How to find the length of the side of a rectangle - ISEE Upper Level Quantitative Reasoning

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Question

Note: Figure NOT drawn to scale

The above figure shows Rhombus .

Which is the greater quantity?

(a)

(b)

Answer

The opposite sides of a parallelogram - a rhombus included - are congruent, so

.

Also, Quadrilateral form a rectangle; since $\overline{AX} || \overline{CY}$ and $\overline{AY} \perp \overline{CY}$ , it follows that $\overline{AY} \perp \overline{AX}$ , and, similarly, $\overline{XC} \perp \overline{CY}$ . Therefore, , and

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Question

The area of a rectangle is 100 square inches. What is the width of the rectangle?

Answer

Without more information we cannot determine the width of a rectangle based on its area alone.

If we know this is a square and therefore that all sides are equal, we could simply take the square root of 100, which is 10.

However, we do not know that all sides are equal. We need to know the length of the rectangle to solve this problem. Therefore the best we can do is see what the equation would look like if we knew the length.

$w\times l=100$

or

$w=\frac{100}{l}$

where is the width and is the length.

Note: the incorrect answer choices are all factors of 100 and possible answers, but we can't be sure which, if any, are the right one.

Compare your answer with the correct one above

Question

A rectangle has perimeter 56. Its length is ; give its width in terms of .

Answer

Substitute in the formula for the perimeter of a rectangle:

$2 \left ( 3x-8 \right ) + 2W = 56$

$2\cdot 3x-2\cdot 8 \right ) + 2W = 56$

$2W \div 2 =\left ( -6x + 72 \right ) \div 2$

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Question

The area of a rectangle is 40; its width is . Which of the following expressions is equal to the length of the rectangle?

Answer

Let be the length. The area of the rectangle is . Replace and solve for :

$L \left ( s+5 \right ) = 40$

$L \left ( s+5 \right ) \div \left ( s+5 \right ) = 40 \div \left ( s+5 \right )$

$L = \frac{40}{s + 5}$

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Question

The perimeter of a rectangle is 70; its width is . Which of the following expressions is equal to the length of the rectangle?

Answer

Let be the length. The perimeter of the rectangle is .

Replace and solve for :

$2L + 2 \left ( t+16 \right ) = 70$

$2L \div 2 =\left ( 38 -2t \right ) \div 2$

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Question

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

Answer

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

We are given a rectangle with a known area and width, and asked to find the length.

To do so, recall this formula:

We know A and w, so simply work backwards to solve for l

$l=\frac{A}{w}=\frac{284ft^2}{16ft}=17.75ft$

So, our answer is 17.75 ft

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale

The above figure shows Rhombus .

Which is the greater quantity?

(a)

(b)

Answer

The opposite sides of a parallelogram - a rhombus included - are congruent, so

.

Also, Quadrilateral form a rectangle; since $\overline{AX} || \overline{CY}$ and $\overline{AY} \perp \overline{CY}$ , it follows that $\overline{AY} \perp \overline{AX}$ , and, similarly, $\overline{XC} \perp \overline{CY}$ . Therefore, , and

Compare your answer with the correct one above

Question

The area of a rectangle is 100 square inches. What is the width of the rectangle?

Answer

Without more information we cannot determine the width of a rectangle based on its area alone.

If we know this is a square and therefore that all sides are equal, we could simply take the square root of 100, which is 10.

However, we do not know that all sides are equal. We need to know the length of the rectangle to solve this problem. Therefore the best we can do is see what the equation would look like if we knew the length.

$w\times l=100$

or

$w=\frac{100}{l}$

where is the width and is the length.

Note: the incorrect answer choices are all factors of 100 and possible answers, but we can't be sure which, if any, are the right one.

Compare your answer with the correct one above

Question

A rectangle has perimeter 56. Its length is ; give its width in terms of .

Answer

Substitute in the formula for the perimeter of a rectangle:

$2 \left ( 3x-8 \right ) + 2W = 56$

$2\cdot 3x-2\cdot 8 \right ) + 2W = 56$

$2W \div 2 =\left ( -6x + 72 \right ) \div 2$

Compare your answer with the correct one above

Question

The area of a rectangle is 40; its width is . Which of the following expressions is equal to the length of the rectangle?

Answer

Let be the length. The area of the rectangle is . Replace and solve for :

$L \left ( s+5 \right ) = 40$

$L \left ( s+5 \right ) \div \left ( s+5 \right ) = 40 \div \left ( s+5 \right )$

$L = \frac{40}{s + 5}$

Compare your answer with the correct one above

Question

The perimeter of a rectangle is 70; its width is . Which of the following expressions is equal to the length of the rectangle?

Answer

Let be the length. The perimeter of the rectangle is .

Replace and solve for :

$2L + 2 \left ( t+16 \right ) = 70$

$2L \div 2 =\left ( 38 -2t \right ) \div 2$

Compare your answer with the correct one above

Question

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

Answer

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

We are given a rectangle with a known area and width, and asked to find the length.

To do so, recall this formula:

We know A and w, so simply work backwards to solve for l

$l=\frac{A}{w}=\frac{284ft^2}{16ft}=17.75ft$

So, our answer is 17.75 ft

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale

The above figure shows Rhombus .

Which is the greater quantity?

(a)

(b)

Answer

The opposite sides of a parallelogram - a rhombus included - are congruent, so

.

Also, Quadrilateral form a rectangle; since $\overline{AX} || \overline{CY}$ and $\overline{AY} \perp \overline{CY}$ , it follows that $\overline{AY} \perp \overline{AX}$ , and, similarly, $\overline{XC} \perp \overline{CY}$ . Therefore, , and

Compare your answer with the correct one above

Question

The area of a rectangle is 100 square inches. What is the width of the rectangle?

Answer

Without more information we cannot determine the width of a rectangle based on its area alone.

If we know this is a square and therefore that all sides are equal, we could simply take the square root of 100, which is 10.

However, we do not know that all sides are equal. We need to know the length of the rectangle to solve this problem. Therefore the best we can do is see what the equation would look like if we knew the length.

$w\times l=100$

or

$w=\frac{100}{l}$

where is the width and is the length.

Note: the incorrect answer choices are all factors of 100 and possible answers, but we can't be sure which, if any, are the right one.

Compare your answer with the correct one above

Question

A rectangle has perimeter 56. Its length is ; give its width in terms of .

Answer

Substitute in the formula for the perimeter of a rectangle:

$2 \left ( 3x-8 \right ) + 2W = 56$

$2\cdot 3x-2\cdot 8 \right ) + 2W = 56$

$2W \div 2 =\left ( -6x + 72 \right ) \div 2$

Compare your answer with the correct one above

Question

The area of a rectangle is 40; its width is . Which of the following expressions is equal to the length of the rectangle?

Answer

Let be the length. The area of the rectangle is . Replace and solve for :

$L \left ( s+5 \right ) = 40$

$L \left ( s+5 \right ) \div \left ( s+5 \right ) = 40 \div \left ( s+5 \right )$

$L = \frac{40}{s + 5}$

Compare your answer with the correct one above

Question

The perimeter of a rectangle is 70; its width is . Which of the following expressions is equal to the length of the rectangle?

Answer

Let be the length. The perimeter of the rectangle is .

Replace and solve for :

$2L + 2 \left ( t+16 \right ) = 70$

$2L \div 2 =\left ( 38 -2t \right ) \div 2$

Compare your answer with the correct one above

Question

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

Answer

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

We are given a rectangle with a known area and width, and asked to find the length.

To do so, recall this formula:

We know A and w, so simply work backwards to solve for l

$l=\frac{A}{w}=\frac{284ft^2}{16ft}=17.75ft$

So, our answer is 17.75 ft

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale

The above figure shows Rhombus .

Which is the greater quantity?

(a)

(b)

Answer

The opposite sides of a parallelogram - a rhombus included - are congruent, so

.

Also, Quadrilateral form a rectangle; since $\overline{AX} || \overline{CY}$ and $\overline{AY} \perp \overline{CY}$ , it follows that $\overline{AY} \perp \overline{AX}$ , and, similarly, $\overline{XC} \perp \overline{CY}$ . Therefore, , and

Compare your answer with the correct one above

Question

The area of a rectangle is 100 square inches. What is the width of the rectangle?

Answer

Without more information we cannot determine the width of a rectangle based on its area alone.

If we know this is a square and therefore that all sides are equal, we could simply take the square root of 100, which is 10.

However, we do not know that all sides are equal. We need to know the length of the rectangle to solve this problem. Therefore the best we can do is see what the equation would look like if we knew the length.

$w\times l=100$

or

$w=\frac{100}{l}$

where is the width and is the length.

Note: the incorrect answer choices are all factors of 100 and possible answers, but we can't be sure which, if any, are the right one.

Compare your answer with the correct one above

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