### All HSPT Quantitative Resources

## Example Questions

### Example Question #1 : How To Make Geometric Comparisons

Examine (a), (b), and (c) to find the best answer:

a) area of a circle with a diameter of

b) area of a circle with a radius of

c) area of a circle with a cirucumference of

**Possible Answers:**

**Correct answer:**

All of these circles have the same diameter, so they all must have the same area:

a) area of a circle with a diameter of

b) area of a circle with a radius of

c) area of a circle with a cirucumference of

### Example Question #1 : Geometric Comparison

Examine (a), (b), and (c) to find the best answer:

a) perimeter of a square with a side length of

b) perimeter of a rectangle with a length of and a width of

c) perimeter of an equailateral triangle with a side length of

**Possible Answers:**

**Correct answer:**

To find perimeter, add up the lengths of all the sides:

a)

b)

c)

(a) and (b) are equal, and they are smaller than (c)

### Example Question #1 : How To Make Geometric Comparisons

Examine (a), (b), and (c) to find the best answer:

a) circumference of a circle with a radius of

b) circumference of a circle with a diameter of

c) diameter of circle with a circumference of

**Possible Answers:**

**Correct answer:**

The formulas to remember for this problem are (Circumference equals diameter times pi) and (diameter equals two pi).

a) circumference of a circle with a radius of

b) circumference of a circle with a diameter of

c) diameter of circle with a circumference of

Therefore (a) is the greatest, followed by (b), then (c).

### Example Question #1 : How To Make Geometric Comparisons

Examine (a), (b), and (c) to find the best answer:

a) the sum of the interior angles of a triangle

b) the sum of the interior angles of a square

c) the total degrees in a circle

**Possible Answers:**

(c) is greater than (a) and (b)

(a) and (b) are equal

(a), (b), and (c) are all equal

(b) and (c) are equal

**Correct answer:**

(b) and (c) are equal

The sum of the interior angles of a triangle is always degrees. For squares, it's always degrees. There are also a total of degrees in a circle. Therefore, (b) and (c) are equal, and they are greater than (a).

### Example Question #2 : How To Make Geometric Comparisons

Examine (a), (b), and (c) to find the best answer:

a) area of a square with side length

b) area of a square with perimeter

c) area of a square with side length

**Possible Answers:**

(a) and (c) are equal

(b) is greater than (a) or (c)

(a), (b), and (c) are all equal

(b) and (c) are equal

**Correct answer:**

(b) and (c) are equal

The area of a square is the side length squared. The perimeter is the side length multiplied by .

(b) and (c) are equal because a side length should be of the perimeter. (a) is the greatest, because the greatest side length leads to the greatest area.

### Example Question #2 : Geometric Comparison

Examine (a), (b), and (c) to find the best answer:

a) area of a square with a perimeter of

b) area of a square with a side length of

c) area of a square with a side length of

**Possible Answers:**

(c) is greater than (a) and (b).

(a) is greater than (b) and (c).

(a) and (b) are equal.

(a) and (c) are equal.

**Correct answer:**

(c) is greater than (a) and (b).

Things to remember here are that area is found by squaring side length and that side length is of the perimeter.

a)

b)

c)

(c) is the greatest, and none of the values are equal.

### Example Question #2 : How To Make Geometric Comparisons

Examine (a), (b), and (c) to find the best answer

a) a square with an area of

b) a square with a side length of

c) a square with a perimeter of

**Possible Answers:**

**Correct answer:**

All of these squares are equal! We can tell because they all have the same side length. For (a), find the square root of the area to find the side length:

For (c), divide the perimeter by four to find the side length:

For (b), we are told that the side length equals .

### Example Question #2 : Geometric Comparison

Examine (a), (b), and (c) to find the best answer:

a) a circle with a radius of

b) a circle with a diameter of

c) a circle with a circumference of

**Possible Answers:**

(a), (b), and (c) have equal area.

(b) has the greatest area.

(c) has the greatest area.

(a) has the greatest area.

**Correct answer:**

(c) has the greatest area.

The circle with the greatest radius is also going to have the greatest area, because .

Use these formulas to find the radius of each circle:

and , so in (b), .

and , so in (c), .

Compare these to (a), with .

Therefore (c) has the largest radius, so it also has the largest area.

### Example Question #4 : How To Make Geometric Comparisons

Examine (a), (b), and (c) to find the best answer:

a) a circle with a circumference of

b) a circle with a radius of

c) a circle with a radius of

**Possible Answers:**

(a) is equal to (b) but not (c).

(a), (b) and (c) are all unequal.

(a) is equal to (c) but not (b).

(a), (b) and (c) are all equal.

**Correct answer:**

(a) is equal to (c) but not (b).

Circumference is found by multiplying the diameter by pi, so the diameter of (a) must be . Radius is half of diameter, so the radius of (a) must be . This means that (a) is equal to (c), but not (b).

### Example Question #4 : How To Make Geometric Comparisons

Examine (a), (b), and (c) to find the best answer:

a) a circle with a radius of

b) a circle with a radius of

c) a circle with an area of

**Possible Answers:**

(a) is equivalent to (c) but not (b)

(a), (b), and (c) are all not equivalent

(a), (b), and (c) are all equivalent

(b) is equivalent to (c) but not (a)

**Correct answer:**

(b) is equivalent to (c) but not (a)

Find the radius of (c) to compare it to (a) and (b).

Since area is , we know that must be and that the square root of must be .

Since the radius of (c) is equal to the radius of (b), the circles are equivalent.

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