Squares - ISEE Middle Level Quantitative Reasoning

Card 0 of 890

Question

Calvin is remodeling his room. He used $\small 32$ feet of molding to put molding around all four walls. Now he wants to paint three of the walls. Each wall is the same width and is $\small 8$ feet tall. If one can of paint covers $\small \small \small 24$ square feet, how many cans of paint will he need to paint three walls.

Answer

When Calvin put up $\small 32$ feet of molding, he figured out the perimeter of the room was $\small 32$ feet. Since he knows that all four walls are the same width, he can use the equation $\small 4s=P$ to determine the length of each side by plugging $\small 32$ in for $\small P$ and solving for $\small s$ .

$\small 4s=32$

In order to solve for $\small s$ , Calvin must divide both sides by four.

The left-hand side simplifies to:

$\small \frac{4s}{4}=s$

The right-hand side simplifies to:

$\small \frac{32}{2}=8$

Now, Calvin knows the width of each room is $\small 8$ feet. Next he must find the area of each wall. To do this, he must multiply the width by the height because the area of a rectangle is found using the equation $\small \small A=l\cdot w$ . Since Calvin now knows that the width of each wall is $\small 8$ feet and that the height of each wall is also $\small 8$ feet, he can multiply the two together to find the area.

$\small 8\cdot 8=64$

Since Calvin wants to find how much paint he needs to cover three walls, he must first find out how many square feet he is covering. If one wall is $\small 64$ square feet, he must multiply that by $\small 3$ .

$\small 64\cdot 3=192$

Calvin is painting $\small 192$ square feet. If one can of paint covers 24 square feet, he must divide the total space ( $\small 192$ square feet) by $\small 24$ .

$\small \frac{192}{24}=8$

Calvin will need $\small 8$ cans of paint.

Compare your answer with the correct one above

Question

Which is the greater quantity?

(a) The surface area of a cube with volume $512 \textrm{ cm} ^{3}$

(b) The surface area of a cube with sidelength $90 \textrm{ mm}$

Answer

We can actually solve this by comparing volumes; the cube with the greater volume has the greater sidelength and, subsequently, the greater surface area.

The volume of the cube in (b) is the cube of 90 millimeters, or 9 centimeters. This is $9 ^{3} = 729 \textrm{ cm}^{3}$ , which is greater than $512\ cm^{3}$ . The cube in (b) has the greater volume, sidelength, and, most importantly, surface area.

Compare your answer with the correct one above

Question

Each side of a square is units long. Which is the greater quantity?

(A) The area of the square

(B)

Answer

The area of a square is the square of its side length:

Using the side length from the question:

$A=\left ( 5t\right ) ^{2} = 5 ^{2} \cdot t ^{2} = 25t ^{2}$

However, it is impossible to tell with certainty which of $25t ^{2}$ and is greater.

For example, if ,

$25t ^{2} = 25 \cdot 2^{2} = 25 \cdot4 = 100$

and

$25t = 25 \cdot 2 = 50$

so $25t ^{2} > 25t$ if .

But if $t = \frac{1}{5}$ ,

$25t ^{2}=25 \cdot \left ( \frac{1}{5} \right )^{2} = 25 \cdot \frac{1}{25} = 1$

and

$25t =25 \cdot \frac{1}{5} = 5$

so $25t ^{2} < 25t$ if $t = \frac{1}{5}$ .

Compare your answer with the correct one above

Question

The sum of the lengths of three sides of a square is one yard. Give its area in square inches.

Answer

A square has four sides of the same length.

One yard is equal to 36 inches, so each side of the square has length

$36 \div 3 = 12$ inches.

Its area is the square of the sidelength, or

$12^{2} = 12 \times 12 = 144$ square inches.

Compare your answer with the correct one above

Question

The sum of the lengths of three sides of a square is 3,900 centimeters. Give its area in square meters.

Answer

100 centimeters are equal to one meter, so 3,900 centimeters are equal to

$3,900 \div 100 = 39$ meters.

A square has four sides of the same length. Since the sum of the lengths of three of the congruent sides is 3,900 centimeters, or 39 meters, each side measures

$39 \div 3 = 13$ meters.

The area of the square is the square of the sidelength, or

$13^{2} =13 \times 13 = 169$ square meters.

Compare your answer with the correct one above

Question

A square has a side with a length of 5. What is the area of the square?

Answer

The area formula for a square is length times width. Keep in mind that all of a square's sides are equal.

$A=l\times w=s\times s$

So, if one side of a square equals 5, all of the other sides must also equal 5. You will find the area of the square by multiplying two of its sides:

$A=s\times s=5 \times 5 = 25$

Compare your answer with the correct one above

Question

One square mile is equivalent to 640 acres. Which of the following is the greater quantity?

(a) The area of a square plot of land whose perimeter measures one mile

(b) 160 acres

Answer

A square plot of land with perimeter one mile has as its sidelength one fourth of this, or $\frac{1}{4}$ mile; its area is the square of this, or

$\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ square miles.

One square mile is equivalent to 640 acres, so $\frac{1}{16}$ square miles is equivalent to

$\frac{1}{16} \times 640 = \frac{640}{16} = 40$ acres.

This makes (b) greater.

Compare your answer with the correct one above

Question

One square kilometer is equal to 100 hectares.

Which is the greater quantity?

(a) The area of a rectangular plot of land 500 meters in length and 200 meters in width

(b) One hectare

Answer

One kilometer is equal to 1,000 meters, so divide each dimension of the plot in meters by 1,000 to convert to kilometers:

$500 \div 1,000= 0.5$ kilometers

$200 \div 1,000 = 0.2$ kilometers

Multiply the dimensions to get the area in square kilometers:

$0.5 \times 0.2 = 0.1$ square kilometers

Since one square kilometer is equal to 100 hectares, multiply this by 100 to convert to hectares:

$0.1 \times 100 = 10$ hectares

This makes (a) the greater.

Compare your answer with the correct one above

Question

Which is the greater quantity?

(a) The perimeter of an equilateral triangle with sidelength 30 inches

(b) The perimeter of a square with sidelength 2 feet

Answer

Each figure has sides that are congruent, so in each case, multiply the sidelength by the number of sides.

(a) The triangle has perimeter $30 \times 3 = 90$ inches

(b) 2 feet are equal to 24 inches, so the square has sidelength $24 \times 4 = 96$ inches.

The square has the greater perimeter.

Compare your answer with the correct one above

Question

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 3 inches and 4 inches

(b) One foot

Answer

The length of the hypotenuse of a triangle with legs 3 inches and 4 inches long is calculated using the Pythagorean Theorem, setting :

$c = \sqrt{a^{2}+b^{2}} = \sqrt{3^{2}+4^{2}} = \sqrt{9+16} = \sqrt{25} =5$

The hypotenuse is 5 inches long. The perimeter is therefore inches, which is equal to one foot.

Compare your answer with the correct one above

Question

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 5 feet and 12 feet

(b) 8 yards

Answer

The length of the hypotenuse of a triangle with legs 5 feet and 12 feet is calculated using the Pythagorean Theorem, setting :

$c = \sqrt{a^{2}+b^{2}} = \sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt{169} =13$

The hypotenuse is 13 feet long. The perimeter is feet, which is equal to 10 yards.

Compare your answer with the correct one above

Question

Which is the greater quantity?

(a) The perimeter of a right triangle with hypotenuse of length 25 centimeters and one leg of length 7 centimeters

(b) One-half of a meter

Answer

The length of the second leg of the triangle can be calculated using the Pythagorean Theorem, setting :

$b = \sqrt{c^{2}-a^{2}} = \sqrt{25^{2}-7^{2}} = \sqrt{625-49}= \sqrt{576} = 24$

The second leg has length 24 centimeters, so the perimeter of the triangle is

centimeters.

One-half of a meter is one-half of 100 centimeters, or 50 centimeters, so (a) is greater.

Compare your answer with the correct one above

Question

$\Delta ABC$ is an equilateral triangle; .

Rectangle ;

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of Rectangle

Answer

(a) The perimeter of the equilateral triangle is $25 \times 3 = 75$ .

(b) ; are of unknown value, but they are equal, so we will call their common length .

Rectangle has perimeter

.

Without knowing , it cannot be determined with certainty which figure has the longer perimeter. For example:

If , then $70 + 2N = 70 + 2 \cdot 1 = 70 + 2 = 72 < 75$

If , then $70 + 2N = 70 + 2 \cdot 3 = 70 + 6 = 76>75$

Compare your answer with the correct one above

Question

$\Delta ABC$ is an isosceles triangle; $\Delta DEF$ is an equilateral triangle

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of $\Delta DEF$

Answer

(a) As an isosceles triangle, $\Delta ABC$ , by definition, has two congruent sides. $AB \neq BC$ , so either :

in which case the perimeter of $\Delta ABC$ is

or

in which case the perimeter of $\Delta ABC$ is

(b) $\Delta DEF$ is an equilateral triangle, so, by definition, all of its sides are congruent; its perimeter is $35 \times 3 = 105$ .

Regardless of the length of $\overline{AC}$ , $\Delta ABC$ has the greater perimeter.

Compare your answer with the correct one above

Question

$\Delta ABC$ and $\Delta DEF$ are right triangles, with right angles $\angle B , \angle E$ , respectively.

Which is the greater quantity?

(a)

(b)

Answer

(a) $\overline{AC}$ is the hypotenuse of $\Delta ABC$ , so by the Pythagorean Theorem,

$\left (AC \right )^{2} =\left ( AB \right ) ^{2} + \left ( B C\right )^{2}$

$\left (AC \right )^{2} =6^{2} + 8^{2}$

$\left (AC \right )^{2} =36 + 64 = 100$

$AC = \sqrt{100} = 10$

(b) $\overline{EF}$ is a leg of $\Delta DEF$ , whose hypotenuse is $\overline{DF}$ , so by the Pythagorean Theorem,

$\left (EF \right )^{2} =\left ( DF \right ) ^{2} - \left ( DE\right )^{2}$

$\left (EF \right )^{2} =26^{2} - 24^{2}$

$\left (EF \right )^{2} =676 -576 = 100$

$EF= \sqrt{100} = 10$

Compare your answer with the correct one above

Question

$\Delta ABC$ is a right triangle with hypotenuse $\overline{AC}$ 10 inches long.

The lengths of $\overline{AB}$ and $\overline{BC}$ , in inches, can both be expressed as integers.

Which is the greater quantity?

(a) $2 \textrm{ feet}$

(b) The perimeter of $\Delta ABC$

Answer

By the Pythagorean Theorem,

$(AB)^{2}+ (BC)^{2}= (AC)^{2}$

$(AB)^{2}+ (BC)^{2}= 10^{2}$

$(AB)^{2}+ (BC)^{2}= 100$

By trial and error, it can be determined that the only two positive integers that can replace and to make this equation true are 6 and 8, in either order:

$6^{2}+ 8^{2}= 100$

Add the three side lengths to get the perimeter inches, which is equal to 2 feet.

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale.

Refer to the above triangle. An insect walks directly from A to B, then directly from B to C. What fraction of the perimeter of the triangle has he walked?

Answer

By the Pythagorean Theorem, the distance from A to C, which we will call , is equal to

$D = \sqrt{ 3^{2}+ 4^{2}} = \sqrt{ 9+ 16} = \sqrt{ 25} = 5$

The perimeter of the triangle is . The insect has traveled units, or $\frac{7}{12}$ of the perimeter.

Compare your answer with the correct one above

Question

An equilateral triangle has perimeter two yards. Which is the greater quantity?

(A) The length of one side of the triangle

(B) 28 inches

Answer

An equilateral triangle has three sides of equal length; the perimeter of this triangle is two yards, which is equal to $2 \times 36 = 72$ inches. One side has length $72 \div 3 = 24$ inches, which is less than 28 inches, so (B) is greater.

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale.

Refer to the above triangle. An insect walks directly from B to A, then directly from A to C. What percent of the perimeter of the triangle has he walked?

Answer

By the Pythagorean Theorem, the distance from A to C, which we will call , is equal to

$D = \sqrt{ 3^{2}+ 4^{2}} = \sqrt{ 9+ 16} = \sqrt{ 25} = 5$

The perimeter of the triangle is . The insect has traveled units out of 12, which is

$\frac{8}{12} \times 100 = 66 \frac{2}{3} %$ of the perimeter.

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale

Refer to the above triangle. An insect walks directly from B to C, then directly from C to A. What fraction of the perimeter of the triangle has he walked?

Answer

By the Pythagorean Theorem, the distance from B to C, which we will call , is equal to

$D = \sqrt{13^{2}- 5^{2}} = \sqrt{169- 25 }= \sqrt{144} = 12$

The perimeter of the triangle is

.

The insect has traveled units, which is

$\frac{25}{30} = \frac{25 \div 5}{30 \div 5} = \frac{5}{6}$

of the perimeter.

Compare your answer with the correct one above

Tap the card to reveal the answer