Praxis: Math (Numbers and Operations)
An electronics store is offering a 20% discount on a pair of noise-cancelling headphones that normally cost $169.79. What is the price of the headphones, before tax, after the discount is applied?
$135.83 Convert the percentage to a decimal and multiply it by the original cost of the headphones to find the amount of the discount: 20% = 0.2 and (0.2)(169.79) = 33.96. Subtract the discount from the original price to find the new price: 169.79 - 33.96 = 135.83.
Sabrina finds a coat on sale for 18% off the original price of $85. She computes her potential savings in the following way: $85 × 0.18 Which of the following methods could Sabrina also use to correctly determine the potential savings?
$85× 18/100
Which of the following points on a number line is the greatest distance from .5?
-1.5 -1.5 is 2 units away from .5 on the number line.
Which number is not natural?
0 Natural numbers are whole counting numbers that begin with 1. Zero is not a natural number.
Simplify the expression: 30 - 2 × 50 + 70
0 To simplify the equation, follow the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction (PEMDAS) and work left to right. The steps to simplify the expression would be: 30 - 2 × 50 + 70 = 30 -100 + 70 = -70 + 70 = 0.
Any number raised to the zero power always equals
1
Unit Fraction
1 over any rational number. The inverse of a whole number. 1/2, 1/3, 1/19
There is a 15% increase in tuition at the local university for next fall. If the current tuition is $3,500 per semester, which equation could be used to find x, the new tuition for the fall?
1.15 • 3500 = x
If the number 180 is written as the product of its prime factors in the form a²b²c, what is the numerical value of a + b + c, where c = 5 and a and b do not equal 1?
10 The prime factors are the numbers that, when multiplied together, equal a number. In this problem there are three prime factors, where two are squared. We know that c = 5 so the equation (a²)(b²)(c) can be written (a²)(b²)(5) = 180. We can then divide by 5 and simplify this to (a²)(b²) = 36. Take the square root from each side simplifies the equation further to (a)(b) = 6. We know that a and b do not equal 1, so they must equal 2 and 3. So (2²)(3²)(5) = 180 and a + b + c is 2 + 3 +5 = 10.
Last week, a round-trip plane ticket from New York City to Chicago cost $118. Today, the same ticket costs $130. What is the percent change of the ticket price from last week to today?
10% increase (look at notes)
Is 101 prime or composite?
101 is also close to 100, so its square root should also be close to 10. To see if 101 is prime or composite, division should be used to test each prime number less than 10 (2, 3, 5, and 7). 101÷2 yields 50.5, a decimal; 101÷3 yields 33.666..., a decimal; 101÷5 yields 20.2, a decimal; 101÷7 yields ~ 14.428..., a decimal. Therefore, 101 is a prime number, with no natural number factors besides 1 and itself.
In Anytown ISD, 13 out of every 20 students ride the bus. Which ratio compares the number or students who ride the bus to those who do not?
13:7 This is the ratio that compares riders to non-riders. 13 out of 20 ride the bus. This means that the complement of this relationship, those who do not ride the bus, is 20 - 13 = 7. So, the ratio of those who ride the bus to those who do not ride the bus is 13:7. When writing ratios, remember that order is important and matters. 13:7 is not the same as 7:13.
Put these fractions, decimals and percentages in order from greatest to least: 15%, 0.34, 245%, 2 ¾, 1.5, 1/15
2 ¾, 245%, 1.5, 0.34, 15%, 1/15 It is easiest to convert these numbers to decimals and then order them. 15% = 0.15, 0.34 = 0.34, 245% = 2.45, 2 ¾ = 2.75, 1.5 = 1.5, 1/15 = 0.067
What is 5/18 as a percent?
27.78% To convert a fraction to a percent, divide the numerator by the denominator and multiply by 100. 5 ÷ 18 = 0.2778 × 100 = 27.78%
Which two numbers have a product of 56 and a quotient of 14?
28 and 2 28 and 2 multiply to 56, and 2 divides into 28 with a quotient of 14.
A pancake recipe requires 1 tablespoon of baking powder per 2 cups of flour. If 2 cups of flour make 4 pancakes, how many tablespoons of baking powder are needed to make 12 pancakes?
3 1 tablespoon of baking powder per 2 cups of flour makes 4 pancakes -> 1 tablespoon of baking powder is used to make 4 pancakes. To make 12 pancakes, 3 tablespoons of baking powder are needed. To find this, divide 12 by 4 to the proportional increase in pancakes. 12 / 4 = 3 -> because the number of pancakes is 3 times as many as the recipe size of 4; multiply 1 tablespoon by 3 to get 3 tablespoons of baking powder.
Which of the following is the product of 2 odd numbers and 1 even number, each of which is greater than 1?
30 The prime factorization of 30 is 2 × 3 × 5 which satisfies the need for 2 odd numbers and an even number.
A field trip to the planetarium is being planned. Mrs. Weir is the principal and needs to know how many buses to take. If each bus will hold 16 people, which equation should she use to reserve the correct number?
350/16 ≈ 22 This is an example of a situation in which an overestimation is required. A bus can carry only 16 people. So, 21 buses will not allow for everyone to have a ride and it does not make sense to take a fraction of a bus. Therefore, 22 buses must be taken.
Jenny is baking cookies. Each dozen cookies requires 2 cups of flour and 0.75 lbs of butter. How many cookies can she make with 6 cups of flour and 3 lbs of butter?
36 She can make 36 cookies. 6 cups of flour allows for 6/2 = 3 dozen cookies to be made.
Order these least to greatest: (¼)3 3^-4 4^3
3^-4 (¼)3 4^3 correct 3^-4 = ⅓ × ⅓ × ⅓ × ⅓ = 1/81 (¼)3 = ¼ × ¼ × ¼ = 1/64 4^3 = 4 × 4 × 4 = 64
It took Julie ¾ of an hour to run 3½ miles. What is her average speed in miles per hour?
4 ⅔ miles per hour 3 ½ needs to be converted to an improper fraction: 7/2 then divide 7/2 by 3/4 * to divide a fraction multiply by the reciprocal 7/2 x 4/3
Which of these is an example of the multiplicative inverse property?
4(¼) = 1 This is an example of the multiplicative inverse property. Any number multiplied by the inverse of itself equals 1.
Tom wants to mentally calculate a 20% tip on his bill of $40. Which of the following is best for Tom to use in the mental calculation of the tip?
40 × .1 × 2 Tom can quickly find 10% of 40 and then double it. In this case the answer is $8 because 10% of 40 is 4 and 4 × 2 is 8.
If the number 888 is written as a product of its prime factors in the form a3bc, what is the numerical value of a + b + c?
42 The prime factorization of 888 is 23 \times× 3 \times× 37. 42 is the correct answer because 2 + 3 + 37 is 42.
A gas pump can pump a quarter gallon of gas every five seconds. If a person is filling up an empty gas tank that can hold 18 gallons of gas, how long will it take the gas pump to fill the empty gas tank?
6 minutes If a gas pump can pump a quarter of a gallon every five seconds, then the pump can deliver a gallon of gas every 20 seconds, and 3 gallons of gas every minute. If the tank is 18 gallons, then it will take 6 minutes (3 gallons per minute × 6 minutes = 18 gallons).
What is the scientific notation of 674.9723?
6.749723 × 102 6.749723 × 102 is correct because the 2 notes that the decimal is moved 2 places to the right which is the original number.
A runner is running a 10k race. The runner completes 30% of the race in 20 minutes. If the runner continues at the same pace, what will her final time be?
67 minutes To find the answer to this question, set up a ratio; remember that 30% = .3 and 100% = 1. Therefore, (20 / .3) = (x / 1) When you cross multiply to solve for x, you get the equation .3x = 20. Divide each side by .3 to isolate x and the answer is 66.66666. The best answer choice is 67 minutes.
What value does 8 represent in the number 2.86 x 104?
8000 2.86 x 104 equals 28600 when expanded correctly.
What is the standard form of: 80,000 + 4,000 + 500 + 90 + 2?
84,592
Is 91 prime or composite?
91 is close to 100, so its square root will be close to 10 To see if 91 is prime or composite, division should be used to test each prime number less than 10 (2, 3, 5, and 7). 91÷2 yields 45.5, a decimal; 91÷3 yields 30.333..., a decimal; 91÷5 yields 18.2, a decimal; 91÷7 yields exactly 13. Therefore, 91 is a composite number, composed of the two factors 7 and 13.
The bald eagle population in New Jersey stood at 82 mating pairs in 2010 and, thanks to conservation efforts, had increased to 161 pairs in 2015. What is the percent increase in the number of mating pairs?
96% To find the percent increase, the difference between the numbers is divided by the original number and multiplied by 100. 161-82/82 x 100%
Which symbol most accurately reflects the relationship between the two numbers below? 0.7 ☐ 4/5
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Ratio
A comparison that shows the relative size of two or more values. The ratio of boys to girls is: 4 to 5; 4:5; 4/5; 0.8.
Improper Fraction
A fraction where the numerator is larger than the denominator 3/2 7/4
Order of Operations (PEMDAS)
A set order in which multi-step equations must be solved Parenthesis, Exponents, Multiplication and Division (L to R), Addition and Subtraction (L to R)
Proportion
A statement that two ratios are equivalent. 2/3 = 4/6
greater than symbol
A symbol that points away from a greater number or expression. "eats big one"
Mixed Number
A whole number with a fraction 3 1/2 4 2/5
Fraction Composition
Adding fractions to come up with a larger one ¼ + ¼ + ¼ = ¾
Algorithm: Opposite Change Rule
Algorithm for solving addition. Add a number to one adden to make a number ending in zero and subtract the same amount from the other adden.
Algorithm: Partial Sums Addition
Algorithm for solving addition. Add each place value separately and then find the total
Algorithm: Column Addition
Algorithm for solving addition. Align numbers based on place value and then add each column
Algorithm: Partial Quotients
Algorithm for solving division. First estimate the answer using base-10 numbers and then keep estimating until the remainder is less than the dividend
Algorithm: Column Division
Algorithm for solving division. Separate the problem into columns and work with each place value separately
Algorithm: Lattice Multiplication
Algorithm for solving multiplication. Draw a lattice with diagonals in each box Write one factor across the top and one along the right side (58 x 213) Multiply each number and write the answer in the box with the tens digit on top (Top left box is 5 x 2 = 10) Add the diagonals beginning in the bottom right corner (4) If the sum is greater than 10, move the 1 to the next diagonal The answer will be along the left side and bottom (red arrow)
Algorithm: Partial Products
Algorithm for solving multiplication. Expand each factor into hundreds, tens, ones, etc. and then multiply and add
Algorithm: Modified Repeated Addition
Algorithm for solving multiplication. Using the concept that multiplication is repeated addition, students use base-10 numbers to multiply and add
Algorithm: Same Change Rule
Algorithm for solving subtraction. Add or subtract the same amount from both numerals and then subtract
Algorithm: Trade First
Algorithm for solving subtraction. Align numbers vertically based on place value and then ensure all top numbers are equal or larger than the bottom numbers
Algorithm: Counting Up
Algorithm for solving subtraction. Start with the number that is being subtracted and count up to the original amount with each step looking for the next base-10 number
Algorithm: Left-to-Right Subtraction
Algorithm for solving subtraction. Subtract each place value separately beginning with the largest and working to the smallest
Distributive Property
An number in front of a group of terms will multiply all terms in the grouping individually Multiplying a value by a quantity involving addition. a(b+c) = ab + ac
Associative Property
An operation is associative if regrouping the terms does not change the outcome (a+b) + c = a + (b+c)
Fraction Decomposition
Breaking down a fraction into smaller fractions that total to the original 2/3 = 1/3 + 1/3
Example: 4 ⅖ - 2 ¾ = ?
First change the numbers to improper fractions: 4 ⅖ = 22/5 because 4×5+2 = 22 2 ¾ = 11/4 because 2×4+3 = 11 Then use the the least common multiple (4 and 5 is 20) to get common denominators: 22/5 × 4/4 = 88/20 11/4 × 5/5 = 55/20 Then subtract: 88/20 - 55/20 = 33/20 Finally reduce back to a mixed number: 33/20 = 1 13/20
Diving Fractions
First flip the second and change the sign to multiplication
Associative Properties
Grouping symbols don't matter.
What is the least common multiple of 6 and 8?
Method 1: Multiples of 6: 6, 12, 18, 24, 30, 36, 42 Multiples of 8: 8, 16, 24, 32, 40, 48 The first multiple the two numbers have in common is 24. - Method 2: 6 = 2×3 8 = 2×2×4 Ignore the one common factor of 2. Multiply the remaining factors. 2×3×4=24 In this example, listing multiples is easy because the LCM is an early multiple. In larger numbers, using the prime factorization method is faster.
What is the difference between -52 and (-5)2?
One answer is positive and one answer is negative. -52 = -(5 × 5) = -25 because the negative sign is not inside the parentheses. (-5)2 = (-5 × -5) = 25 because the negative sign is inside the parentheses.
Commutative Properties
Order doesn't matter.
Using Proportions to Solve Problems:
Set up the proportion (interpret word problem if needed) Find the cross-products Simplify the cross-products Isolate the variable Find the answer
In the number 22,672,131, what is the value of the digit 7?
Seventy thousand The value of the digit is the number and the place value it holds. Since the digit is in the ten thousands place, it has a place value of 10,000. Since it is a 7, it's value is 70,000, seventy thousand.
What is the prime factorization of 224?
Since 224 is divisible by 2, start first by dividing by 2. 2-112 2-56 2-28 2-14 2-7 In this example since there are multiple factors that are 2, we must express the factorization using an exponent. 2×2×2×2×2 = 25 . The prime factorization of 224 is 25 × 7
When considering the addition problem 1/3 + 3/8, which of the following statements is true?
The Least Common Denominator = 24 Because 3 and 8 are relatively prime their least common denominator (LCD) can be found using the formula 3 • 8 = 24. Therefore, 24 is the LCD.
This equation demonstrates which of the following properties? (4 × 7) × 8 = 4 × (7 × 8)
The associative property of multiplication Multiplication can be done in any order so the numbers can be associated in any way without changing the outcome.
Absolute Value
The distance a number is from zero; always a positive number Absolute Value of 5 and -5 is 5
What is the greatest odd factor of 2,496?
The prime factorization of 2,496 is 26 × 3 × 13. When multiplying the odd numbers (3 and 13), the greatest odd factor is 39
Magnitude
The size of a number
Multiplying Fractions
To multiply fractions, multiply the numerators and multiply the denominators
Area Models
Using the expanded form of a number, each number is multiplied together and the products are added to find the final answer This area model shows 42 × 37
Volume of a Cylinder Formula
V=Bh
Factors
Values that are multiplied to get another number. Some factors of 12 are 3 and 4 because 3 x 4 = 12
Rounding
a number is simplified to its closest multiple of 10, 100, 1,000, etc. 26 rounds to 30
Multiplicative Identity
a number that, when multiplied by x, yields x. one or forms of one such as x/x 6x1 = 6
Exponential Form
a number written with a base and an exponent 5,232 would be written as 5×10^3 + 2×10^2 + 3×10^1 + 2×10^0
Prime Factor
a prime number or term that can be multiplied by another to get a number. 2 x 6 = 12, 2 is a prime factor
Algorithm
a specific set of steps that allows the user to reach the correct answer every time
A tennis ball has a diameter of about 3 inches. What is the approximate volume of a cylindrical container if it holds three tennis balls?
about 64 in³ To find the volume of a cylinder, the B (area of the base) is multiplied by the height. The tennis ball can is three tennis balls high or about 9 inches. B, the area of the base, would be the area of the circle with the diameter of the tennis ball, or 3 inches. If the diameter is 3 inches, the radius would be 1.5 inches and the area would be: B = A of circular base = πr² = π(1.5)² = π(2.25) ≈ 7.07 in². So, the volume of the cylinder would be: V = Bh ≈ 7.07(9) = 63.63 in³. 64 in³ is the best approximate answer to this question.
Benchmark Fraction
an easily remembered fraction that can be used to make problems simpler 1/10, 1/4, 1/2, etc.
Neither subtraction or division
are commutative or associative
Expanded Form
break apart each digit in the number and show the digits true value 4,358 = 4000 + 300 + 50 + 8
Commutative Property
changing the order of terms does not change the outcome a + b = b + a
Addition and multiplication are both
commutative and associative.
Reducing Fractions
dividing the numerator and denominator by any common factors to put the fraction in lowest terms 2/4 = 1/2 3/9 = 1/3
Rectangular Arrays
dots or objects that show the end product of 2 numbers; the objects are arranged so there are an equal number of items in each row and column These rectangular array show 4 × 3
Base 10 Number system
each place location for a number has a value that is a power of 10 10, 100, 1000, 10000
Which of the following represents 8,435.21 in written form?
eight thousand four hundred thirty-five and twenty-one hundredths
A common procedure used to find the prime factorization of a value is to use a
factor tree When creating a factor tree, select any pair of factors of the original value. For instance, to make a factor tree for 70, choose either 7 and 10, or the pair 2 and 35, or the pair consisting of 5 with 14.
Decimal Fractions
fractions with a denominator of 10 1/10 = 0.1
10^2
hundred 10^-2 : hundredths
Exponents
indicate repeated multiplication and are written above and after the number in question in the form: x^n where n represents the number of times multiplication is repeated. 3^2 =3 ×3
What is the place value of the "5" in the number 15,436,129?
millions
To convert a decimal into a percent
multiply by 100 and put the % symbol behind the number.
Note that creating proportional ratios is only true with
multiplying/dividing
Prime Numbers
natural numbers greater than 1 that have no numbers that will divide into them without a remainder. The set of known prime numbers begins at 2 and also includes 3, 5, 7, 11, 13, 17, 19, 23, 29, 31... 7 = 7 × 1
Composite numbers
natural numbers that have numbers that divide into them 4,6,8,9 ...
10^0
one
Irrational Numbers
real numbers that CANNOT be represented exactly as a ratio of two integers. pi (π)
To convert a percent into a decimal
rewrite it over 100 and simplify. What is 24.3% expressed in decimal form? Rewrite as 24.3/100 Dividing by 100 moves the decimal two places to the left, 0.243
A student-athlete can run 10 yards in 6 seconds. Which equation shows the number of yards that can be run in s seconds?
s/0.6 If 10 yards can be run in 6 seconds, 1 yard can be run in 0.6 seconds. Therefore, the total number of seconds divided by 0.6 will give the number of yards run.
Scientific Notation
takes large or small numbers and simplifies them to a digit between 1 and 10 that is multiplied by a base-ten number 4,200,000 would be written as 4.2 × 10^6 0.000426 would be written as 4.26 × 10^-4 5,232 would be written as 5.232 × 10^3
10^1
ten 10^-1 : tenths
Greatest common factor (GCF) / Greatest Common Divisor (GCD)
the largest number that will divide evenly into two or more numbers For 12 and 15, GCF = 3
Remainders
the number remaining that cannot be divided as it is less than the divisor 13/2 = 6R1 The remainder is 1
Prime Factorization
the process of writing a number in terms of its prime factors 12 = 2x2x3
Real Numbers
the set of rational and irrational numbers
Least Common Multiple (LCM)
the smallest number two or more numbers will divide into evenly For 12 and 15, LCM = 60
10^3
thousand 10^-3 :thousandths
Relatively Prime
two numbers are relatively prime if they share no common factors 34 and 15 34 and 15 are relatively prime because 34 = 2 × 17 and 15 = 3 × 5, and there are no factors in common
Ignore the remainder and the answer is only the quotient:
when you need a whole number AND the remainder can be lost. A group of children are preparing trays of cookies for their teachers. They want each teacher to receive the same number of cookies. The children should take the number of cookies they have and divide it by the number of teachers. Any remaining cookies would be "thrown out" so as to keep the trays of cookies even.
Add one to the quotient for the solution:
when you need a whole number AND the remainder needs to be kept. A group of students are taking a field trip and need buses to get there. The students would take the total number of students going on the trip and divide by how many students a bus can hold. If the number doesn't divide evenly, the students would need to add one whole bus in order to carry all of the remaining students, even if the remainder is only 1 student, as all students need a seat on a bus to attend the field trip.
Use the quotient and remainder to find the answer:
when you need to know both. A teacher has a box of books she wants to give to her students at the end of the year. She wants to know how many books she can give to each student and how many will be left over to keep for next year. She would take the total number of books in the box and divide it by the number of students in her class. The quotient would be how many books each student can get and the remainder is the number of books she will keep.
Use the remainder as the answer:
when you want what is "left over". A farmer is putting apples into baskets to sell. He wants to put a dozen apples in each basket and plans to eat any extra apples for a snack. The farmer wants to know how many apples he can eat for a snack. The farmer would divide his total number of apples by 12 and the remainder would be his answer.
Inverse operations are helpful as students solve
word problems. Example 1: If a student owes a friend $12.00 and has $5.00, how much more would she need? 5 + ? = 12 and the inverse is 12 - 5, therefore the answer is $7.00. Example 2: If there are 24 students in a class and the teacher would like to separate them into equal groups of 4, how many groups will there be? 24 ÷ 4 = 6 because the inverse is 6 × 4 = 24.