All ISEE Upper Level Quantitative Resources
Example Questions
Example Question #1 : Quadrilaterals
The area of a square is .
Give the perimeter of the square.
The length of one side of a square is the square root of its area. The polynomial representing the area of the square can be recognized as a perfect square trinomial:
Therefore, the square root of the area is
,
which is the length of one side.
The perimeter of the square is four times this length, or
.
Example Question #1 : Squares
The perimeters of six squares form an arithmetic sequence. The second-smallest square has sides that are two inches longer than those of the smallest square.
Which, if either, is the greater quantity?
(a) The perimeter of the third-smallest square
(b) The length of one side of the largest square
(a) and (b) are equal
(a) is greater
(b) is greater
It is impossible to tell which is greater from the information given
(a) is greater
Let the length of one side of the first square be . Then the length of one side of the second-smallest square is , and the perimeters of the squares are
and
This makes the common difference of the perimeters 8 units.
The perimeters of the squares being in arithmetic progression, the perimeter of the th-smallest square is
Since , this becomes
The perimeter of the third-smallest square is
The perimeter of the largest, or sixth-smallest, square is
The length of one side of this square is one fourth of this, or
Therefore, we are comparing and .
Since a perimeter must be positive,
;
also, .
Therefore, regardless of the value of ,
,
and
,
making (a) the greater quantity.
Example Question #2 : Squares
The sidelength of a square is . Give its perimeter in terms of .
The perimeter of a square is four times the length of a side, which here is :
Example Question #2 : Squares
A diagonal of a square has length . Give its perimeter.
The length of a side of a square can be determined by dividing the length of a diagonal by - that is, . A diagonal has length , so the sidelength is
Multiply this by four to get the perimeter:
Example Question #1 : Squares
The perimeters of six squares form an arithmetic sequence. The smallest square has area 9; the second smallest square has area 25. Give the perimeter of the largest of the six squares.
None of the other responses gives the correct answer.
The two smallest squares have areas 9 and 25, so their sidelengths are the square roots of these, or, respectively, 3 and 5. Their perimeters are the sidelengths multiplied by four, or, respectively, 12 and 20. Therefore, in the arithmetic sequence,
and the common difference is .
The perimeter of the th smallest square is
Setting , the perimeter of the largest (or sixth-smallest) square is
.
Example Question #1 : Squares
The perimeter of a square is one yard. Which is the greater quantity?
(a) The area of the square
(b) square foot
(a) is greater.
(a) and (b) are equal.
It is impossible to tell form the information given.
(b) is greater.
(a) is greater.
One yard is equal to three feet, so the length of one side of a square with this perimeter is feet. The area of the square is square feet. , making (a) greater.
Example Question #1 : How To Find The Area Of A Square
Square 1 is inscribed inside a circle. The circle is inscribed inside Square 2.
Which is the greater quantity?
(a) Twice the area of Square 1
(b) The area of Square 2
(a) and (b) are equal.
(a) is greater.
(b) is greater.
It is impossible to tell from the information given.
(a) and (b) are equal.
Let be the sidelength of Square 1. Then the length of a diagonal of this square - which is times this sidelength, or , by the Theorem - is the same as the diameter of this circle, which, in turn, is equal to the sidelength of Square 2.
Therefore, Square 1 has area , and Square 2 has area , twice that of Square 1.
Example Question #2 : How To Find The Area Of A Square
Which is the greater quantity?
(A) The area of a square with sidelength one foot
(B) The area of a rectangle with length nine inches and height fourteen inches
(A) and (B) are equal
(B) is greater
It is impossible to determine which is greater from the information given
(A) is greater
(A) is greater
The area of a square is the square of its sidelength, which here is 12 inches:
square inches.
The area of a rectangle is its length multiplied by its height, which, respectively, are 9 inches and 14 inches:
square inches.
The square has the greater area.
Example Question #2 : How To Find The Area Of A Square
A square lawn has sidelength twenty yards. Give its area in square feet.
20 yards converts to feet. The area of a square is the square of its sidelength, so the area in square feet is square feet.
Example Question #2 : How To Find The Area Of A Square
Rectangle A and Square B both have perimeter 2 meters. Rectangle A has width 25 centimeters. The area of Rectangle A is what percent of the area of Square B?
The perimeter of a rectangle can be given by the formula
Rectangle A has perimeter 2 meters, which is equal to 200 centimeters, and width 25 centimeters, so the length is:
The dimensions of Rectangle A are 75 centimeters and 25 centimeters, so its area is
square centimeters.
The sidelength of a square is one-fourth its perimeter, which here is
centimeters; its area is therefore
square centimeters.
The area of Rectangle A is therefore
that of Square B.