Basic Skills/ TAP Test -- Mathematics Boot Camp Study Guide

Basic Skills/ TAP Test -- Mathematics Boot Camp Study Guide
The Basic Skills/ TAP Test of mathematics examines skills required to demonstrate quantitative
literacy at the college level through the application of mathematical methods and reasoning to
the solution of real-world problems.
Comparison Basic Skills to TAP:
Mathematics: BS300 - 42 multiple choice; TAP400- 50 multiple choice online
TAP Study Guide:
Websites for additional help:
Dr. Jim Olsen, WIU: skills.htm
Praxis Pre-professional Skills Test:
Math Dictionary:
Basic Math Help -
ACT practice test (math):
ACT practice test booklet p. 17-20 (can be printed) (answers at the end of p. 20)
ACT practice test booklet explanations for test questions
Discovery Education:
Free Math Help:
Click “algebra” then “see all 258 skills”
Either “enter your problem” or view the “example problems”
Purple math: “tutoring and lessons” or “quizzes and
Suggested General Strategies for the Math Test:
Read the test standards and objectives. This is a source of information about topics
covered by the test. This can give a more specific idea of the knowledge and skills
you will need to demonstrate on the test.
As you are answering the questions, remember to read the directions and the test
questions carefully. Read all response options. Remember that the test questions
call for the "best answer." Do not choose what appears to be the first reasonable
response option; read and evaluate all choices to find the best answer.
Draw a quick sketch to illustrate the given information and help you “see” how the
pieces of information fit together.
If you’re running out of time and cannot quickly determine the best answer to a
question, try to eliminate as many of the options as possible. Then, guess among the
remaining answer choices.
Remember if a problem is long, looks hard or is a topic that gives you trouble, skip it
and save for the end. If the problem is more than four lines long, skip it and save it
till the end.
Always go to the end of a word problem. Read the question. Study the diagram and
think about the implications.
Answer their question, not your question. When answering, make sure to answer
the question asked. If the question asks for “length”, don’t give the value for
Plug-in Technique. Work backwards. If appropriate, try substituting the answers
into the problem. Start from the middle choice and work up/down depending on
your initial result.
Always check your answer for: Does it make sense? Does it answer the question?
Are your computations correct?
Use any remaining time at the end of the session to check the accuracy of your
work. Go back to the questions that gave you difficulty and verify your work on
PRACTICE, PRACTICE, PRACTICE! There are many problems on websites to work on.
You may want to print them out so you can annotate!
Key Math Vocabulary:
Acute angle -- an angle measuring between zero and 90 degrees
Acute triangle -- a triangle with three acute angles
Alternate exterior angles -- congruent angles formed by two parallel lines cut by a transversal;
located on opposite sides of the transversal outside the parallel lines
Alternate interior angles -- congruent angles formed by two parallel lines cut by a transversal;
located on opposite sides of the transversal between the parallel lines
Bar graph – a diagram representing the frequency distribution for nominal or discrete data.
It consists of a sequence of bars, or rectangles, corresponding to the possible values, and the
length of each is proportional to the frequency.
For more info:
Box-and-whisker plot – a diagram constructed from a set of numerical data showing a box
that indicates the middle 50% of the marked observations together with lines, sometime
called ‘whiskers’, that go out from the quartile to the most extreme data value in that
direction which is not more than 1.5 times the Inter Quartile Range from the quartile.
For more info:
Circle -- a 2-D figure; the set of all points at a given distance from the center
Circle graph – a graph for categorical data. The proportion of elements belonging to each
category is proportionally represented as a pie-shaped sector of a circle. Sometimes called a
pie chart.
For more info:
Circumference -- the perimeter of a circle
Clusters of data – a portion of high concentration in a data set.
For more info:
Complementary angles -- two or more angles that sum to 90 degrees
Cone -- a 3-D figure with a circular or non-circular base and rounding sides rising to a point
Congruent -- of the same size and shape
Corresponding angles -- congruent angles formed by two parallel lines cut by a transversal;
positioned on matching "corners" of intersections
Cylinder -- a 3-D figure with circular or non-circular parallel congruent bases and curving sides
Data – the observations gathered from an experiment, survey or observational study.
For more info:
Diagonal -- a line segment connecting two non-adjacent vertices of a polygon
Equiangular triangle -- a triangle with three congruent angles
Equilateral triangle -- a triangle with three congruent sides
Histogram – A bar graph presenting the frequencies of occurrence of data points. Sometimes
called a frequency histogram
For more info:
Isosceles trapezoid -- a trapezoid with two congruent legs
Isosceles triangle -- a triangle with two congruent legs
Kite -- a quadrilateral with two pair of adjacent congruent sides
Lateral faces -- the polygon shapes making up the sides of a polyhedron
Line-of-best-fit – the line that best represents the trend that the points in a scatter plot
For more info:
Line plot – a line graph that orders the data along a real number line. Also called a dot plot.
For more info:
Mean – the mean is an appropriate location measure for interval or ratio variables. Suppose
there are N individuals in the population and x denotes an interval or ratio variable. Let
values for the ith individual be denoted by i x . The mean of x is the number
For more info:
Median – suppose the observations in a set of numerical data are ranked in ascending order.
The median is the middle observation if there are an odd number of observations, and is the
average of the two middlemost observations if there are an even number of observations.
For more info:
Mode –The mode is the most frequently occurring value in a set of discrete data. There can be
more than one mode if two or more values are equally common.
For more info:
Obtuse angle -- an angle measuring between 90 and 180 degrees
Obtuse triangle -- a triangle with one obtuse angle
Odds – a way of representing the likelihood of an event's occurrence. The odds m:n in favor
of an event means we expect the event will occur m times for every n times it does not occur.
For more info:
Parallel lines -- lines that never intersect in a plane
Parallelogram -- a quadrilateral with two pair of parallel sides
Perimeter -- the distance around a 2-D figure
Perpendicular -- meeting at a 90 degree angle
Polygon -- a simple closed two-dimensional shape made of line segments
Prism -- a 3-D figure with parallel congruent polygon bases and sides made of parallelograms
Probability – the chance/likelihood that a particular event (or set of events) will occur
expressed on a scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage
between 0 and 100%.
For more info:
Pyramid -- a 3-D figure with a polygon for a base and triangular faces meeting at a point at the
Pythagorean Theorem -- the sum of the squares of the legs of a right triangle equals the square
of the hypotenuse; a2 + b2 = c2 where c is always the hypotenuse side
Radius -- the distance from the center to the circle
Range – the range of a sample (or a data set) is a measure of the spread or the dispersion of
the observations. It is the difference between the largest and the smallest observed value.
For more info
Rectangle -- a parallelogram with a right angle
Regular -- 2-D or 3-D figure having all congruent sides and all congruent angles
Rhombus -- a quadrilateral with four congruent sides
Right angle -- an angle measuring exactly 90 degrees
Right triangle -- a triangle with one right angle
Sample – a subset of a population that is obtained through some process, possibly random
selection or selection based on a certain set of criteria, for the purposes of investigating the
properties of the underlying parent population.
For more info:
Sphere -- a 3-D figure; the set of all points at a given distance from the center
Square -- a quadrilateral with all congruent sides and all congruent angles
Stem-and-leaf plot – a semi-graphical method used to represent numerical data, in which the
first (leftmost) digit of each data value is a stem and the rest of the digits of the number are
the leaves. For more info:
Straight angle -- measuring exactly 180 degrees; makes a straight line
Supplementary angles -- two or more angles that sum to 180 degrees
Transversal -- a line that crosses a pair of parallel lines
Trapezoid -- a quadrilateral with one pair of parallel sides
Tree diagram – A tree diagram displays all the possible outcomes of an event.
For more info:
Variable – a quantity that varies. For example, the weight of a randomly chosen member of a
football team is such a variable. Variables are usually represented by letters.
For more info:
Venn diagram – a graphic means of showing intersection and union of sets by representing
them as bounded regions.
For more info:
Vertical angles -- congruent opposite angles formed by two intersecting lines
Standard 0014
Solve problems involving integers, fractions, decimals, and units of measurement.
The following are examples of content that may be covered under this standard.
 Solve problems involving integers, fractions, and decimals, including percentages.
 Solve problems involving ratios and proportions.
 Solve problems involving units of measurement, including U.S. customary and metric
measurements, and conversions, including scientific notation (e.g., 6.05 x 10 8).
 Use estimation skills to solve problems.
Practice Test Samples
1. In a park, the ratio of ducks to geese is 16 to 9. How many of the 300 birds are geese?
To solve: The ratio tells me that, of every 16 + 9 = 25 birds, 9 are geese. That is, 9/25 of
the birds are geese. Then there are ( 9/25 )(300) = 108 geese
2. Convert 3 gallons to quarts.
To solve: Quarts are smaller than gallons; every gallon has four quarts. Since I'm
converting from a larger unit (gallons) to a smaller unit (quarts), my answer needs to be
a bigger number. So I multiply:(3)(4) = 12 quarts
3. Convert 7920 yards to miles
To solve: Miles are bigger than yards; there are 1760 yards in every mile. Since
I'm converting from a smaller unit (yards) to a bigger unit (miles), my answer needs to
be a smaller number. So I divide: 7920 ÷ 1760 = 4.5 miles
4. The original price of a shirt was $20. It was decreased to $15. What is the percent
To solve: The absolute decrease is 20 - 15 = $5
The percent decrease is the absolute decrease divided by the original price (part/whole).
5 / 20 = 0.25 then convert to a percent = 25%
5. Mary has a monthly salary of $2400. She spends $380 per month on food. What percent of
her monthly salary does she spend on food?
To solve: The part of her salary that is spent on food is $380 out of her monthly salary of
$2400… percent = part / whole = 380 / 2400 = 0.16 (rounded to 2 decimal places) then
convert to a percent = 16%
6. Solve and write (3.4 x 106) (4.2 x 103) in scientific notation.
To solve: (3.4)(4.2) x 10 6+3 = 14.28 x 109 = 1.428 x 1010
7. Solve and write (3 x 104)3 in scientific notation.
To solve: 33 x 104x3 = 27 x 1012 = 2.7 x 1013
8. A rectangle has a length of 20 mm and a width of 15 mm. What is the perimeter of the
rectangle in cm?
To solve: add two widths and two lengths (20 + 20 + 15 +15 = 70mm) then convert the
70 mm into cm (70mm ÷ 10) = 7cm
9. A walking path is 402 m from start to finish. If the path is walked 5 times how far to the
nearest kilometer is walked?
To solve: multiple 402 x 5 = 2010m then convert the m to km = 2.010 km then round to
the nearest km = 2 km
10. Lilly rides 1.3 miles from her house to school then 1.68 miles from school to swim
practice and finally she rides 2.2 miles from swim practice back to her house. She does this 5
days a week. Approximately how far does she ride in one week?
To solve: round each distance (1.3 →1.0), (1.68 → 2.0), (2.2 →2.0) ….so, in one day she
rides 5 miles (1+2+2), 5 miles x 5 days = 25 miles per week.
11. Lilly has 24 yards of ribbon for 18 presents she has to wrap. Each present gets the same
amount of ribbon. How much ribbon will each present get?
To solve: first, convert 24 yards to inches…24 yards x 3 = 72 feet, 72 feet x 12 = 864
inches. Now divide the 864 inches by 18 and that equals 48 inches or the answers might
be in only feet (4 feet) or the answer could be in yards and feet (I yard 1 foot)
12. Jessica owns Bubbles pet store, she needs to replace the tubing in the 22 fish tanks at the
store. Each tank needs a length of tubing that is 1 foot 8 inches. Plus the owner wants an
additional 5 feet of tubing to save for repairs. What is the total length of tubing Jessica needs
to buy for his store?
To solve: convert 1 foot 8 inches into all inches, 12 + 8 = 20 inches, then multiple 20
inches by 22 tanks and get 440 inches needed for the tank. Next, Convert the extra 5
feet to inches (5 x 12 = 60 inches) now add the inches needed for the tank and the extra
inches together: 440 + 60 = 500 inches. BUT the answer probably won’t be in only
inches so, divide 500 by 12 to see how many feet can be taken out of the inches: 500 ÷
12 = 41.66 so we know 41 feet can be taken out. Since 41 feet = 492 inches we know
that 500 inches = 41 feet 8 inches if that isn’t a choice then you may have to convert
that to yards so 41 feet divide by 3 is 13.66 so we know that 13 yards can be taken out,
13 x 3 is 39 which means we would have 2 feet left over. Another way to write the
answer is 13 yards 2 feet 8 inches
Standard 0015
Apply mathematical reasoning skills to analyze patterns and solve problems.
The following are examples of content that may be covered under this standard.
 Draw conclusions using inductive reasoning.
 Draw conclusions using deductive reasoning.
 Identify errors in mathematical explanations.
*To help you understand the difference between inductive and deductive reasoning…
Inductive reasoning is from a specific case or cases and deriving a general rule. It draws
inferences from observations in order to make generalizations.
Inference can be done in four stages:
o Observation: collect facts, without bias.
o Analysis: classify the facts, identifying patterns o of regularity.
o Inference: From the patterns, infer generalizations about the relations
between the facts.
o Confirmation: Testing the inference through further observation.
Inductive reasoning strategies
o Derive a general rule in an accepted area and then apply the rule in the
area where you want the situation or person to behave.
o Give lots of detail, then explain what it all means.
o Talk about the benefits of the parts and only get to the overall benefits
o Take what has happened and give a plausible explanation for why it has
Deductive reasoning starts with a general case and deduces specific instances. Deductive
reasoning starts with an assumed hypothesis or theory.
 Deductive reasoning assumes that the basic law from which an argument is
made is applicable in all cases. This can let you take a rule and apply it perhaps
where it was not really meant to be applied.
 Deductive conclusions can be valid or invalid.
Practice Test Samples
1. Look carefully at the following figures. Then, use inductive reasoning to make a conjecture
about the next figure in the pattern.
To solve: look carefully
at the pattern; you
should came up with
this figure:
2. Look at the patterns below. Draw the next figure.
To solve: The trick is to see that one dot is always placed between and
above two dots. Also, the next figure always has one more dot at the
very bottom row keeping this in mind; your next figure should look like
3. Write the next three numbers in the sequence 7, -7, 14, -42, 168...
To solve: Since we are bouncing back and forth between positive and negative numbers,
a pattern in addition doesn't look promising. Let's check out multiplication. At first
glance, I would say that a negative number is probably what we are looking for here,
since it does alternate signs. It doesn't appear to be the same number each time,
because 7 times -1 is -7, but -7 times -2 = 14. It looks like we have 7 (-1), -7 (-2), 14 (-3),
-42 (-4), 168, ... Aha, we have a pattern in multiplication - we are multiplying by the
next negative integer. Since 168(-5) = -840, -840(-6) = 5040, 5040(-7) = -35280. Final
Answer: -840, 5040, -35280
4. Write the next three numbers in the sequence 3, 5, 9, 15, 23…
To solve: see if there is some pattern in addition…we are not adding the same number
each time to get to the next number. But, make note of what you do add to each
number 3 +2, 5 +4, 9+6, 15 +8, 23, see the pattern? You are always adding the next
even number. Since 23+10 = 33, 33 +12 = 45, 45 +14 = 59. Final Answer: 33, 45, 59
5. Use the statements #1-5 below to answer the question that follows:
1. All people wearing hats have blonde hair.
2. Some of the people have red hair.
3. All people who have blonde hair like hamburgers.
4. People who have red hair like pizza.
5. Keith has blonde hair.
Which of the following statements MUST be true?
a. Keith likes hamburgers.
b. Keith has red hair.
c. Keith likes pizza.
d. Keith is wearing a hat.
To solve: with deductive reasoning, you need to be a 100% sure. Since statement 3 says
that ALL people who have blonde hair like hamburgers and Keith has blonde hair, then
statement a, Keith likes hamburgers, is a 100% guarantee.
6. Jerry, Kevin, Todd and Mark all live on the first floor of an apartment complex. One is a
manager, one is a computer programmer, one is a singer, and the other is a teacher. Use the
statements below to answer the question Which is the manager?
A. Jerry and Todd eat lunch with the singer.
B. Kevin and Mark carpool with the manager.
C. Todd watches CSI with the manger and the singer.
To solve: You can use a process of elimination on this problem.
Statement A, Jerry and Todd eat lunch with the singer, doesn’t let us definitively
eliminate anyone from being the manager.
However, statement B, Kevin and Mark carpool with the manager, eliminates Kevin and
Mark from being the manager. And statement C, Todd watches CSI with the manger
and the singer, eliminates Todd. The only one that could be (100%, without a doubt)
the manager is Jerry.
7. Four friends - Suzy, John, Sally, and Tom - each has his or her own hobby. One collect
coins, one sews, one cooks, and one plays in a band, not necessarily in that order.
Use the statements below to answer: Who is the friend that likes to sew?
A. Suzy and John always eat lunch with the friend that plays in the band.
B. Sally and Tom carpool with the one who likes to sew.
C. John and the friend that likes to cook visited the one who likes to sew.
To solve: use a process of elimination on this problem. Statement A, Suzy and John
always eat lunch with the friend that plays in the band, doesn't let us definitively
eliminate anyone from being the one who likes to sew.
However, statement B, Sally and Tom carpool with the one who likes to sew, eliminates
Sally and Tom from being the one who likes to sew.
Statement C, John and the friend that likes to cook visited the one who likes to sew,
eliminates John.
The only one that could be (100%, without a doubt) the one who likes to sew is Suzy.
8. Write the next five figures in the pattern.
To solve: It looks like several things change throughout this sequence. It starts with one
line and then one circle then it has two lines and two circles and then three lines. So, it
is alternating between lines and circles and each time it alternates it adds one more of
that figure.
So the next five figures would be:
Note that we stopped at the fifth one, if we would have continued; there would be a
total of four lines that follow the three circles.
Standard 0016
Solve problems involving algebra and geometry:
The following are examples of content that may be covered under this standard.
 Graph numbers or number relationships.
 Find the value of the unknown in a given one-variable equation.
 Express one variable in terms of a second variable in two-variable equation.
 Solve problems involving lines and angles.
 Solve problems involving two-and three-dimensional geometric figures (e.g. perimeter
and area problems, volume and surface area problems).
Practice Test Samples
1. Graph the following ordered pairs of numbers in the
coordinate plane: (4, -6), (-3, 5)
To solve: Recall: each set of parentheses represents
the location of a point in the x-y (coordinate) plane.
The first number represents a location to the right or
left of the origin (0,0). The second number represents
a location above or below the origin. Start at the
origin and move accordingly.
2. Find the roots of y = 3x² − 5x – 2 and graph.
To solve: Recognize this as a second degree equation.
Anticipate a parabola-shaped graph. Generate a table of values
by selecting any Real number for an “x” value, substitute it in
for x in the equation, and calculate the corresponding value for
y. After finding several sets of values, locate the corresponding
points on the graph. Sketch the graph, by drawing a smooth
curve passing through the points. Remember to select both
positive and negative values for x, and using x = 0 is always
helpful. **Another way to solve the problem is to factor the
equation like this: 3x² − 5x − 2 = (3x + 1)(x − 2). Then set each
half equal to 0 to find the roots…if 3x + 1 = 0 then one root is
−1/3, if x – 2 = 0 then the other root is 2.
The roots are -1/3 and 2
3. Solve for x in 4x – 28 = 2x + 14
To solve: Recognize this as a first degree equation; Anticipate no more than 1 solution;
Isolate the variable on one side of the equation, combine like terms, and simplify.
4x – 28 = 2x + 14…add 28 to each side and get 4x = 2x + 42, then subtract 2x from each
side and get 2x = 42, finally divide each side by 2 and get x = 21
4. Solve for x in x² + 4x – 28 = -7
To solve: Recognize this as a second degree equation; Anticipate the possibility of 2
solutions; Get one side of the equation equal to zero, factor, set each factor to zero and
solve for the variable as in the Strategy for a above.
or: x2 + 4x – 28 = -7…add 7 to each side and get x2 + 4x – 21, now think you two numbers
that when multiplied give you -21 and when added give you 4..yes the two numbers are
-3 and 7 so the new expression is (x-3)(x+7), the final set to solve for x is to set each side
equal to 0… if x – 3 = 0 then x= 3 and if x + 7 = 0 the x = -7 so the final answer is
x = 3 and -7
5. In the diagram below, assume that line l is parallel to line m. The two parallel lines are
intersected by a transversal line p One angle measure is given in the diagram. Find the
measures of the other angles.
To solve: Recall that angles such as that labeled 125⁰ is a vertical angle with
c, so it is
125⁰ too. Angle B is supplementary with the angle labeled 125⁰, so together they are
180⁰…(180-125 = 55) Recall that angles such as c and d are called alternate interior
angles and will have the same measure. Angles such a and e are corresponding angles
and will have the same measure. Remember, this reasoning only works if the lines are
angles a, b, e, f = 55o angles c, d, g = 125o
6. A certain rectangle has an area of 28 m². Its length is 2.5 m. Find it width.
To solve: Quickly sketch a diagram and label everything you know. Recall that
the area of a rectangle is length × width, so if only one dimension is known,
divide the area by that dimension to get the other. (hint: sometimes we think of
length as being longer than width. In math it’s not a problem if the length turns
out to be shorter than the width.)
7. By attaching the edges of the circular sector of paper, a right circular cone is formed.
What is the lateral surface area of the cone shown below? (This question adapted from ICTS Study Guide, 2006)
To solve: Recall that a sector is part of a circle with 2 sides that are radii of
the circle and that lateral surface area is the area of the region. The formula
for area of a circle: A = r². Since our sector is 40⁰ and a full circular area
would be 360⁰, we have
= of a circle. A full circle with a radius of 8
inches would have an area of ·8², which is 64 in². To answer the question,
we would need to find 1/9 of this amount. Notice that this question does not
ask for the total surface area. If it were needed, we would have been told on
the area of the base of the cone too. The answer is D.
Standard 0017
Understand concepts and procedures related to data analysis and statistics.
The following are examples of content that may be covered under this standard.
 Interpret information form tables, line graphs, bar graphs, histograms, pictograms, and
pie charts.
 Recognize appropriate representations of various data in graphic form (discrete and
 Demonstrate an understanding of fundamental statistical concepts (e.g., mean,
correlation, standard deviation).
 Interpret graphic and nongraphic representations of frequency distributions,
percentiles, central tendency, variability, and correlation.
Practice Test Samples
1. The scores of a group of graduate students are shown below in a graph.
What is the range of the data?
To solve: Range is the difference between the largest and smallest data (100-0)= 100.
2. The graphs show the growth in the use of cellular phones. Why do they look different?
The data is different.
The vertical scale is different.
The horizontal scale is different.
The number of people is increasing
To solve: Check the scale on each… b) The vertical scale is different
3. According to the graph, approximately how many students needed 20 minutes to locate
suitable library materials?
To solve: From 20 on the horizontal axis, go vertically until you hit the graph.
Then go horizontally to the vertical axis and read the value… d) 50
Use the circle graph to solve problems 4-5.
The circle graph shows the percent of the total population of a small town is 9,520 living in the
given types of housing.
4. Find the number of people who live in condos and apartments to the nearest whole
a) 4,094
b) 5,712
c) 5,710
d) 6,000
To solve: Add the percents for condos and apartments (17% + 43% = 60%) and
multiply by total population (9,520 x 60%) = b) 5715
5. Find the number of degrees in the sector of the circle that represents people who live in
townhomes. Round your answer to the nearest degree.
b) 21.6
c) 21
d) 216
To solve: Multiply the number of degrees (360°) in a circle by .06 (360° x .06) =
b) 21.6
Use the stem and leaf plot to solve problems 6 - 7
The weights (in pounds) of a group of high school students are listed below in a stem-and-leaf
6. What is the median of the data?
To solve: The median is the
data point which is in the center of the
data AFTER it is arranged from low to high.
Median = 130
7. What is the mode of the data?
To solve: The mode is the data point which
occurs most frequently.
Mode = 142
8. The following scatterplot displays temperatures recorded in various locations at
different latitudes on a particular summer day. What can be said about the correlation?
a) Negative correlation
b) Positive correlation
c) No correlation
d) no correlation
To solve: Since the data is approximately linear, consider the relationship
between the values…a) Negative correlation
Standard 0018
Solve applied problems using a combination of mathematical skills (including word problems
involving one and two variables).
The following are examples of content that may be covered under this standard.
 Apply combinations of mathematical skills to solve a series of related problems.
 Identify an equation to solve word problems involving one and two variables.
 Apply number concepts and geometric principles to solve practical problems.
 Apply statistical principles to analyze patterns and trends in data.
Practice Test Samples
1. Solve for x and y in 3x + 2y = 3 and x = 3y – 10
To solve: Replace x in the first equation with its equivalent, (3y - 10) from the second
3x + 2y = 3
3(3y - 10) + 2y = 3
9y - 30 + 2y = 3
11y = 33
Top equation.
Replaced x with (3y - 10).
Multiplied out.
Divide each side by 11 to get answer.
Now that y has a value, you can plug that value in either equation and find a value for x.
Because the second equation has already been solved for x, it will be easier to plug 3 in
for y in that equation.
x = 3(3) - 10
x = 9 - 10
x = -1
The solution is the ordered pair (-1,3).
2. Jess has a garden that is 5.2 feet by 3.8 feet. If 3 flowers fit into each square foot, about
how many flowers will she be able to plant in the area?
To solve: 5.2 → 5 feet, 3.8 → 4, area is L x W so, 5 x 4 = 20 square feet and if 3 flowers
can go in each square foot, 20 x 3 = 60 flowers
Solve for x in 3x = 2x + 1
To solve: By subtracting 2x from each side, the variables are all on the same side of the
equation. By combining like terms, the variable is isolated, and the equation is simplified
and solved. X = 1
Find the value of c.
To solve: a2 + b2 = c2 Write the Pythagorean Theorem
Then plug in
52 + 122 = c2
169 = c2 Solve for c
c = 13
5. A rectangle has a length that is one more than 2 times its width. The perimeter of the
rectangle is 26 feet. Find the length of the rectangle and the area of the rectangle.
To solve: First draw a picture:
2X + 1
for the length : the width can be represented as X and then the length would be 2X + 1,
next add all the 4 sides together (X) + (X) +(2X + 1) + (2X + 1) = 6X + 2, now set that = to
the perimeter of 26, 6X + 2 = 26, simplify by subtracting two from each side 6X = 24,
then divide each side by 6 and get X = 4, the width is 4 and then the length is 9 feet.
Draw another picture:
4 feet
9 feet
for the area: Now that you know the width (4) and the length (9) multiple them
together to get the area. Area = Length x Width 9 x 4 = 36, the area is 36 square feet.
6. The 24 students in Mr. Madison’s class earn stickers for good work and behavior. During
the first week the least amount of stickers earned by a student was 10 and the most amount
of stickers earned by a student was 30. Which of the following could be a reasonable
estimate of the total number of stickers Mr. Madison gave out during the first week of
a) 240
b) 480
c) 720
d) 960
To solve: find the average between the least amount and the greatest amount of
stickers (10 + 30) ÷ 2 = 20. Then multiple the 20 stickers by the 24 students to
get b) 480
5. A rectangular solid measures 3 feet by 4 feet. The volume of the box is 48 cubic feet. If y
represents the height of the box, then which equation represents the volume?
3 + 4y = 48
7y = 48
12y = 48
12 + y = 48
To solve: remember Volume = height x length x width … c)12y = 48
6. In the figure to the right, AB = BC and the measure of
What is the measure of B?
C is 40 .
To solve: remember if AB = BC then angles A and C are equal. And also
remember that the sum of interior angles in any triangle is 180 o so,
180o – (40o + 40o) = c) 100o
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