outside the planter is \(40\) square yards.
5403223
A \(5\)-by-\(5\) square array can be enlarged in two ways. Plan A adds one complete row of \(5\) unit squares. Plan B adds one complete column of \(5\) unit squares. Compare the new areas. Does either plan create a larger area?

Hints

- Write the dimensions made by each plan. - Compare the two products rather than relying on the direction of the added strip.

Solution

1. Plan A makes a \(6\)-by-\(5\) rectangle with area \(6 \times 5 = 30\) square units. 2. Plan B makes a \(5\)-by-\(6\) rectangle with area \(5 \times 6 = 30\) square units. 3. Both plans add the same \(5\) square units, so neither new area is larger.

Answer

Both plans create an area of \(30\) square units.
5403273
A rectangle is \(9\) units long and \(3\) units wide. A second rectangle has the same length but twice the width. Find both areas and the increase in area.

Hints

- Change the width before finding the second area. - Compare the two products after keeping the length fixed.

Solution

1. The first area is \(9 \times 3 = 27\) square units. 2. Twice the width is \(2 \times 3 = 6\) units. 3. The second area is \(9 \times 6 = 54\) square units. 4. The increase is \(54 - 27 = 27\) square units.

Answer

The areas are \(27\) and \(54\) square units. The area increases by \(27\) square units.
5403323
A rectangle has one side length of \(6\) units and an area of \(42\) square units. After that rectangle is extended along its other side, its area is \(60\) square units. How many units were added to the other side?

Hints

- Use each area with the unchanged side length to find the corresponding missing dimension. - Compare the original and new dimensions.

Solution

1. The original other side is \(42 \div 6 = 7\) units. 2. The extended other side is \(60 \div 6 = 10\) units. 3. The extension is \(10 - 7 = 3\) units.

Answer

The other side was extended by \(3\) units.
5403373
A \(3\)-by-\(9\) rectangle is cut into two smaller rectangles. The cut makes one piece \(3\) units by \(3\) units and the other piece \(3\) units by \(6\) units. Find the area of each piece and the area of the whole rectangle.

Hints

- Find each smaller rectangle’s area from its side lengths. - Add the two non-overlapping areas to get the whole area. - Check the result using the original \(3\)-by-\(9\) dimensions.

Solution

1. The smaller piece has area \(3 \times 3 = 9\) square units. 2. The larger piece has area \(3 \times 6 = 18\) square units. 3. Add the partial areas: \(9 + 18 = 27\) square units. 4. Check the whole rectangle: \(3 \times 9 = 27\) square units.

Answer

The piece areas are \(9\) square units and \(18\) square units. The whole rectangle has area \(27\) square units.
5403573
A \(6\)-by-\(6\) square is enlarged by attaching a \(2\)-by-\(6\) rectangle along one side. What is the added area, and what is the total area of the new \(8\)-by-\(6\) rectangle?

Hints

- Find the area of the original square and the attached strip separately. - The two parts meet along an edge but do not overlap.

Solution

1. The square’s area is \(6 \times 6 = 36\) square units. 2. The attached rectangle’s area is \(2 \times 6 = 12\) square units. 3. The total area is \(36 + 12 = 48\) square units.

Answer

The added area is \(12\) square units, and the total area is \(48\) square units.
5403633
A \(9\)-by-\(9\) square grid is divided by grid lines every \(3\) units, making equal \(3\)-by-\(3\) blocks. A student says there are \(3 + 3 = 6\) blocks because there are \(3\) rows of blocks and \(3\) columns of blocks. Explain the error. How many blocks are there, what is the area of one block, and what is the area of the whole grid?

Hints

- Think of the blocks as a rectangular array, not as two separate lists. - Find the number of blocks by combining rows and columns. - Check the total area in two ways: from the blocks and from the full grid dimensions.

Solution

1. The blocks form an array with \(3\) rows and \(3\) columns, so the number of blocks is \(3 \times 3 = 9\), not \(3 + 3\). 2. One block has area \(3 \times 3 = 9\) square units. 3. The nine blocks have total area \(9 \times 9 = 81\) square units. 4. This matches the area of the original \(9\)-by-\(9\) grid: \(9 \times 9 = 81\) square units.

Answer

There are \(9\) blocks. Each block has area \(9\) square units, and the whole grid has area \(81\) square units.
5403663
An \(8\)-by-\(6\) rectangle is divided by one vertical center line and one horizontal center line, making \(4\) equal smaller rectangles. What are the dimensions and area of each smaller rectangle? Verify the whole area.

Hints

- Each center line halves one dimension. - Use the area of one part and the number of equal parts to check the whole.

Solution

1. Halving the \(8\)-unit side gives \(4\) units, and halving the \(6\)-unit side gives \(3\) units. 2. Each smaller rectangle is \(4\) units by \(3\) units and has area \(4 \times 3 = 12\) square units. 3. The four parts have total area \(4 \times 12 = 48\) square units, matching \(8 \times 6 = 48\).

Answer

Each smaller rectangle is \(4\) units by \(3\) units with area \(12\) square units. The whole area is \(48\) square units.
5403723
A rectangle has \(7\) rows and \(6\) columns. Two complete columns are removed. Luis says the area decreases by only \(2\) square units because \(2\) columns were removed. Explain his error and find the new area.

Hints

- Determine how many unit squares are in one complete column. - Distinguish the number of columns removed from the area removed.

Solution

1. Each removed column contains \(7\) unit squares. 2. Two columns remove \(2 \times 7 = 14\) square units, not \(2\) square units. 3. Four columns remain, so the new area is \(7 \times 4 = 28\) square units.

Answer

Luis counted columns instead of the unit squares in them. The new area is \(28\) square units.
5403763
A rectangular sheet is \(6\) units by \(8\) units. A \(2\)-unit by \(3\)-unit label is placed on top of the sheet without cutting or removing any paper. A student says the sheet's area is now \(42\) square units. Explain the error. What are the sheet's area and the visible area not covered by the label?

Hints

- Decide whether covering part of a region changes the region itself. - Keep the total area separate from the uncovered visible area.

Solution

1. The sheet's area is \(6 \times 8 = 48\) square units. 2. Placing a label on top does not remove paper, so the sheet's area remains \(48\) square units. 3. The label covers \(2 \times 3 = 6\) square units, leaving \(48 - 6 = 42\) square units visible.

Answer

The student confused the visible uncovered area with the sheet's total area. The sheet's area is \(48\) square units, and \(42\) square units remain visible.
5403803
A tile design has an area of \(18\) square units. Six square units are white, and \(8\) square units are blue. The rest is yellow. What is the yellow area?

Hints

- Combine the areas of the two known colored parts. - The three colors together make the whole design. - Subtract the known area from the total area.

Solution

1. The white and blue parts cover \(6 + 8 = 14\) square units. 2. Subtract the known parts from the whole: \(18 - 14 = 4\) square units.

Answer

The yellow area is \(4\) square units.
5403853
The figure is made from two non-overlapping rectangles. Use the dimensions in the figure to find the area of the whole figure.
Figure for problem 540385

Hints

- Read the dimensions of the two rectangles from the figure. - Find each rectangle's area. - Add the areas because the rectangles do not overlap.

Solution

1. The lower rectangle is \(4\) units by \(3\) units, so its area is \(4 \times 3 = 12\) square units. 2. The upper rectangle is \(3\) units by \(4\) units, so its area is \(3 \times 4 = 12\) square units. 3. Add the non-overlapping areas: \(12 + 12 = 24\) square units.

Answer

The area of the whole figure is \(24\) square units.
5403863
A \(6\)-row by \(9\)-column rectangle has every other column shaded, beginning with the first column. Find the shaded area and the unshaded area.

Hints

- List the alternating column numbers to determine how many are shaded. - Each complete column contains the same number of unit squares.

Solution

1. The shaded columns are \(1\), \(3\), \(5\), \(7\), and \(9\), so \(5\) columns are shaded. 2. The shaded area is \(6 \times 5 = 30\) square units. 3. Four columns are unshaded, so the unshaded area is \(6 \times 4 = 24\) square units.

Answer

The shaded area is \(30\) square units, and the unshaded area is \(24\) square units.
5403953
A square has an area of \(36\) square units. A rectangle has the same area and a width of \(4\) units. Find the square’s side length and the rectangle’s length.

Hints

- Find two equal factors for the square’s area. - Use the known rectangle width as one factor of the same area. - Check that both sets of dimensions give \(36\) square units.

Solution

1. The square’s equal side lengths satisfy \(6 \times 6 = 36\), so its side length is \(6\) units. 2. The rectangle’s length is \(36 \div 4 = 9\) units.

Answer

The square’s side length is \(6\) units, and the rectangle’s length is \(9\) units.
5403993
A \(9\)-row by \(8\)-column rectangle has \(6\) rows shaded. Find the shaded area and the unshaded area.

Hints

- Use the number of shaded rows and the number of columns to find the shaded area. - Subtract the shaded rows from the total number of rows. - Use the remaining rows to find the unshaded area.

Solution

1. The shaded area is \(6 \times 8 = 48\) square units. 2. The remaining \(9 - 6 = 3\) rows are unshaded. 3. The unshaded area is \(3 \times 8 = 24\) square units.

Answer

The shaded area is \(48\) square units, and the unshaded area is \(24\) square units.
5404043
The same \(36\) unit squares are arranged first as a rectangle with \(4\) rows and then as a rectangle with \(6\) rows. How many squares are in each row in the two arrangements?

Hints

- The total number of unit squares stays fixed. - Divide the total area by each row count. - Check each result by multiplying rows by squares per row.

Solution

1. With \(4\) rows, each row has \(36 \div 4 = 9\) squares. 2. With \(6\) rows, each row has \(36 \div 6 = 6\) squares. 3. Both arrangements have area \(36\) square units.

Answer

The first rectangle has \(9\) squares per row, and the second has \(6\) squares per row.
5404283
Rectangle A has \(8\) rows of \(7\) unit squares. Rectangle B has \(5\) rows of \(7\) unit squares. How much greater is the area of Rectangle A? Explain the difference as the area of an extra rectangular strip.

Hints

- Compare both the total areas and the numbers of equal rows. - The rows present in one rectangle but not the other form their own rectangle.

Solution

1. Rectangle A has area \(8 \times 7 = 56\) square units. 2. Rectangle B has area \(5 \times 7 = 35\) square units. 3. The difference is \(56 - 35 = 21\) square units. 4. Rectangle A has \(8 - 5 = 3\) extra rows, forming a \(3\)-by-\(7\) strip with area \(3 \times 7 = 21\) square units.

Answer

Rectangle A’s area is \(21\) square units greater. The difference is a \(3\)-by-\(7\) strip.
5404403
The figure shows a rectangular frame with an open region inside it. Use the dimensions in the figure to find the area of the solid region.
Figure for problem 540440

Hints

- Read the outer rectangle's dimensions from the figure. - Break the open region into two non-overlapping rectangles using the labeled lengths. - Subtract the open area from the outer rectangle's area.

Solution

1. The outer rectangle is \(6\) units by \(4\) units, so its area is \(6 \times 4 = 24\) square units. 2. Split the opening into a \(4\)-by-\(1\) rectangle and a \(3\)-by-\(1\) rectangle. Their areas are \(4\) and \(3\) square units, for \(7\) square units altogether. 3. Subtract the opening from the whole rectangle: \(24 - 7 = 17\) square units.

Answer

The solid region has an area of \(17\) square units.
5404413
A rectangular hall is \(6\) units by \(8\) units. A \(2\)-unit-wide aisle runs along one \(6\)-unit side. What area remains for seating?

Hints

- Find the area of the entire hall. - Find the rectangular aisle’s area. - Subtract the aisle area from the hall area.

Solution

1. The hall’s area is \(6 \times 8 = 48\) square units. 2. The aisle’s area is \(2 \times 6 = 12\) square units. 3. Subtract: \(48 - 12 = 36\) square units.

Answer

The seating area is \(36\) square units.
5404573
A rectangle has an area of \(6\) square units and one side length of \(2\) units. Its other side is tripled. Ava adds \(3\) units to that side and gets a new area of \(12\) square units. Explain Ava’s error and find the correct new area.

Hints

- Find the original missing side first. - “Tripled” means multiplied by \(3\), not increased by \(3\). - Use the unchanged side and the new side to find the new area.

Solution

1. The original other side is \(6 \div 2 = 3\) units. 2. Tripling means multiplying by \(3\), so the new side is \(3 \times 3 = 9\) units. 3. The correct new area is \(2 \times 9 = 18\) square units. 4. Ava treated “tripled” as “increased by \(3\).”

Answer

Ava used addition instead of multiplication. The correct new area is \(18\) square units.
5404613
A rectangular sheet is \(8\) units by \(6\) units. It is folded exactly in half so the \(8\)-unit side becomes \(4\) units while the \(6\)-unit side stays the same. What are the original area and the visible area after folding?

Hints

- Use the dimensions before and after the fold as two rectangles. - Only one dimension changes when the sheet is folded this way.

Solution

1. The original area is \(8 \times 6 = 48\) square units. 2. After folding, the visible rectangle is \(4\) units by \(6\) units. 3. The visible area is \(4 \times 6 = 24\) square units.

Answer

The original area is \(48\) square units, and the visible area after folding is \(24\) square units.
5404633
The figure shows a rectangular board with two non-overlapping rectangular openings. Use the dimensions in the figure to find the area of the board that remains solid.
Figure for problem 540463

Hints

- Read the dimensions of the whole board and both openings from the figure. - Find the three rectangular areas separately. - Subtract both opening areas from the whole-board area.

Solution

1. The whole board is \(9\) units by \(8\) units, so its area is \(9 \times 8 = 72\) square units. 2. The first opening is \(2\) units by \(3\) units, so its area is \(2 \times 3 = 6\) square units. 3. The second opening is \(1\) unit by \(4\) units, so its area is \(1 \times 4 = 4\) square units. 4. Subtract both openings: \(72 - 6 - 4 = 62\) square units.

Answer

The board has \(62\) square units of solid area remaining.
5402923
A rectangle has an area of \(18\) square units. Its side lengths are whole numbers, and one side is twice as long as the other. What are the side lengths?

Hints

- List factor pairs that can form a rectangle with the given area. - Test the relationship between the two side lengths for each pair.

Solution

1. Whole-number factor pairs of \(18\) are \(1\) and \(18\), \(2\) and \(9\), and \(3\) and \(6\). 2. Only \(3\) and \(6\) have one factor that is twice the other. 3. Check: \(3 \times 6 = 18\).

Answer

The side lengths are \(3\) units and \(6\) units.
5403043
A \(6\)-by-\(6\) square is split along a grid line into two nonempty rectangles. A student labels one part as having an area of \(14\) square units. Could that label be correct if both rectangles have whole-number side lengths? Explain.

Hints

- Identify the side length that both smaller rectangles must keep. - Consider the possible whole-number widths of a nonempty part.

Solution

1. Both smaller rectangles keep a side length of \(6\) units. 2. Their other side lengths must be whole numbers from \(1\) through \(5\). 3. The possible areas of one part are \(6 \times 1 = 6\), \(6 \times 2 = 12\), \(6 \times 3 = 18\), \(6 \times 4 = 24\), or \(6 \times 5 = 30\) square units. 4. Since \(14\) is not one of these possible areas, the label cannot be correct.

Answer

No. A part created by the grid-line split must have an area of \(6\), \(12\), \(18\), \(24\), or \(30\) square units, not \(14\) square units.
5403813
A rectangle has area \(72\) square units and one side length of \(8\) units. It is divided into three same-height strips with areas \(16\), \(24\), and \(32\) square units. Find the width of each strip and verify the full rectangle's other side length.

Hints

- Use the shared side length with each strip area to recover a missing width. - Add the adjacent strip widths to rebuild the whole rectangle.

Solution

1. Each strip shares the \(8\)-unit height. 2. The strip widths are \(16 \div 8 = 2\), \(24 \div 8 = 3\), and \(32 \div 8 = 4\) units. 3. Their widths total \(2 + 3 + 4 = 9\) units. 4. The full rectangle is \(8\) by \(9\), and \(8 \times 9 = 72\).

Answer

The strip widths are \(2\), \(3\), and \(4\) units. The full rectangle's other side length is \(9\) units.
5404143
A rectangular array originally has \(6\) equal rows. After one entire row is removed, \(35\) unit squares remain. How many squares were in each row, and what was the area of the original rectangle?

Hints

- Find how many rows remain after one row is removed. - Use the remaining area to determine the equal number of squares in each row. - Restore the removed row when finding the original area.

Solution

1. Removing one of \(6\) rows leaves \(5\) rows. 2. The number of squares in each row is \(35 \div 5 = 7\). 3. The original area is \(6 \times 7 = 42\) square units.

Answer

Each row had \(7\) squares, and the original rectangle had area \(42\) square units.
5404203
A rectangle must use all \(54\) unit squares. It must have more than \(6\) rows but fewer than \(10\) rows. How many rows can it have, and how many squares will be in each row?

Hints

- List the whole-number row counts allowed by the condition. - Test which row count uses all the unit squares in equal rows.

Solution

1. The possible row counts are \(7\), \(8\), and \(9\). 2. Neither \(7\) nor \(8\) divides \(54\) into equal whole-number rows. 3. Nine rows work because \(54 \div 9 = 6\).

Answer

The rectangle can have \(9\) rows with \(6\) squares in each row.
5404233
A \(9\)-by-\(8\) rectangular board will be tiled with \(3\)-by-\(4\) rectangles. Maya places the \(3\)-unit side of each tile along the \(9\)-unit side of the board and the \(4\)-unit side along the \(8\)-unit side. Leo wants to rotate every tile. Which all-one-direction arrangement fits exactly, and how many tiles does it use? Explain why the other orientation does not make a complete grid.

Hints

- Check how many tile side lengths fit along each board side. - An exact rectangular grid cannot leave a partial tile at an edge.

Solution

1. Maya's orientation fits \(9 \div 3 = 3\) tiles across and \(8 \div 4 = 2\) tiles down. 2. Her arrangement uses \(3 \times 2 = 6\) tiles and covers the board exactly. 3. If every tile is rotated, the \(4\)-unit side would have to fit evenly along the \(9\)-unit side, but \(9\) cannot be divided into whole groups of \(4\).

Answer

Maya's orientation fits exactly and uses \(6\) tiles. Rotating every tile does not make a complete grid because \(4\)-unit lengths do not fit evenly along \(9\) units.
5404323
Two rectangles share a side length of \(7\) units and are placed side by side without overlapping. Their areas are \(28\) square units and \(35\) square units. What are their other side lengths? What are the dimensions and area of the combined rectangle?

Hints

- Use each known area with the shared side length to find the missing dimension. - When the rectangles are joined, add the side lengths in the direction they are placed side by side.

Solution

1. The first rectangle's other side is \(28 \div 7 = 4\) units. 2. The second rectangle's other side is \(35 \div 7 = 5\) units. 3. Side by side, the combined width is \(4 + 5 = 9\) units, while the common side remains \(7\) units. 4. The combined area is \(28 + 35 = 63\) square units, which also equals \(7 \times 9 = 63\).

Answer

The other side lengths are \(4\) units and \(5\) units. The combined rectangle is \(7\) units by \(9\) units and has area \(63\) square units.
5404363
List every different rectangle with whole-number side lengths from \(4\) through \(9\) units and area \(36\) square units. Then decide whether any of the rectangles is a square.

Hints

- Find factor pairs whose product is \(36\). - Keep only pairs in which both side lengths are from \(4\) through \(9\). - A rectangle is a square when its two side lengths are equal.

Solution

1. Find factor pairs of \(36\) in which both factors are from \(4\) through \(9\). 2. The pairs are \(4 \times 9\) and \(6 \times 6\). 3. Reversing \(4\) and \(9\) only turns the same rectangle. 4. The \(6\)-by-\(6\) rectangle is a square because its side lengths are equal.

Answer

The dimensions are \(4\) by \(9\) and \(6\) by \(6\). The \(6\)-by-\(6\) rectangle is a square.
5404543
A square has an area of \(64\) square units. It is cut exactly in half to make two congruent rectangles. What are the dimensions and area of each rectangle?

Hints

- First find the equal side length of the square. - Cutting the square in half changes one dimension but not the other.

Solution

1. The square's side length is \(8\) units because \(8 \times 8 = 64\). 2. Cutting one \(8\)-unit side in half gives a length of \(4\) units while the other side remains \(8\) units. 3. Each rectangle is \(4\) units by \(8\) units and has area \(4 \times 8 = 32\) square units.

Answer

Each rectangle is \(4\) units by \(8\) units with an area of \(32\) square units.
5404593
Three congruent rectangles are each \(2\) units by \(3\) units. They are joined without gaps or overlaps to make one larger rectangle. Give two different possible sets of dimensions for the larger rectangle, and verify that both have the correct area.

Hints

- Find the combined area of the three small rectangles. - In one arrangement, keep the \(2\)-unit side fixed and combine the three \(3\)-unit sides. - In another arrangement, keep the \(3\)-unit side fixed and combine the three \(2\)-unit sides.

Solution

1. One small rectangle has area \(2 \times 3 = 6\) square units. 2. Three rectangles have total area \(3 \times 6 = 18\) square units. 3. Placing all three side by side along their \(3\)-unit sides makes a \(2\)-by-\(9\) rectangle with area \(2 \times 9 = 18\). 4. Stacking all three along their \(2\)-unit sides makes a \(6\)-by-\(3\) rectangle with area \(6 \times 3 = 18\).

Answer

Two possible larger rectangles are \(2\) units by \(9\) units and \(6\) units by \(3\) units. Each has area \(18\) square units.
5404663
The figure shows a rectangle surrounded by a border that is \(1\) unit wide. Jordan says the border's area is the same number as the original rectangle's perimeter. Explain what Jordan missed and find the border's area.
Figure for problem 540466

Hints

- Read the inside and outside dimensions from the figure. - Compare what a one-unit-wide strip along each side counts with what happens at the four corners. - You can find the border area by subtracting the inside area from the outside area.

Solution

1. From the figure, the original rectangle is \(7\) units by \(5\) units and the outside rectangle is \(9\) units by \(7\) units. 2. The original perimeter is \(7 + 5 + 7 + 5 = 24\) units. Counting one square along each unit of that boundary misses the four new corner squares in the border. 3. The outside area is \(9 \times 7 = 63\) square units, and the original area is \(7 \times 5 = 35\) square units. 4. The border area is \(63 - 35 = 28\) square units.

Answer

Jordan missed the four corner squares. The border's area is \(28\) square units.

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