Geometry - ISEE Middle Level Quantitative Reasoning

Card 0 of 2265

Question

Give the equation of the line through point that has slope $-\frac{1}{5}$ .

Answer

Use the point-slope formula with $m = -\frac{1}{5}, x_{1}= 1, y_{1} = 4$

$y -y _{1} = m (x -x _{1} )$

$y -4 = -\frac{1}{5} (x -1 )$

$y -\frac{20}{5} = -\frac{1}{5} x + \frac{1}{5}$

$y = -\frac{1}{5} x + \frac{21}{5}$

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Question

Which is the greater quantity?

(A) The slope of the line

(B) The slope of the line

Answer

Rewrite each in the slope-intercept form, ; will be the slope.

$- 2y \div (-2) =\left ( -4x+ 10 \right )\div (-2)$

The slope of the line of is

The slope of the line of is also

The slopes are equal.

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Question

Which is the greater quantity?

(A) The slope of the line

(B) The slope of the line

Answer

Rewrite each in the slope-intercept form, ; will be the slope.

The slope of this line is .

The slope of this line is .

Since , (A) is greater.

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Question

and are positive integers, and . Which is the greater quantity?

(a) The slope of the line on the coordinate plane through the points and .

(b) The slope of the line on the coordinate plane through the points and .

Answer

The slope of a line through the points and can be found by setting

$x_{1} = A-1,y_{1} = B - 1, x_{2} = A+1, y_{2} = B+ 1$

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$= \frac{ (B+ 1)- (B- 1)}{ (A+ 1)- (A- 1)}$

$= \frac{ B-B+ 1+1}{ A-A + 1 + 1}$

$= \frac{ 2}{ 2}$

The slope of a line through the points and can be found similarly:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$= \frac{(A+ 1)- (A- 1) }{ (B+ 1)- (B- 1) }$

$= \frac{A-A + 1+1}{B-B + 1 + 1}$

$= \frac{ 2}{ 2}$

The lines have the same slope.

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Question

A line passes through the points with coordinates and , where . Which expression is equal to the slope of the line?

Answer

The slope of a line through the points and , can be found by setting

$x_{1} = -A,y_{1} =B, x_{2} = A, y_{2} = B$ :

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$m = \frac{B-B}{A - (-A)} = \frac{0}{A+A} = \frac{0}{2A} = 0$

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Question

Choose the best answer from the four choices given.

The point (15, 6) is on which of the following lines?

Answer

For this problem, simply plug in the values for the point (15,6) into the different equations (15 for the -value and 6 for the -value) to see which one fits.

$(6)=\frac{1}{2}(15)-7$ (NO)

$(6)=\frac{2}{3}(15)-4$ (YES!)

$(6)=\frac{-1}{2}(15)+7$ (NO)

$(6)=\frac{-2}{3}(15)+4$ (NO)

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Question

Choose the best answer from the four choices given.

What is the point of intersection for the following two lines?

$y=\frac{3}{4}x-11$

$y=\frac{-5}{6}x+8$

Answer

At the intersection point of the two lines the - and - values for each equation will be the same. Thus, we can set the two equations as equal to each other:

$\frac{3}{4}x-11=\frac{-5}{6}x+8$

$\frac{3}{4}x=\frac{-5}{6}x+19$

$\frac{9}{12}x=\frac{-10}{12}x+19$

$\frac{19}{12}x=19$

$(\frac{12}{19})(\frac{19}{12}x)=19 (\frac{12}{19})$

$\large x=12$

$y=\frac{3}{4}(12)-11= -2$

point of intersection $=\left ( 12,-2 \right )$

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Question

Choose the best answer from the four choices given.

What is the -intercept of the line represented by the equation

$y=\frac{7}{9}x-3\ ?$

Answer

In the formula , the y-intercept is represented by (because if you set to zero, you are left with ).

Thus, to find the -intercept, set the value to zero and solve for .

$0=\frac{7}{9}x-3$

$\frac{-7}{9}x=-3$

$(\frac{-9}{7})\frac{-7}{9}x=-3 (\frac{-9}{7})$

$x=\frac{27}{7}=3\frac{6}{7}$

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Question

The ordered pair is in which quadrant?

Answer

There are four quadrants in the coordinate plane. Quadrant I is the top right, and they are numbered counter-clockwise. Since the x-coordinate is , you go to the left one unit (starting from the origin). Since the y-coordinate is , you go upwards four units. Therefore, you are in Quadrant II.

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Question

If angles s and r add up to 180 degrees, which of the following best describes them?

Answer

Two angles that are supplementary add up to 180 degrees. They cannot both be acute, nor can they both be obtuse. Therefore, "Supplementary" is the correct answer.

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Question

The lines of the equations

and

intersect at a point .

Which is the greater quantity?

(a)

(b)

Answer

If and , we can substitute in the second equation as follows:

$4x \div 4 = 8 \div 4$

Substitute:

$y= 3 \cdot 2$

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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.

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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.

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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 $t = \frac{1}{5}$ .

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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.

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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.

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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:

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

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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.

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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.

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Question

Using the information given in each question, compare the quantity in Column A to the quantity in Column B.

A certain rectangle is seven times as long as it is wide.

Column A Column B

the rectangle's the rectangle's

perimeter area

(in units) (in square units)

Answer

This type of problem reminds us to be wary of simply plugging in numbers (which works with certain problems). If you were to choose 1 and 7 here, the perimeter would be larger; if you chose 10 and 70, the area would be much larger.

To solve this problem with variables:

$area=w\cdot (7w)= 7w^{2}$

From here we can see that smaller values of will lead to a larger perimeter, while larger values of will lead to a larger area.

The answer cannot be determined.

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