### All ISEE Middle Level Quantitative Resources

## Example Questions

### Example Question #1 : Plane Geometry

Calvin is remodeling his room. He used 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 feet tall. If one can of paint covers square feet, how many cans of paint will he need to paint three walls.

**Possible Answers:**

**Correct answer:**

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

In order to solve for , Calvin must divide both sides by four.

The left-hand side simplifies to:

The right-hand side simplifies to:

Now, Calvin knows the width of each room is 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 . Since Calvin now knows that the width of each wall is feet and that the height of each wall is also feet, he can multiply the two together to find the area.

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 square feet, he must multiply that by .

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

Calvin will need cans of paint.

### Example Question #1 : Quadrilaterals

Which is the greater quantity?

(a) The surface area of a cube with volume

(b) The surface area of a cube with sidelength

**Possible Answers:**

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

**Correct answer:**

(b) is greater

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 , which is greater than . The cube in (b) has the greater volume, sidelength, and, most importantly, surface area.

### Example Question #1 : Plane Geometry

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

**Possible Answers:**

**Correct 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

inches.

Its area is the square of the sidelength, or

square inches.

### Example Question #1 : Squares

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

**Possible Answers:**

**Correct answer:**

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

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

meters.

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

square meters.

### Example Question #1 : Quadrilaterals

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

(A) The area of the square

(B)

**Possible Answers:**

(A) and (B) are equal

(A) is greater

It is impossible to determine which is greater from the information given

(B) is greater

**Correct answer:**

It is impossible to determine which is greater from the information given

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

Using the side length from the question:

However, it is impossible to tell with certainty which of and is greater.

For example, if ,

and

so if .

But if ,

and

so if .

### Example Question #1 : Quadrilaterals

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

**Possible Answers:**

**Correct answer:**

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

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:

### Example Question #1 : Squares

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

**Possible Answers:**

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

A square plot of land with perimeter one mile has as its sidelength one fourth of this, or mile; its area is the square of this, or

square miles.

One square mile is equivalent to 640 acres, so square miles is equivalent to

acres.

This makes (b) greater.

### Example Question #1 : Plane Geometry

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

**Possible Answers:**

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(a) is the greater quantity

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

kilometers

kilometers

Multiply the dimensions to get the area in square kilometers:

square kilometers

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

hectares

This makes (a) the greater.

### Example Question #1 : Plane Geometry

What is the perimeter of a square with area 196 square inches?

**Possible Answers:**

It cannot be determined from the information given.

**Correct answer:**

A square with area 196 square inches has sidelength inches, and therefore has perimeter inches

### Example Question #1 : How To Find Perimeter

If a square has sides measuring , what is the perimeter of the square, in simplest form?

**Possible Answers:**

**Correct answer:**

To find the perimeter of a square, you must add together all the sides. In this case, we are adding four times.

Since all of the denominators are the same, there is no need to find a commond denominator, so we add together the numerators. This gives us .

Since both the numerator and denomator are divisible by four, we must simplify this fraction.

The perimeter of the square is .

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