### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : How To Find The Length Of A Radius

The area of Circle B is four times that of Circle A. The area of Circle C is four times that of Circle B. Which is the greater quantity?

(a) Twice the radius of Circle B

(b) The sum of the radius of Circle A and the radius of Circle C

**Possible Answers:**

(a) is greater.

(a) and (b) are equal.

It cannot be determined from the information given.

(b) is greater.

**Correct answer:**

(b) is greater.

Let be the radius of Circle A. Then its area is .

The area of Circle B is , so the radius of Circle B is twice that of Circle A; by a similar argument, the radius of Circle C is twice that of Circle B, or .

(a) Twice the radius of circle B is .

(b) The sum of the radii of Circles A and B is .

This makes (b) greater.

### Example Question #2 : How To Find The Length Of A Radius

The time is now 1:45 PM. Since noon, the tip of the minute hand of a large clock has moved feet. How long is the minute hand of the clock?

**Possible Answers:**

**Correct answer:**

Every hour, the tip of the minute hand travels the circumference of a circle. Between noon and 1:45 PM, one and three-fourths hours pass, so the tip travels or times this circumference. The length of the minute hand is the radius of this circle , and the circumference of the circle is , so the distance the tip travels is this, or

Set this equal to feet:

feet.

This is equivalent to 1 foot 4 inches.

### Example Question #3 : How To Find The Length Of A Radius

The tip of the minute hand of a giant clock has traveled feet since noon. It is now 2:30 PM. Which is the greater quantity?

(A) The length of the minute hand

(B) Three yards

**Possible Answers:**

(B) is greater

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

(A) is greater

(A) and (B) are equal

**Correct answer:**

(B) is greater

Betwen noon and 2:30 PM, the minute hand has made two and one-half revolutions; that is, the tip of minute hand has traveled the circumference of its circle two and one-half times. Therefore,

feet.

The radius of this circle is the length of the minute hand. We can use the circumference formula to find this:

The minute hand is eight feet long, which is less than three yards (nine feet), so (B0 is greater.

### Example Question #11 : Circles

Compare the two quantities:

Quantity A: The radius of a circle with area of

Quantity B: The radius of a circle with circumference of

**Possible Answers:**

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

The quantity in Column B is greater.

The two quantities are equal.

**Correct answer:**

The quantity in Column B is greater.

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

For our two quantities, we have:

**Quantity A**

Therefore,

Taking the square root of both sides, we get:

**Quantity B**

Therefore,

Therefore, quantity B is greater.

### Example Question #5 : How To Find The Length Of A Radius

Compare the two quantities:

Quantity A: The radius of a circle with area of

Quantity B: The radius of a circle with circumference of

**Possible Answers:**

The quantity in Column A is greater.

The quantity in Column B is greater.

The relationship cannot be determined from the information given.

The two quantities are equal.

**Correct answer:**

The two quantities are equal.

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

For our two quantities, we have:

**Quantity A**

Therefore,

Taking the square root of both sides, we get:

**Quantity B**

Therefore,

Therefore, the two quantities are equal.

### Example Question #1 : How To Find The Length Of A Radius

If the diameter of a circle is equal to , then what is the value of the radius?

**Possible Answers:**

**Correct answer:**

Given that the radius is equal to half the diameter, the value of the radius would be equal to divided by 2. This gives us:

### Example Question #7 : How To Find The Length Of A Radius

The area of a circle is . Give its radius in terms of .

(Assume is positive.)

**Possible Answers:**

**Correct answer:**

The relation between the area of a circle and its radius is given by the formula

Since

:

We solve for :

Since is positive, as is :

### Example Question #1 : Radius

The areas of five circles form an arithmetic sequence. The smallest circle has radius 4; the second smallest circle has radius 8. Give the radius of the largest circle.

**Possible Answers:**

**Correct answer:**

The area of a circle with radius is . Therefore, the areas of the circles with radii 4 and 8, respectively, are

and

The areas form an arithmetic sequence, the common difference of which is

.

The circles will have areas:

Since the area of the largest circle is , we can find the radius as follows:

The radius can be calculated now:

### Example Question #1 : How To Find The Area Of A Circle

Circle B has a radius as long as that of Circle A.

Which is the greater quantity?

(a) The area of Circle A

(b) Twice the area of Circle B

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(a) is greater.

If we call the radius of Circle A , then the radius of Circle B is .

The areas of the circles are:

(a)

(b)

Twice the area of Circle B is

,

making (a) the greater number.

### Example Question #2 : How To Find The Area Of A Circle

Circle 1 is inscribed inside a square. The square is inscribed inside Circle 2.

Which is the greater quantity?

(a) Twice the area of Circle 1

(b) The area of Circle 2

**Possible Answers:**

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(a) and (b) are equal.

If the radius of Circle 1 is , then the square will have sidelength equal to the diameter of the circle, or . Circle 2 will have as its diameter the length of a diagonal of the square, which by the Theorem is times that, or . The radius of Circle 2 will therefore be half that, or .

The area of Circle 1 will be . The area of Circle 2 will be , twice that of Circle 1.

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