The point that is 2 units to the left of the $y$ -axis and 4 units above the $x$ -axis has coordinates.

Ankit G.

Numerade Educator

If $x$ is positive and $y$ is negative, then the point $(x, y)$ is in Quadrant _____.

Ankit G.

Numerade Educator

The distance between the points $(a, b)$ and $(c, d)$ is _____. So the distance between (1,2) and (7,10) is _____.

Ankit G.

Numerade Educator

The point midway between $(a, b)$ and $(c, d)$ is _____. So the point midway between (1,2) and (7,10) is ______.

Ankit G.

Numerade Educator

Refer to the following figure.

Find the coordinates of the points shown.

Ankit G.

Numerade Educator

Refer to the following figure.

List the points that lie in Quadrants I and III.

Ankit G.

Numerade Educator

Plot the given points in a coordinate plane.

$$(0,5),(-1,0),(-1,-2),\left(\frac{1}{2}, \frac{2}{3}\right)$$

Ankit G.

Numerade Educator

Plot the given points in a coordinate plane.

$$(-5,0),(2,0),(2.6,-1.3),(-2.5,3.5)$$

Ankit G.

Numerade Educator

Sketch the region given by the set.

$$\{(x, y) \mid x \geq 2\}$$

Ankit G.

Numerade Educator

Sketch the region given by the set.

$$\{(x, y) \mid 0 \leq y \leq 2\}$$

Ankit G.

Numerade Educator

Sketch the region given by the set.

$$\{(x, y) \mid x \geq 1 \text { and } y<3\}$$

Ankit G.

Numerade Educator

Sketch the region given by the set.

$$\{(x, y) \mid x<2 \text { and } y \geq 1\}$$

Ankit G.

Numerade Educator

Sketch the region given by the set.

$$\{(x, y) \mid-1<x<1 \text { and }-2<y<2\}$$

Ankit G.

Numerade Educator

Sketch the region given by the set.

$$\{(x, y) \mid-3 \leq x \leq 3 \text { and }-1 \leq y \leq 1\}$$

Ankit G.

Numerade Educator

A pair of points is graphed. (a) Find the distance between them. (b) Find the midpoint of the segment that joins them.

Ankit G.

Numerade Educator

Ankit G.

Numerade Educator

Ankit G.

Numerade Educator

Ankit G.

Numerade Educator

A pair of points is given. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them.

$$(0,8),(6,16)$$

Ankit G.

Numerade Educator

A pair of points is given. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them.

$$(-2,5),(10,0)$$

Ankit G.

Numerade Educator

A pair of points is given. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them.

$$(3,-2),(-4,5)$$

Ankit G.

Numerade Educator

A pair of points is given. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them.

$$(-1,1),(-6,-3)$$

Ankit G.

Numerade Educator

A pair of points is given. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them.

$$(6,-2),(-6,2)$$

Ankit G.

Numerade Educator

A pair of points is given. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them.

$$(0,-6),(5,0)$$

Ankit G.

Numerade Educator

In these exercises we find the areas of plane figures.

Draw the rectangle with vertices $A(1,3), B(5,3), C(1,-3),$ and $D(5,-3)$ on a coordinate plane. Find the area of the rectangle.

Ankit G.

Numerade Educator

In these exercises we find the areas of plane figures.

Draw the parallelogram with vertices $A(1,2), B(5,2)$, $C(3,6),$ and $D(7,6)$ on a coordinate plane. Find the area of the parallelogram.

Ankit G.

Numerade Educator

In these exercises we find the areas of plane figures.

Plot the points $A(1,0), B(5,0), C(4,3),$ and $D(2,3)$ on a coordinate plane. Draw the segments $A B, B C, C D$, and DA. What kind of quadrilateral is $A B C D$, and what is its area?

Ankit G.

Numerade Educator

In these exercises we find the areas of plane figures.

Plot the points $P(5,1), Q(0,6),$ and $R(-5,1)$ on a coordinate plane. Where must the point $S$ be located so that the quadrilateral $P Q R S$ is a square? Find the area of this square.

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula.

Which of the points $A(6,7)$ or $B(-5,8)$ is closer to the origin?

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula.

Which of the points $C(-6,3)$ or $D(3,0)$ is closer to the point $E(-2,1) ?$

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula.

Which of the points $P(3,1)$ or $Q(-1,3)$ is closer to the point $R(-1,-1) ?$

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula.

(a) Show that the points (7,3) and (3,7) are the same distance from the origin.

(b) Show that the points $(a, b)$ and $(b, a)$ are the same distance from the origin.

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula.

Show that the triangle with vertices $A(0,2), B(-3,-1),$ and $C(-4,3)$ is isosceles.

Ankit G.

Numerade Educator

In these exercises we use the converse of the Pythagorean Theorem (see page 277 ) to show that the given triangle is a right triangle.

Refer to triangle $A B C$ in the figure below.

(a) Show that triangle $A B C$ is a right triangle by using the converse of the Pythagorean Theorem.

(b) Find the area of triangle $A B C$.

Ankit G.

Numerade Educator

In these exercises we use the converse of the Pythagorean Theorem (see page 277 ) to show that the given triangle is a right triangle.

Show that the triangle with vertices $A(6,-7), B(11,-3)$, and $C(2,-2)$ is a right triangle by using the converse of the Pythagorean Theorem. Find the area of the triangle.

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula.

Show that the points $A(-2,9), B(4,6), C(1,0),$ and $D(-5,3)$ are the vertices of a square.

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula.

Show that the points $A(-1,3), B(3,11),$ and $C(5,15)$ are collinear by showing that $d(A, B)+d(B, C)=d(A, C)$.

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula.

Find a point on the $y$ -axis that is equidistant from the points (5,-5) and (1,1).

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula and the Midpoint Formula.

Find the lengths of the medians of the triangle with vertices $A(1,0), B(3,6),$ and $C(8,2),$ (A median is a line segment from a vertex to the midpoint of the opposite side.)

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula and the Midpoint Formula.

Plot the points $P(-1,-4), Q(1,1),$ and $R(4,2)$ on a coordinate plane. Where should the point $S$ be located so that the figure $P Q R S$ is a parallelogram?

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula and the Midpoint Formula.

If $M(6,8)$ is the midpoint of the line segment $A B$ and if $A$ has coordinates $(2,3),$ find the coordinates of $B$.

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula and the Midpoint Formula.

(a) Sketch the parallelogram with vertices $A(-2,-1)$, $B(4,2), C(7,7),$ and $D(1,4)$.

(b) Find the midpoints of the diagonals of this parallelogram.

(c) From part (b) show that the diagonals bisect each other.

Ankit G.

Numerade Educator

In these exercises we use the Distance Formula and the Midpoint Formula.

The point $M$ in the figure is the midpoint of the line segment $A B$. Show that $M$ is equidistant from the vertices of triangle $A B C$.

Ankit G.

Numerade Educator

Suppose that each point in the coordinate plane is shifted 3 units to the right and 2 units upward.

(a) The point (5,3) is shifted to what new point?

(b) The point $(a, b)$ is shifted to what new point?

(c) What point is shifted to (3,4)$?$

(d) Triangle $A B C$ in the figure has been shifted to triangle $A^{\prime} B^{\prime} C^{\prime} .$ Find the coordinates of the points $A^{\prime}, B^{\prime},$ and $C^{\prime}$.

Ankit G.

Numerade Educator

Suppose that the $y$ -axis acts as a mirror that reflects each point to the right of it into a point to the left of it.

(a) The point (3,7) is reflected to what point?

(b) The point $(a, b)$ is reflected to what point?

(c) What point is reflected to (-4,-1)$?$

(d) Triangle $A B C$ in the figure is reflected to triangle $A^{\prime} B^{\prime} C^{\prime}$ Find the coordinates of the points $A^{\prime}, B^{\prime},$ and $C^{\prime}$.

Ankit G.

Numerade Educator

A city has streets that run north and south and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points $A$ and $B$ is 7 blocks- that is, 3 blocks east and 4 blocks north. To find the straightline distance $d$, we must use the Distance Formula.

(a) Find the straight-line distance (in blocks) between $A$ and $B$.

(b) Find the walking distance and the straight-line distance between the corner of $4 \mathrm{th}$ St. and $2 \mathrm{nd}$ Ave. and the corner of 11 th $\mathrm{St}$. and 26 th Ave.

(c) What must be true about the points $P$ and $Q$ if the walking distance between $P$ and $Q$ equals the straight-line distance between $P$ and $Q ?$

Ankit G.

Numerade Educator

Two friends live in the city described in Exercise $53,$ one at the corner of 3 rd St. and 7 th Ave. and the other at the corner of 27 th St. and 17 th Ave. They frequently meet at a coffee shop halfway between their homes.

(a) At what intersection is the coffee shop located?

(b) How far must each of them walk to get to the coffee shop?

Ankit G.

Numerade Educator

The graph shows the pressure experienced by an ocean diver at two different depths. Find and interpret the midpoint of the line segment shown in the graph.

Ankit G.

Numerade Educator

Plot the points $M(6,8)$ and $A(2,3)$ on a coordinate plane. If $M$ is the midpoint of the line segment $A B,$ find the coordinates of $B .$ Write a brief description of the steps you took to find $B$ and your reasons for taking them.

Ankit G.

Numerade Educator

Plot the points $P(0,3)$, $Q(2,2),$ and $R(5,3)$ on a coordinate plane. Where should the point $S$ be located so that the figure $P Q R S$ is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.

Ankit G.

Numerade Educator

Write each fraction in lowest terms. If the fraction is already in lowest terms, so indicate. SEE EXAMPLE 2. (OBJECTIVE 1)

$$\frac{33}{55}$$

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Write each fraction in lowest terms. If the fraction is already in lowest terms, so indicate. SEE EXAMPLE 2. (OBJECTIVE 1)

$$\frac{27}{18}$$

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Write each fraction in lowest terms. If the fraction is already in lowest terms, so indicate. SEE EXAMPLE 2. (OBJECTIVE 1)

$$\frac{35}{14}$$

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Write each fraction in lowest terms. If the fraction is already in lowest terms, so indicate. SEE EXAMPLE 2. (OBJECTIVE 1)

$$\frac{72}{64}$$

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Write each fraction in lowest terms. If the fraction is already in lowest terms, so indicate. SEE EXAMPLE 2. (OBJECTIVE 1)

$$\frac{26}{21}$$

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Perform each multiplication. Simplify each result when possible. SEE EXAMPLE 3. (OBJECTIVE 2).

$$\frac{1}{3} \cdot \frac{2}{5}$$

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Perform each multiplication. Simplify each result when possible. SEE EXAMPLE 3. (OBJECTIVE 2).

$$\frac{3}{4} \cdot \frac{5}{7}$$

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Perform each multiplication. Simplify each result when possible. SEE EXAMPLE 3. (OBJECTIVE 2).

$$\frac{4}{3} \cdot \frac{6}{5}$$

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Perform each multiplication. Simplify each result when possible. SEE EXAMPLE 3. (OBJECTIVE 2).

$$\frac{7}{8} \cdot \frac{6}{15}$$

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Perform each multiplication. Simplify each result when possible. SEE EXAMPLE 3. (OBJECTIVE 2).

$$12 \cdot \frac{5}{6}$$

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Perform each multiplication. Simplify each result when possible. SEE EXAMPLE 3. (OBJECTIVE 2).

$$10 \cdot \frac{5}{12}$$

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Perform each multiplication. Simplify each result when possible. SEE EXAMPLE 3. (OBJECTIVE 2).

$$\frac{10}{21} \cdot 14$$

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Perform each multiplication. Simplify each result when possible. SEE EXAMPLE 3. (OBJECTIVE 2).

$$\frac{5}{24} \cdot 16$$

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Perform each division. Simplify each result when possible. SEE EXAMPLE 5. (OBJECTIVE 2)

$$\frac{2}{5} \div \frac{3}{2}$$

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Perform each division. Simplify each result when possible. SEE EXAMPLE 5. (OBJECTIVE 2)

$$\frac{4}{5} \div \frac{3}{7}$$

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Perform each division. Simplify each result when possible. SEE EXAMPLE 5. (OBJECTIVE 2)

$$\frac{3}{4} \div \frac{6}{5}$$

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Perform each division. Simplify each result when possible. SEE EXAMPLE 5. (OBJECTIVE 2)

$$\frac{3}{8} \div \frac{15}{28}$$

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Perform each division. Simplify each result when possible. SEE EXAMPLE 5. (OBJECTIVE 2)

$$9 \div \frac{3}{8}$$

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Perform each division. Simplify each result when possible. SEE EXAMPLE 5. (OBJECTIVE 2)

$$23 \div \frac{46}{5}$$

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Perform each division. Simplify each result when possible. SEE EXAMPLE 5. (OBJECTIVE 2)

$$\frac{54}{20} \div 3$$

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Perform each division. Simplify each result when possible. SEE EXAMPLE 5. (OBJECTIVE 2)

$$\frac{39}{27} \div 13$$

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Perform each operation. Simplify each result when possible. See EXAMPLES 6-7.(OBJECTIVE 3)

$$\frac{3}{5}+\frac{3}{5}$$

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Perform each operation. Simplify each result when possible. See EXAMPLES 6-7.(OBJECTIVE 3)

$$\frac{4}{7}-\frac{2}{7}$$

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Perform each operation. Simplify each result when possible. See EXAMPLES 6-7.(OBJECTIVE 3)

$$\frac{5}{17}-\frac{3}{17}$$

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Perform each operation. Simplify each result when possible. See EXAMPLES 6-7.(OBJECTIVE 3)

$$\frac{2}{11}+\frac{9}{11}$$

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Perform each operation. Simplify each result when possible. See EXAMPLES 6-7.(OBJECTIVE 3)

$$\frac{1}{42}+\frac{1}{6}$$

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Perform each operation. Simplify each result when possible. See EXAMPLES 6-7.(OBJECTIVE 3)

$$\frac{17}{25}-\frac{2}{5}$$

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Perform each operation. Simplify each result when possible. See EXAMPLES 6-7.(OBJECTIVE 3)

$$\frac{7}{10}-\frac{1}{14}$$

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Perform each operation. Simplify each result when possible. See EXAMPLES 6-7.(OBJECTIVE 3)

$$\frac{8}{25}+\frac{1}{10}$$

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Perform each operation. Simplify each result when possible. SEE EXAMPLE 8.(OBJECTIVE 4)

$$4 \frac{3}{5}+\frac{3}{5}$$

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Perform each operation. Simplify each result when possible. SEE EXAMPLE 8.(OBJECTIVE 4)

$$2 \frac{1}{8}+\frac{3}{8}$$

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Perform each operation. Simplify each result when possible. SEE EXAMPLE 8.(OBJECTIVE 4)

$$3 \frac{1}{3}-1 \frac{2}{3}$$

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Perform each operation. Simplify each result when possible. SEE EXAMPLE 8.(OBJECTIVE 4)

$$6 \frac{1}{5}-4 \frac{2}{5}$$

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Perform each operation. Simplify each result when possible. SEE EXAMPLE 8.(OBJECTIVE 4)

$$3 \frac{3}{4}-2 \frac{1}{2}$$

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Perform each operation. Simplify each result when possible. SEE EXAMPLE 8.(OBJECTIVE 4)

$$15 \frac{5}{6}+11 \frac{5}{8}$$

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Perform each operation. Simplify each result when possible. SEE EXAMPLE 8.(OBJECTIVE 4)

$$8 \frac{2}{9}-7 \frac{2}{3}$$

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Perform each operation. Simplify each result when possible. SEE EXAMPLE 8.(OBJECTIVE 4)

$$3 \frac{4}{5}-3 \frac{1}{10}$$

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Change each fraction to decimal form and determine whether the decimal is a terminating or repeating

decimal. (OBJECTIVE 5)

$$\frac{3}{5}$$

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Change each fraction to decimal form and determine whether the decimal is a terminating or repeating

decimal. (OBJECTIVE 5)

$$\frac{5}{9}$$

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Change each fraction to decimal form and determine whether the decimal is a terminating or repeating

decimal. (OBJECTIVE 5)

$$\frac{9}{22}$$

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Change each fraction to decimal form and determine whether the decimal is a terminating or repeating

decimal. (OBJECTIVE 5)

$$\frac{8}{5}$$

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Perform each operation. SEE EXAMPLES 10, 11, and 12.(OBJECTIVE 5)

$$43.54+315.7$$

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Perform each operation. SEE EXAMPLES 10, 11, and 12.(OBJECTIVE 5)

$$345.213-27.35$$

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Perform each operation. SEE EXAMPLES 10, 11, and 12.(OBJECTIVE 5)

$$67.235-22.45$$

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Perform each operation. SEE EXAMPLES 10, 11, and 12.(OBJECTIVE 5)

$$21.36+4.573$$

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Perform each operation. SEE EXAMPLES 10, 11, and 12.(OBJECTIVE 5)

$$7.2 \cdot 15.6$$

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Perform each operation. SEE EXAMPLES 10, 11, and 12.(OBJECTIVE 5)

$$4.21 \cdot 2.73$$

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Perform each operation. SEE EXAMPLES 10, 11, and 12.(OBJECTIVE 5)

$$0 . 2 3 \longdiv { 1 . 0 4 6 5 }$$

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Perform each operation. SEE EXAMPLES 10, 11, and 12.(OBJECTIVE 5)

$$4 . 7 \longdiv { 1 0 . 8 5 7 }$$

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Round each of the following to two decimal places and then to three decimal places. SEE EXAMPLE 13. (OBJECTIVE 6)

$$496.2583$$

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Round each of the following to two decimal places and then to three decimal places. SEE EXAMPLE 13. (OBJECTIVE 6)

$$13.0547$$

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Round each of the following to two decimal places and then to three decimal places. SEE EXAMPLE 13. (OBJECTIVE 6)

$$6,025.3982$$

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Round each of the following to two decimal places and then to three decimal places. SEE EXAMPLE 13. (OBJECTIVE 6)

$$1.6048$$

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Use a calculator to perform each operation and round each answer to two decimal places.

$$474.81+23.4532$$

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Use a calculator to perform each operation and round each answer to two decimal places.

$$843.45213-712.765$$

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Use a calculator to perform each operation and round each answer to two decimal places.

$$25.25 \cdot 132.179$$

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Use a calculator to perform each operation and round each answer to two decimal places.

$$234.874 \cdot 242.46473$$

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Use a calculator to perform each operation and round each answer to two decimal places.

$$0 . 4 5 6 )\overline { 4 . 5 6 9 4 3 2 3 }$$

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Use a calculator to perform each operation and round each answer to two decimal places.

$$4 3 . 2 2 5 )\overline { 3 2 . 4 6 5 7 4 8 }$$

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Use a calculator to perform each operation and round each answer to two decimal places.

$$55.77443-0.568245$$

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Use a calculator to perform each operation and round each answer to two decimal places.

$$0.62317+1.3316$$

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SEE EXAMPLES 4, 9, AND 14. (OBJECTIVE 7)

A farmer has plowed $12 \frac{1}{3}$ acres of a $43 \frac{1}{2}$ -acre field. How much more needs to be plowed?

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SEE EXAMPLES 4, 9, AND 14. (OBJECTIVE 7)

The four sides of a garden measure $7 \frac{2}{3}$ feet, $15 \frac{1}{4}$ feet, $19 \frac{1}{2}$ feet, and $10 \frac{3}{4}$ feet. Find the length of the fence needed to enclose the garden.

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SEE EXAMPLES 4, 9, AND 14. (OBJECTIVE 7)

A designer needs $4 \frac{1}{3}$ yards of material for each dress he makes. How much material will he need to make 15 dresses?

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SEE EXAMPLES 4, 9, AND 14. (OBJECTIVE 7)

Each lap around a stadium track is $\frac{1}{4}$ mile. How many laps would a runner have to complete to run 26 miles?

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SEE EXAMPLES 4, 9, AND 14. (OBJECTIVE 7)

After hurricane damage estimated at 187.75 dollar million, a county sought relief from three agencies. Local agencies gave 46.8 million dollar and state agencies gave 72.5 million dollar. How much must the federal government contribute to make up the difference?

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SEE EXAMPLES 4, 9, AND 14. (OBJECTIVE 7)

$26.5 \%$ of the 12,419,000 citizens of Illinois are nonwhite. How many are nonwhite?

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The following circle graph shows the various sources of retirement income for a typical retired person. Use this information in Exercises $141-142.$

If a retiree has 36,000 dollar of income, how much is expected to come from pensions and Social Security?

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The following circle graph shows the various sources of retirement income for a typical retired person. Use this information in Exercises $141-142.$

If a retiree has 52,000 dollar of income, how much is expected to come from earned income?

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In the manufacture of active-matrix color LCD computer displays, many units must be rejected as defective. If $23 \%$ of a production run of 17,500 units is defective, how many units are acceptable?

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Almost all of the water must be removed when food is preserved by freeze-drying. Find the weight of the water removed from 750 pounds of a food that is $36 \%$ water.

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This year, sales at Positronics Corporation totaled 18.7 million dollar. If the projection of $12 \%$ annual growth is true, what will be next year's sales?

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In tryouts for the Olympics, a speed skater had times of $44.47,43.24,42.77,$ and 42.05 seconds. Find the average time. Give the result to the nearest hundredth. (Hint: Add the numbers and divide by 4.)

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Samuel drove his car 16,275.3 miles last year, averaging 25.5 miles per gallon of gasoline. If the average cost of gasoline was 3.45 dollar per gallon, find the fuel cost to drive the car.

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A woman earns 48,712.32 dollar in taxable income. She must pay $15 \%$ tax on the first 23,000 dollar and $28 \%$ on the rest. In addition, she must pay a Social Security tax of $15.4 \%$ on the total amount. How much tax will she need to pay?

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A rectangular parking lot is 253.5 feet long and 178.5 feet wide. A 55 -gallon drum of asphalt sealer covers 4,000 square feet and costs 97.50 dollar. Find the cost to seal the parking lot. (Sealer can be purchased only in full drums.)

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Each TV a retailer buys costs 3.25 dollar per day for warehouse storage. What does it cost to store 37 TVs for three weeks?

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A manufacturer of computer memory boards has a profit of 37.50 dollar on each standard-capacity memory board, and 57.35 dollar on each high-capacity board. The sales department has orders for 2,530 standard boards and 1,670 high-capacity boards. Which order will produce the greater profit?

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A Holstein cow will produce 7,600 pounds of milk each year, with a $3 \frac{1}{2} \%$ butterfat content. Each year, a Guernsey cow will produce about 6,500 pounds of milk that is $5 \%$ butterfat. Which cow produces more butterfat?

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Each year, a typical dairy cow will eat 12,000 pounds of food that is $57 \%$ silage. To feed 30 cows, how much silage will a farmer use in a year?

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Two carpenters bid on a home remodeling project. The first bids 9,350 dollar for the entire job. The second will work for 27.50 dollar per hour, plus 4,500 dollar for materials. He estimates that the job will take 150 hours. Which carpenter has the lower bid?

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A high-efficiency home heating system can be installed for 4,170 dollar, with an average monthly heating bill of 57.50 dollar. A regular furnace can be installed for 1,730 dollar, but monthly heating bills average 107.75 dollar. After three years, which system has cost more altogether?

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Refer to Exercise $155 .$ Decide which furnace system will have cost more after five years.

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Explain how you would decide which of two decimal fractions is the larger.

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In what situations would it be better to leave an answer in the form of an improper fraction?

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When would it be better to change an improper-fraction answer into a mixed number?

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Can the product of two proper fractions be larger than either of the fractions?

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How does the product of one proper and one improper fraction compare with the two factors?

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