math
so changing from mixed to improper is important. This method works:
A b/c = Axc + b/c
If there are 3 red balls, 7 green balls, and 8 yellow balls in a sock, what is the probability of randomly pulling out a ball that is not red?
15/18 = 5/6 =83.3%
Write as an algebraic expression: What is 21 less than double the square of a number?
\[2x^{2}-21\] Explanation: The square of a number is \(x^{2}\).Double that is \(2x^{2}\).21 less than that is \(2x^{2}-21\).
The three angles in a triangle have measures of \(x, 2x, \text{ and } 3x\). What is the measure of the smallest angle (in degrees)?
\[30^\circ\] \(x+2x+3x=180^\circ\)\(6x=180^\circ\)\(x=30^\circ\)
Which is larger, 3/8 or 0.36038 or 0.360?
\[\dfrac{3}{8}\] If you change \(\dfrac{3}{8}\) to a decimal by dividing \(3\) by \(8\), you will get \(0.375\), which is larger than \(0.360\).
The numbers 1 through 13 are written on slips of paper and placed in a bag. If one slip is picked at random, what is the probability that it will not be odd?
\[\dfrac{6}{13} \text { or } 46.2 \%\] The numbers that are not odd are 2, 4, 6, 8, 10, and 12. There are 6 of them, so the probability of picking one of them is \(\dfrac{6}{13}\).
A student, Jordan, scores 37 out of 55 on Test 1, 47 out of 72 on Test 2, and 41 out of 64 on Test 3. What is the highest score, in percent, Jordan earned? -66% -67% -65% -68%
67% Test 1: 37÷55≈67% Test 2: 47÷72≈65% Test 3: 41÷64≈64%
weight
Again, you have to be aware of both measurement systems. Weight in U.S. customary units is usually done in ounces (oz.), pounds (lb.), and tons (T.). Common weights in the metric system are grams (g) and kilograms (kg), though these units are more correctly known as units of mass.
Formulas for Circles
Area Use A=πr2A=πr2 to calculate the area of a circle. Know the formula and use π=3.14π=3.14. For example, what is the area of a circle with a diameter of 12 cm? A=3.14×(6 cm)2A=3.14×(6 cm)2A=3.14×36 cm2A=3.14×36 cm2A=113.04 cm2A=113.04 cm2 Circumference Use C=2πrC=2πr or C=πdC=πd to calculate the circumference of a circle. Again, know the formula and use π=3.14π=3.14. For example. What is the circumference of a bolt that is 0.50 inches in diameter? C=πdC=πdC=3.14×0.50 inchesC=3.14×0.50 inchesC=1.57 inches
Calculate Percent of Change—
The population of Smallville went from 7,200 down to 6,500. What was its percent of change? Subtract to find the difference and divide that by the original amount. 7200-6500=700 700/7200=0.0972 0.0972=9.72%
Other Polygons
The sum of angles in a pentagon is 540∘540∘.The sum of angles in a hexagon is 720∘720∘.The sum of angles in an octagon is 1080∘1080∘.
What is the sum of \(\dfrac{13}{20} \text{ and } \dfrac {7}{10}\)?
\[1\dfrac{7}{20}\] The lowest common denominator is \(20\). \[\dfrac{13}{20} + \dfrac{7}{10}\cdot \dfrac{2}{2}\] \[\dfrac{13}{20} + \dfrac{14}{20}\] \[\dfrac{27}{20}\] \[1\dfrac{7}{20}\]
Which of the following statements is/are true? a) If x causes y, a scatter plot of y vs. x would be very likely to show a correlation.b) If a scatter plot of y vs. x shows a correlation, it is very likely that x causes y.c) If a scatter plot of y vs. x shows no correlation, it is very likely that x causes y. -a,c -b -a -b,c
a Correlation does not imply causation, so rule out answer (b). Likewise, if there is no correlation, then you have no reason to think there is a cause and effect relationship. That rules out c.
Does this equation have one solution, no solution, or many solutions? \[2(3x-x)=4x+3\]
no solution If you try to solve the equation, here is what happens: \(2(3x-x)=4x+3\)\(2(2x)=4x+3\)\(4x=4x+3\)\(0=3\) Since no value of x can make \(0=3\), there is no solution.
True or false: A figure with 4 equal sides and 4 right angles is a type of rectangle.
true "A figure with 4 equal sides and 4 right angles" defines a square, but a square is a type of rectangle. Rectangles are figures with opposite sides equal and 4 right angles. The square also fits this definition. A square is a specialized rectangle.
Change a Percent To a Decimal
All that is needed is to move the decimal two places to the left. 12% = 0.12 4.5% = 0.045
Which of these shows a pair of equivalent expressions? -10y; 11y+9y/10 -10% of x ; x/0.10 -21n-7(n) ; 2n x 7 - 2/3t + 5t/9 ; 4t/9 + 5t/9
21n-7(n) ; 2n x 7
Which one of these numbers is not prime? {23,47,79,91}
91 7×13=91 7×13=91, so 91 91 is not prime.
You are given a number in decimal form and need to convert it to a percent. What is the correct way to do this?
Move the decimal 2 places to the right. A percent expresses a quantity based on a whole amount of 100; the word percent can be thought of as per one hundred. For example, 35% means 35 out of 100, or \(\frac{35}{100}\), which is 0.35. The decimal number is the percentage divided by 100. To divide by 100, move the decimal 2 places to the left. For example, to convert 62% to a decimal, drop the percent sign to give 62.0, then move the decimal place two places to the left to give 0.62.
Rates With Time
To get a unit rate, divide the given quantity by the given time. r=q/t This was shown in the previous section. Q can stand for any quantity, such as distance, dollars, liters, calories, or whatever is being measured. Depending on what you are looking for, you can easily rearrange the given formula two different ways: Q=rt or t=Q/r
For example, how much would 24 cubic feet of water weigh?
Q=ru Q= 62.4/1ft^3 x 24ft^3 = 14976.6 lbs
Volume of a Rectangular Prism
The volume of a rectangular prism (box) can be had using A=length×width×heightA=length×width×height, or A=lwhA=lwh for short. For example, what is the volume of a box that is 0.60 m by 0.80 m by 1.10 m? V=0.60 m×0.80 m×1.10 mV=0.60 m×0.80 m×1.10 m V=0.528 m3V=0.528 m3
What is the name for numbers divisible only by the number 1 and the number itself?
prime numbers
solve for x: 3x^2−11=21+x^2 - x=±√5 - x= ±4 - x=±6 - x=±√8or ±2√2
x= ±4 3x^2 - 11 = 21 + x^2 3x^2-x^2=21+11 2x^2=32 x^2=16 x= ±4
Half of a certain number equals 5 less than 3 times the number. What is the number? -x=2 -x=5/2 -x=-2 -x=10/7
x=2 start with 1/2x=3x-5 x/2=3x-5 (multiply both sides by 2) x=6x-10 Subtract 6x from both sides) -5x=-10 (divide both sides by -5) x=2
Subtracting Integers
Change the sign of the second integer and add the resulting integers, following the rules of addition. 20−(−6)=20+(6)=2620−(−6)=20+(6)=26 −19−(−7)=−19+(7)=−12−19−(−7)=−19+(7)=−12 −32−(4)=−32+(−4)=−36
Proportion problems usually involve three given quantities and one unknown quantity. For example, suppose it takes 1 cup of sugar to make 24 molasses cookies. How much sugar will it take to make 84 cookies? Set up a proportion using the three given quantities and a variable.
1/24 = x/84 1cup/24 cookies = x cups/84 cookies 84 x 1cup/24cookies = 84 x Xcups/ 84 cookies 84x 1/24=x x= 3.5 cups
A right triangle has sides of 5 cm, 12 cm, and 13 cm. What is the area of the triangle? - 30cm^2 - 44cm^2 - 36cm^2 - 28cm^2
30cm^2 The longest side, 13 cm, is the hypotenuse, so the 5 cm and 12 cm sides must be the legs. They are perpendicular, so one can be the base and the other can be the altitude. It doesn't matter which leg you use for the base. A= 1/2 bh A= 1/2 x 12 x 5
given △ABC ≅ △DEF DE=15 EF=12 DF=9 What is the perimeter of △ABC -12 -18 -24 -36
36
A carpet layer can install carpet at a rate of 12yd2 per hour. How long will it take him to do a 16.5 ft x 1818 ft room and a 9 ft x 27ft hall? -5.00 hours -4.00 hours -6.50 hours -1.67 hours
5.00 hours 5.5 yd x 6yd = 33 yd^2 3yd x 9yd = 27 yd^2 total area= 33yd^2 + 27yd^2 = 60yd^2 1hour/12yd^2 = x/60yd^2 x=5.00hours
James is planning a long-distance bicycle trip that will be a total of 840 miles. He determines he can travel an average of 160 miles each day. He will arrive on the ____ day of the trip. Which choice represents the correct number to fill in the blank? -6th -7th -5th -4th
6th To find the total number of days traveled given the total mileage and the miles per day traveled, divide the total mileage by the number of days.This can be seen by looking at the measurable units. miles/ miles/day =miles x days/mile = days The given unit of the answer is days, so that is what we require. This shows that division was the correct operation to do. The result is 5.25 days.He will arrive during the 6th day, because, after 5 days, he still has 0.25 (1/4) of a day of travel remaining. He will arrive after 1/4 of the 6th day has elapsed.
Solving Problems with Decimals
A common decimal problem is converting a decimal to a fraction. What you do is use place value as a denominator and the decimal digits (without the decimal point) as a numerator and reduce it to the lowest terms. An example is 0.20, which is 20 hundredths. 20/100=1/5 A mixed number example is 6.025, which is 6 and 25 thousandths. 6 25/1000= 6 1/40
Vertical Angles—
A pair of opposite congruent angles formed by intersecting lines
Explain the difference between similar and congruent figures.
Congruent figures are exactly the same size and shape. Similar figures are the same shape, but different sizes. In congruent figures, the lengths of the sides and the angle measurements are the same in both figures. In similar figures, the angles are the same in the figures, but the sides are different lengths (but in the same ratio). The shape is the same but the size is different.
Converting Units
Converting metric units is done by simply moving the decimal point. The trick is to know which way and how far. For example, to go from 5.3 kg to g units, remember that a kilogram is 1000 g. If you're converting to grams, you are going to a smaller unit, so there will be more of them. To get more of them, move the decimal point to the right, in this case, three places: 5.3 kg = 5.300 g.
Concepts Concerning Integers
Here's a reminder, though it's pretty elementary, that you should know how to add, subtract, multiply, and divide positive and negative integers, along with positive and negative decimals and percents. Also elementary is the concept of even and odd integers. A couple of additional concepts are listed below. Prime Numbers—If an integer has no factors other than itself and 1, it is a prime number. For example: 13 has no factors other than 1 and 13, so 13 is prime. Perfect Squares— It is worthwhile to know the squares of at least the integers 1 through 12, i.e. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144.
Multiplying And Dividing Integers—
If both integers have the same sign, multiply (or divide) and give the result a positive sign. (−25)⋅(−6)=150(−25)⋅(−6)=150 If the integers have different signs, multiply (or divide) and give the result a negative sign.(14)⋅(−3)=−42
Solving Quadratic Equations
If the highest power of x (or whatever variable) in an equation is 2, it is a quadratic equation. The simplest quadratics look like this: x2=64x2=64. To solve it, ask yourself what number, when squared, gives you 64? It's 8, of course, but don't forget it could also be -8. A more formal way to describe how to solve it is to say "take the square root of both sides of the equation." √x^2=√64 x=±8 2x^2-6(+6)=44(+6) 2x^2=50 2x^2(/2)=50(/2) x^2=25 √x^2=V25 x=±5
What does the standard deviation tell you about a data set or population?
It is a measure of the statistical variability of that data or population. The standard deviation measures how far on average the data points are from the mean. It is a way to characterize how spread out or tight the data set is.
When asked to add the polynomials \((x^{2} + 2)\) and \((2x - 7)\), James gave the answer as \((4x - 5)\). Why is this wrong?
James tried to combine the unlike terms \(x^{2}\) and \(2x\) by adding. Unlike terms cannot be added or subtracted. The terms are unlike because the variable is raised to a different power in each. The correct answer is \(x^{2} + 2x -5\).
Properties of Plane Shapes
Plane shapes are those that totally lie in one flat surface. Many of them have 3 or more line segments for sides, and they are called polygons. Polygons that have all sides and all angles congruent are called regular polygons. Circles and ellipses are examples of plane shapes that aren't polygons.
As an example,suppose a painter can paint a wall at a rate of 5 square ft per minute. How long will it take him to paint a wall that is 9 ft by 15 ft?
The 9 ft by 15 ft wall is 135 ft^{2}135 ft^{2}. That will be our quantity, Q. t= Q/r t= 135/5 t=27minutes
Order of Operations
Remember to follow the order of operations (PEMDAS): Do everything in parentheses (P), left to right.Evaluate any exponents (E), left to right.Do all multiplication and division (MD), in order, left to right.Then do all addition and subtraction (AS), in order, from left to right. A good way to remember: "Please Excuse My Dear Aunt Sally"
Properties of Whole Numbers
Solving even the simplest problems calls for knowing properties of whole numbers such as the ones listed below. You should be very familiar with them. Some of them can also help you evaluate whether an answer you get makes sense and is actually correct.
Other Rates—
Some ratios that can be treated as rates don't involve time. Density is one. The weight of 1 cubic foot of water is 62.4 lb. A slightly different way to say this is to say that water weighs 62.4 lb for every 1 cubic foot, or that its density is 62.4 lb/ 1ft^3 or 62.4 lb/ft^3
Change a Ratio To a Percent—
Suppose a student gets 37 out of 44 possible points on a math test. What is her percentage correct? Divide 37 by 44 and change to a percent. 37/44=0.841-8.41%
Integers
Take the set of whole numbers and combine them with the set of their negatives, (except there is no negative zero) and get the set of integers extending infinitely in both positive and negative directions.
temperature
The Fahrenheit temperature is well known to those of us in the U.S., in which 32∘F32∘F is the freezing point of water and 212∘F212∘F is boiling. The Celsius scale has 0∘C0∘C as freezing and 100∘C100∘C is boiling. There are formulas for converting from one system to another that you don't have to know, but should know how to use them, given that they are pretty basic algebraic equations.
Time
The metric (SI) base unit of time is the second (s), and prefixes define other time units, such as millisecond (ms), kilosecond (ks), and so forth. Minutes, hours and days are used in both systems.
Rectangle
The perimeter of a rectangle is the sum of the four sides: P=2l+2wP=2l+2w.The area of a rectangle is the product of length and width. A=lwA=lw. p=2l+2w A=lw
Selecting Operations
There are common clue words or phrases that show up in math problems that can help you decide what operation(s) need(s) to be done. Total, sum, perimeter, in all can indicate addition.Difference, exceeds, how much more, how much less, can indicate subtraction.Product, times, area, total, indicate multiplication, though total can mean addition too.Quotient, average, ratio, can indicate division. Many problems need more than one step to solve. If you can't figure out the final answer, figure out anything you can as a first step and you may be able to use that quantity to figure out the final answer.
Ordering Fractions
There will be an on-screen calculator on this test, so the easiest way to order fractions is to change each fraction to a decimal, and order them as above. The only difference is that you will be in charge of rounding each decimal rather than having them given. The method above will still work, as long as you don't do something silly like round everything to a single digit. When in doubt, keep a few extra decimal places when you round.
Linear Relationships
Word problems can be based on a linear relationship. You can tell that two quantities have a linear relationship if you see that when one of them keeps changing by a constant amount, the other one also changes by a constant, though different, amount. Here's an example: Suppose a corn plant is 12 cm high and grows 2.5 cm each day. (Each time the number of days increases by 1, the height increases by 2.5 cm.) A graph of this over a period of days would be a straight line. An expression for the growth (not the height, just the growth) of the plant, in terms of d, the number of days would be: 2.5 d2.5 d A word equation for the height would be: height = beginning height + growthh=12+2.5 dh=12+2.5 d How tall would the plant be after 30 days?h=12+2.5⋅30=87 cm
length
You will find that these U.S. customary units of length are found in many math problems: inches, feet, yards, and miles. Know how to convert between these units. In the metric system, all lengths are based on the meter (m), and the following units are often seen: millimeters (mm), centimeters (cm), maybe decimeters (dm), and kilometers (km). Know what the prefixes mean. There are lots more, but you're not as likely to run into them.
What is the range of this data set? \(\{51, 59, 59, 78, 88, 95\}\)
\[44\] The range is the difference between the largest and the smallest data value. \(95-51=44\)
Triangles
equilateral-60,60,60 right-90 isosceles-two equal sides obtuse-less than 90 scalene-3 differed sides acute tirangle-greater than 90 Triangles have 3 sides. The sum of their angles is 180∘180∘. A scalene triangle has no congruent sides or angles. An acute triangle has all angles less than 90∘90∘. An isosceles triangle has 2 congruent sides and 2 congruent angles. An equilateral triangle has 3 congruent sides and 3 congruent angles. A right triangle has one right angle. An obtuse triangle has one angle larger than 90∘90∘.
"When mitten sales increase, so do cases of frostbite." True or false? Because of this correlation, it is logical to conclude that mittens somehow cause frostbite.
false The opening statement indicates that there is a correlation, but a correlation is not enough evidence to conclude that there is a cause and effect between mittens and frostbite. (Correlation does not imply causation.)
On a test, students were asked to supply an equivalent way to express the ratio 8:6. Four of the responses are listed in the attached document. Which of them are correct? student A- 4/3 student B- 6:8 student C- 3/2 Student D- 8/6
students A and D Ratios can also be written as fractions with the first item in the ratio as the numerator and the second item as the denominator, as student D has done.Student A has given a fraction equivalent to the original ratio and is also correct.The order of the ratio must be preserved, so student B is incorrect.The answer given by student C is not an equivalent fraction to the original ratio.
A linear equation has the form y=mx+b. If you replace y, m, and b with numbers, you have an equation in x that can be solved. Here is a simple example:
31=2x+5 (-5) 31 = 2x +5 (-5) (/2) 26=2x(/2) 13=x
Another method is to use a conversion factor. Use 1000g/1kg to multiply times the given number:
5.3 KG x 1000g/ 1 KG= 5.300g Notice how the kg units cancel out. That's how you know you have the conversion factor right side up. If you were going from g to kg, you would use the factor 1kg/1000g
Adjacent Angles
Angles that have a common side and a common vertex (corner point).
Sarah is running a yard sale. In the first four hours, she made sales of $15, $21, $12, and $9. She was hoping for an average of $15 an hour but is a little short of that goal at $14.25. How much in sales must she have in the fifth hour to bring the average up to $15 per hour? -$18 -$15.75 -$20.25 -$75
$18 An average is calculated by dividing a total by the number of items.In this case, the total sales divided by the number of hours, to give an average of sales per hour. In this case, we know the number of hours will be 5. She wants an average of $15 per hour, so that means that total sales/ 5=15 or total sales = 5x15 or $175 She has $57 in sales after 4 hours, so she must get $18 more in sales to reach $75 by the end of 5 hours.
solve for x: 2(x-8) +4 =3x-3 -x=5 -x=-2 x=-9 x=3
-9
A clothing store is having a sale with a 10 percent discount on boots and a 15 percent discount on gloves. The usual cost of a pair of boots is b and the usual cost of a pair of gloves is g. Which of these is an expression for the total cost of 1 pair of boots and 2 pairs of gloves during this sale? -0.9+1.7g -0.1b+0.3g -2.6bg -0.75(b+g)
0.9+1.7g The discounted price of the boots is b-0.1b=0.9bb-0.1b=0.9b.Similarly, the discounted price of a pair of gloves is g-0.15g=0.85gg-0.15g=0.85g.The total cost of 1 pair of boots and 2 pairs of gloves during the sale is 0.9b+2(0.85g)=0.9b+1.7g0.9b+2(0.85g)=0.9b+1.7g.
If these numbers are arranged from lowest to highest, which number will be in the third position? {1/14, 0.032, 0.1007, 1/10}
1/10 Arrange the decimals 0.032, 0.071, 0.10, 0.10070.032, 0.071, 0.10, 0.1007You'll see that 0.10 is in the third place and since it equals 1/10, that is the answer. If you have trouble ordering decimals, move each decimal point four places to the right.320, 710, 1000, 1007
Jack's socks are loose in a drawer. There are 4 blue socks, 8 black socks, and 4 brown socks. He randomly grabs a blue sock from the drawer. If he randomly grabs a second sock, what is the probability that it will also be blue? -1/4 -1/5 -1/16 -3/16
1/5 The probability of a random event is the number of successful possibilities divided by the total number of possibilities.After Jack has taken the first sock, there are 15 remaining as the total number of possible socks.Of these, there are 3 remaining blue socks, which represent successful events.The probability is then: 3/15=1/5
A sequence of numbers is represented by a1,a2,a3,a4,a5...... in the first term, a1=3 and an+1= 2an-2 what is a4 an+1 stands for the following term -10 -15 -28 -11
10 if an+1=2an-2 a2=2a1-2 a2=2(3)-2 a2=4 a3=2a2-2 a3=2(4)-2 23=6 a4=2a3-2 a4=2(6)-2 a4=10
Choose the answer that would be a counterexample to the statement: Odd three digit numbers are always prime. -113 -129 -137 -101
129 129=3×43 129=3×43, so it is not prime.
Converting U.S. units to other U.S. units, or from one system to another, can also be done with a conversion factor. For example to go from 1320 ft to miles, remember that 1 mile is 5280 ft and do this:
1320FT x 1 mile/ 5280FT =0.25 mi
An aquarium measures 75 cm long by 40 cm wide by 50 cm tall. If it is filled to a depth of 45 cm, how many liters of water are in it? (1 liter=1000 cm^3) -135L -140L -90L -169L
135L V= 75cm x 40cm x 45cm (Notice that we don't use the figure 50 in the calculations. That is how tall the aquarium is, but it is only filled to 45 cm. The figure 50 is a piece of unneeded information. You may see this provided in other problems.) v= 135,000cm^3 v=135000 cm^3 x 1L/1000cm^3 v=135L
When adding fractions with unlike denominators, it is best to use the least common multiple of the denominators as the common denominator. What common denominator is best to use for adding the fractions 5/6 and 7/9 -36 -15 -54 -18
18 54 and 36 are common multiples of 6 and 9, but they are not the least common multiple.To find the least common multiple, examine the prime factors of 6 and 9:6 = 2 x 3 and 9 = 3 x 36 and 9 share one instance of the prime factor 3, so the least common multiple will contain one 3 coming from the overlap and the other 3 coming from the prime factorization of 9.Multiply the prime factors together to compute the LCM:2 x 3 x 3 = 18
A farmer wants to get an estimate of insect damage in his corn crop, so he randomly selects 75 plants from different areas in the cornfield and inspects each plant for damage. Of those 75 plants, 9 of them show an infestation of corn borers. He knows that there are about 31,000 plants per acre and he has a 50 acre field. How many infested plants would you expect to exist in the 50 acre field? - 97,000 plants - 186,000 plants -117,000 plants - 202,000 plants
186,000 plants Start by calculating the total number of plants. 31,000×50=1,550,000 Set up a proportion to calculate the number of infested plants 9/75 = x/1,550,000 x=186000
Mary is mixing a custom paint color that requires 1 part of coloring for every 4 parts of white paint. How much coloring will Mary need for a total of 12 gallons of colored paint? -3 gallons -2 2/5 gallons -9 3/5 gallons -1 gallon
2 2/5 gallons The ratio of coloring to white is 1:4.A total of 12 gallons of colored paint are needed.Let c represent the amount of coloring in the 12 gallons.The amount of white is then 12 - c. c/12-c = 1/4 12-c=4c 12=5c 5c=12 c=12/5 = 2 2/5 Because of the 1:4 ratio, 3 gallons may seem like a correct answer (divide 12 by 4). It is not correct because the total amount of paint is 12 gallons. If the problem stated that 12 gallons of white paint were used, then 3 gallons would have been correct. But, if you use 3 gallons of colored paint plus the 12 gallons of white paint, you would use 15 gallons of paint and that is not what the problem specifies. These problems can be tricky. Be sure to read them carefully.
Jim buys 24 feet of fencing to protect his garden. He will be making a rectangular garden. What is the area of the largest rectangular garden that Jim can make with this amount of fencing? -he cannot form a rectangle using that length -35 square feet -any rectangle he makes from a given length of fencing will have the same area -36 square feet
36 square feet The 24 feet of fencing is the perimeter of his garden. For a rectangle with width w and length l, the perimeter P is, p=2w+2l In this case 2w+2l=24, or w+l=12. He must use a length and width of the rectangle that adds up to 12 feet. The possible combinations are 1 x 11, 2 x 10, 3 x 9, 4 x 8, 5 x 7, and 6 x 6.The combination with the highest product, which is the area, is 6 x 6.A square is a special form of rectangle. The area is 6 x 6 = 36 square feet.
An end of the year exam consisting of math, science, and language arts questions has the following breakdown: the math and science sections each account for 35% of the overall score, and the language arts section accounts for 30% of the overall score. If Nicholas scored an 89 on the math section, an 85 on the science section, and a 76 on the language arts section, what is his overall score? Round your answer to a whole number. -81 -83 -85 -84
84 This is a weighted average question. Multiply each score by the percentage of the total that it accounts for: weighted math score: 89⋅0.35=31.1589⋅0.35=31.15weighted science score: 85⋅0.35=29.7585⋅0.35=29.75weighted language arts score: 76⋅0.3=22.8 The overall score is the sum of the weighted scores: 31.15+29.75+22.8=83.731.15+29.75+22.8=83.7, rounded to the nearest whole number, is 84.
Rates are a special use of ratios. A few examples of rates are speed (miles per hour), salary (dollars per week), flow rate (liters per minute), and nutrition (calories per gram). Most often, rates are written as unit rates, meaning the number that comes after the "per" is 1. For example, if you drive 90 miles in 2 hours, your unit speed rate is
90miles/2hours 45miles/1hour
What is the difference between a ratio and a proportion?
A ratio compares 2 quantities in a given order. A proportion is a statement that the ratios are the same.
Triangle
A=1/2bh
Drawing Inferences
An inference is a statement made about something that can't be observed directly. It's a conclusion based on looking at a portion (sample) of a population and assuming that what you see in that sample applies to the population as a whole. For example, suppose that a wildlife biologist wants to determine how many diseased deer are in the state of Michigan. They randomly trap 400 deer (the sample) around the state and find that 3 of them are diseased. They infer that the ratio 3:400 will hold true for the whole population of 1,000,000 deer.Using a proportion, they calculate the likely number of diseased deer. 3/400=x/1000000 x=7500
All About Decimals
Decimals and fractions also exist on the number line that was shown under integers. They actually fill the space between the integers (along with the irrational numbers, which we won't be discussing.) Understanding decimals is based on understanding place value. Starting at the decimal point, each digit to the right represents the next lower fractional division of ten, i.e. 1/10 ( 0.1), 1/100 (0.01), 1/1000 (0.001) 1/10000 (0.0001)
Finding Equivalent Expressions
Equivalent expressions are expressions that express the same quantity, but are not in the same form (they don't look the same). You run into these all the time when you are simplifying algebraic expressions or using the commutative, associative, or distributive properties. For example: 3×x×5, 3⋅5⋅x, and 15x3×x×5, 3⋅5⋅x, and 15x are all equivalent expressions. Other examples are: 5×85×8 is equivalent to 8+8+8+8+88+8+8+8+8 (x−2)(x+2)(x−2)(x+2) is equivalent to x2−4
Congruency
For line segments and angles, congruent means having the same measures.For polygons, congruent means all corresponding sides have the same measure and all corresponding angles have the same measure. The corresponding sides and angles are the ones that are in matching positions in each triangle. In the given figure, corresponding angle pairs are:∠A and ∠D∠A and ∠D, ∠B and ∠B∠B and ∠B, ∠C and ∠F∠C and ∠F. Corresponding side pairs are: AB¯¯¯¯¯¯¯¯ and DE¯¯¯¯¯¯¯¯AB¯ and DE¯, BC¯¯¯¯¯¯¯¯ and EF¯¯¯¯¯¯¯¯BC¯ and EF¯, AC¯¯¯¯¯¯¯¯ and DF¯¯¯¯¯¯¯¯AC¯ and DF¯ The symbol for "is congruent to" is ≅≅. In the figure above, △ABC≅△DEF
Ordering Decimals
Here's a quick trick to put a group of decimals in order. Find the number that has the most decimal places. Let's say it has four decimal places. Move the decimal point on all the numbers four places to the right, adding zeros if you need to. Presto, no more decimal places, only whole numbers. Now it's easy. For example: Order these numbers from highest to lowest: 0.107, 1.29, 0.0566, 0.020After moving the decimal point four places: 1070, 12,900, 566, 200Final order: 12,900, 1070, 566, 200 →→ 1.29, 0.107, 0.0566, 0.020
Adding Integers
If both integers have the same sign, add them and give the result the sign that was common to both. −11+(−7)=−18−11+(−7)=−18 27+4=31 If the integers have different signs, take the absolute value of both and subtract the smaller integer from the larger one. Give the result the original sign of the larger one. 13+(−4)=?13+(−4)=? 13−(4)=913−(4)=9 −15+8=?−15+8=? Think (15)−(8)=7(15)−(8)=7 and make the 77 negative. Result: −7
Similarity
In casual use, the word similar means that two things are a lot alike. In geometry, it means they are the same exact shape, but not the same size. More exactly, two similar figures have all their corresponding angles congruent. The two pentagons above are similar, but not congruent.
Angles Formed By Parallel Lines
In the diagram we have parallel lines m and n that are intersected by line k, called a transversal. Notice the pattern of the angle measures. The angles that are adjacent to each other will always add up to 180∘180∘.The angles that are opposite each other (vertical angles) are always congruent.The angle pattern around line m exactly matches the pattern around line n.
Identifying Relevant Information
Know what you have.— Read the problem. Then read it again. Write down all given numbers with their units. Identify each one with a variable: t for time, c for cost, l for length, etc. Know what you want.— Write down what the problem is asking for. Label it with an appropriate variable. Try to figure out the units the answer should have. Example: P = ____ meters. Look at known formulas or theorems.— Many problems have standard formulas that could apply: d=rt, A=lw, C=2πr, y=mx+bd=rt, A=lw, C=2πr, y=mx+b. Look at the variables that are given and wanted and try to match them to a formula. Make it easier.— Often the numbers in a problem are unfriendly. Try putting simpler numbers in and solving the problem with those. Your brain will work better with those and may see a way to do the problem with the original numbers. Make a drawing.— If the problem is geometric, a labeled drawing is almost a necessity. Be aware that the problem may have extra information that isn't actually needed, though most test questions seem to give only what is needed.
Angles in Parallelograms
Opposite angles of a parallelogram are congruent —The diagram shows an example of a parallelogram, where, like all parallelograms, opposite angles are congruent, and consecutive angles add up to 180∘180∘. (They are supplementary.)
Ordering Percents
Percents can easily be changed to decimals by moving the decimal point two places to the left. When they are decimals, put them in order as shown above.
Ordering Numbers
Putting different types of numbers in order is discussed in the next three sections, but what if you are given a mixture of all three types to order? For example: 2/5, 0.331, 3/4, 2.9% Handle this group by changing each one to a decimal (you will have a calculator) and following the method below.
Quadrilaterals
Quadrilaterals have 4 sides. The sum of their angles is 360∘360∘. Parallelograms have both pairs of opposite sides parallel. Rectangles are parallelograms that have 4 right angles. Squares are rectangles that have 4 congruent sides. Rhombuses are parallelograms with 4 congruent sides. (So a square is also a rhombus.) Trapezoids have only 1 pair of opposite sides parallel.
Calculate a Percent Of a Quantity
Remember that, in this context, "of" means multiply. What is 3.8 % of 15,200? Change the percent to a decimal and multiply. 0.038 x 15200 = 577.6
volume
Take each length unit above, cube it, and you have volume units. This is done in both U.S. customary and metric systems. The units may be abbreviated with a cu. in front of the unit (cu. ft. for cubic feet) or an exponent of 3 for the length unit (ft3 or m3ft3 or m3). In addition to cubic units, the metric system has liters (L) and milliliters (mL), and the U.S. system has pints, quarts, and gallons.
Associative Property
The associative property has to do with grouping, meaning using parentheses. Does it matter where you put the parentheses? It can. In Addition— (6+2)+118+11 19 6+(2+11)6+13 19
Commutative Property
The commutative property has to do with order. Order matters in some operations, but not in others. If order doesn't matter, we say that that operation has a commutative property. In Addition—Does 8+11=11+88+11=11+8? Yes.This is the commutative property of addition: a+b=b+aa+b=b+a (same results either way) In Subtraction—Does 25−13=13−2525−13=13−25? No.The commutative property doesn't exist for subtraction. (different results) In Multiplication—Does 7×9=9×77×9=9×7? Yes.This is the commutative property for multiplication: a⋅b=b⋅aa⋅b=b⋅a (same results) In Division—Does 8÷2=2÷88÷2=2÷8 ? No.The commutative property doesn't exist for division. (different results) Summary: Addition and multiplication are commutative because you get the same results with either order.
How many correct solutions exist for the equation \(x^{2} = 36\)?
There are two solutions: 6 and -6. The square root of a number is a number that, when multiplied by itself, gives the original number as the answer. So, 6 x 6 and (-6) x (-6) both give the answer 36. In the context of a word problem, the negative answer may not make sense and can be omitted.
Probability
To calculate probability, you need only two things. First, you need to know how many possibilities there are for some event. For example, how many different possibilities are there when you roll a normal die?The answer is six because the die has exactly six different dot patterns. Second, you need to know how many possibilities there are to satisfy some requirement. For example, when you roll a die, how many different even numbers can show up? The answer is three: 2, 4, and 6. Let's call those winners. To get the probability of rolling an even number, divide the number of winners by the total number of possibilities: 3/6=1/2=50%
Which of these show(s) a correct statement? a) xy(5t X 2) -5xyt X 2xy b) ab(2x - 3)= 2abx - 3ab c) 6rt(m + 2n) =12mnrt d) 4d(9c/8c) = 36cd/ 32dp
b The distributive property does not work over multiplication or division, so that rules out choices a and d. The distributive property does work over addition, but choice c should be 6mrt(m+2n)=6mrt+12nrt, not 12mnrt
Someone says that all horses eat oats. You point out that wild horses don't eat oats. You have refuted their claim by providing a ____.
counterexample A counterexample is a statement that is the opposite of what is being claimed. If the counterexample is true, then the claim must be false.
Angles
less than 90- actue equal to 90- right greater than 90 but not 180- obtuse equal to 180-straight
Which measure of center gives a value where 50% of the data points are greater than the average and 50% of the data points are below the average?
the median The mean is the arithmetic average of the data. If the data is skewed, with many low values or many high values, the mean will not be in the middle of the data and the median will be a more accurate representation of data's center.
True or false? The angles in a small regular pentagon are congruent to the angles of a large regular pentagon.
true A regular pentagon has five congruent angles that are always the same size (\(108^{\circ}\), if you'd like to know) no matter the size of the sides.
