Download Number Basics Decimals - Bakersfield Christian High School

The verbal answers to all of the following questions should be memorized before completion of
pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1.
Number Basics
What are the first six place values to the left of the decimal point?
Ones, tens, hundreds, one thousands, ten thousands, hundred thousands.
What are the main place value groupings to the left of the decimal point?
Ones, thousands, millions, billions, trillions, etc.
What are the first three place values to the right of the decimal point?
Tenths, hundredths, thousandths.
How do you round numbers?
Look one digit to the right of the place you are rounding to. If that digit is a 0-4, round down. If that
digit is a 5-9, round up.
examples: Round 87 to the nearest ten. 90
Round 354,918 to the nearest ten thousand. 350,000
Round 45.0836 to the nearest tenth. 45.1
How do you compare numbers?
On a number line, larger numbers are to the right and smaller numbers are to the left.
What is the meaning of an exponent?
The base needs to be multiplied by itself the number of times of the exponent.
example: 25 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 32
What is the rule for order of operations?
PEMDAS. Parenthesis first, exponents next, then multiplication and division left to right, finally
addition and subtraction left to right.
example: 24 − 5 + (8 − 2 ) = 24 − 5 + 6 = 16 − 5 + 6 = 11 + 6 = 17
How do you approximate a square root of a number?
The square root of a given number will be a number between the square roots of the two closest perfect
square numbers.
example: The 12 will be a number between the 9 = 3 and the 16 = 4.
Decimals
How do you add decimals?
Align the decimal points. Add right to left, carrying as necessary.
12.8
example: 12.8 + 9 =
= 21.8
+9.
Pre-Algebra (page 1/9)
How do you subtract decimals?
Align the decimal points. Subtract right to left, borrowing as necessary.
3 13
example: 43.8 − 9 = 4 3.8 = 4 3 .8 = 34.8
−9.
−9.
How do you multiply decimals?
Multiply normally. The sum of the number of decimal places gives the new number of decimal places.
example: .03 × 1.002 = 0.03006
How do you multiply decimals by powers of ten?
Shift the decimal place to the right by the numbers of zeros in the power of ten.
example: 12.405 × 10,000 = 124,050.
Where are the dividend, divisor, and quotient located in a division problem?
quotient
divisor dividend
How do you divide decimals?
Eliminate decimals in the divisor. Shift the decimal point the same number of places in the dividend.
Divide normally.
7.9
example: .02 .158 = 2 15.8 = 2 15.8
How do you divide decimals by powers of ten?
Shift the decimal place to the left by the numbers of zeros in the power of ten.
example: 12.405 ÷ 10,000 = 0.0012405
How do you write a number in scientific notation?
Starting at the left, put a decimal point to the right of the 1st nonzero number. Count the number places
the decimal point moved. If the number got smaller, the exponent on the ten will be positive. If the
number got larger, the exponent on the ten will be negative.
examples: a) 48,000 = 4.8 × 104
b) 0.0000048 = 4.8 × 10−6
Signed Numbers
What does absolute value do to a number?
Absolute value makes numbers positive.
examples: 5 = 5, −8 = 8
How do you add signed numbers?
If the numbers are the same sign, add the numbers and keep the sign. If the numbers are different signs,
subtract the numbers and keep the sign of the larger magnitude number.
examples: 3 + −8 = −5,
− 4 + −2 = −6
Pre-Algebra (page 2/9)
How do you subtract signed numbers?
(Subtracting a negative number is the same as adding a positive number. Subtracting a positive number
is the same as adding a negative number.) If the numbers are the same sign, add the numbers and keep
the sign. If the numbers are different signs, subtract the numbers and keep the sign of the larger
magnitude number.
examples: 3 − 8 = 3 + −8 = −5,
− 8 − 2 = −8 + −2 = −10, 6 − −7 = 6 + 7 = 13
How do you multiply signed numbers?
Multiply normally. If the numbers have the same sign, the answer is positive. If the numbers have
different signs, the answer is negative.
examples: 2 ⋅ −4 = −8,
− 3 ⋅ −6 = 18,
− 5 ⋅ 4 = −20
How do you divide signed numbers?
Divide normally. If the numbers have the same sign, the answer is positive. If the numbers have
different signs, the answer is negative.
examples: −32 ÷ −4 = 8,
− 30 ÷ 2 = −15, 12 ÷ −2 = −6
Factors and Multiples
What is a factor?
A factor is a number that divides evenly into another number.
example: 4 is a factor of 20.
What is the greatest common factor of two numbers?
The largest number that divides evenly into both numbers.
example: 12 is the greatest common factor of 24 and 36.
What is a prime number?
A number having exactly two factors.
example: 11 is a prime number since 11 is divisible evenly by only 1 and 11.
What is a composite number?
A number having more than two factors.
example: 10 is a composite number since its factors are 1, 2, 5, and 10.
What is the prime factorization of a number?
A number written as the product of only prime numbers.
example: The prime factorization of 80 is 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 = 24 ⋅ 5
What is a multiple?
A number obtained by multiplying a given number by an integer.
example: 5, 10, and 15 are multiples of 5.
What is the least common multiple of two numbers?
The smallest positive number that is evenly divisible by both numbers.
example: 30 is the least common multiple of 10 and 15.
Pre-Algebra (page 3/9)
Fractions and Mixed Numbers
When is the value of a fraction equivalent to one?
When the numerator and denominator are the same.
examples: All of the following fractions are equivalent to one.
3 10 17
3 10 17
, ,
How do you simplify fractions?
Divide the numerator and denominator by all common factors.
36 36 ÷ 2 18 18 ÷ 3 6
=
=
=
=
example:
78 78 ÷ 2 39 39 ÷ 3 13
What is an improper fraction?
A fraction in which the value of the numerator is greater than or equal to the value of the denominator.
8
6
example: The following two fractions are both improper. and
5
6
How do you compare fractions?
Find a common denominator and compare the numerators.
3
1
3 3
1 7
9
7
3 1

→ ⋅ and ⋅ 
→
and

→ >
example: and
7
3
7 3
3 7
21
21
7 3
How do you add fractions?
Find a common denominator. Add the numerators. Leave the denominator the same. Simplify.
1 3 1 5 3 3 5
9 14 7
+
=
=
example: + = ⋅ + ⋅ =
6 10 6 5 10 3 30 30 30 15
How do you subtract fractions?
Find a common denominator. Subtract the numerators. Leave the denominator the same. Simplify.
5 1 5 5 1 4 25 4 21
−
=
example: − = ⋅ − ⋅ =
8 10 8 5 10 4 40 40 40
How do you add a whole number to a fraction?
Find a common denominator. Add the fractions
3
3 20 23
=
example: + 5 = +
4
4 4
4
How do you multiply fractions?
Cancel common factors. Multiply the numerators. Multiply the denominators.
15 4 3 1 3
⋅ = ⋅ =
example:
32 35 8 7 56
How do you divide fractions?
Multiply by the reciprocal fraction.
1 3 1 4 1 2 2
example: ÷ = ⋅ = ⋅ =
6 4 6 3 3 3 9
Pre-Algebra (page 4/9)
How do you convert a mixed number into an improper fraction?
Multiply the denominator by the whole number and add the numerator. Leave the denominator the
same.
7 3 ⋅ 8 + 7 31
=
example: 3 =
8
8
8
How do you convert an improper fraction into a mixed number?
Divide the numerator by the denominator. The result is the whole number and the remainder is the
numerator of the fraction. The denominator stays the same.
5R1
21
1
example:
= 4 21 = 5
4
4
How do you convert a fraction into a decimal?
Divide the numerator by the denominator.
0.6
3
example: = 5 3.0 = 0.6
5
How do you convert a mixed number into a decimal?
The whole number stays the same. Convert the fraction into a decimal and combine it with the whole
number.
0.25
1
1
example: 3 = 3 + = 3 + 4 1.00 = 3 + 0.25 = 3.25
4
4
How do you convert a decimal into a mixed number?
The part to the left of the decimal becomes the whole number. The part to the right of the decimal is
written as a fraction and then simplified.
85
85 ÷ 5
17
= 12 +
= 12
example: 12.85 = 12 +
100
100 ÷ 5
20
How do you add mixed numbers?
Add the fraction parts (carrying if necessary). Add the whole number parts.
3
7
3 7
6 7
13
5
5
5
5
example: 1 + 5 = 1 + 5 + + = 1 + 5 + + = 6 + = 6 + 1 = 6 + 1 + = 7 + = 7
4
8
4 8
8 8
8
8
8
8
8
How do you subtract mixed numbers?
Subtract the fraction parts (borrowing if necessary). Subtract the whole number parts.
3
9
6
9
16
9
7
example: 4 − 2 = 4 − 2 = 3 − 2 = 1
5
10
10
10
10
10 10
How do you multiply mixed numbers?
Convert to improper fractions. Multiply the improper fractions. Convert back to mixed numbers.
6 R1
2 4 5 19 1 19 19
1
example: 1 ⋅ 3 = ⋅ = ⋅ =
= 3 19 = 6
3 5 3 5 3 1
3
3
Pre-Algebra (page 5/9)
How do you divide mixed numbers?
Convert to improper fractions. Divide the improper fractions. Convert back to mixed numbers.
1R11
1
3 25 7 25 4 25 1 25
11
example: 3 ÷ 1 =
÷ =
⋅ =
⋅ =
= 14 25
=1
8
4 8 4 8 7 2 7 14
14
Ratio, Proportion, and Percentage
What is a ratio?
A comparison of two numbers.
example: 3:8 is a ratio. 3 8 is a ratio.
What is a proportion?
Two ratios that are equal.
3 6
example: = .
4 8
How do you solve a proportion?
Cross multiply and divide.
3 x
3 ⋅ 20 3 ⋅ 5
x=
=
= 15
example: =
4 20
4
1
What is a percentage?
A number representing a part out of 100.
example: 23% means 23 parts out of 100.
How do you convert a percentage to a decimal?
Divide the percentage by 100.
example: 45% is the same as 45 ÷ 100 which is 0.45 as a decimal.
How do you convert a percentage to a fraction?
Write the number with a denominator of 100.
60
6 3
= =
example: 60% =
100 10 5
How do you convert a decimal into a percentage?
Multiply by 100 and add the percentage symbol.
example: 0.4 written as a percentage is 40%.
What is the formula for percentage increase and percentage decrease?
The change divided by the original and then multiplied by 100%.
example: Last week 10 students said they loved math. This week 13 students say they love math. What
is the percentage increase in math loving students? The change is 3 students so the percentage increase
3
⋅ 100% = 30% increase.
is
10
Pre-Algebra (page 6/9)
How do you set up a proportion equation for solving a percent problem?
is
%
=
of 100
12
x
12 ⋅ 100 12 ⋅ 10 4 ⋅ 10
=
, x=
=
=
= 40%
example: What percentage of 30 is 12?
30 100
30
3
1
How are the following units related?
a) feet and inches? b) feet and yards? c) feet and miles? d) millimeters and centimeters?
e) centimeters and meters? f) millimeters and meters? g) kilometers and meters?
a) 1 foot equals 12 inches. b) 3 feet equals 1 yard. c) 5280 feet equals 1 mile.
d) 10 millimeters equals 1 centimeter. e) 100 centimeters equals 1 meter.
f) 1000 millimeters equals 1 meter. g) 1 kilometer equals 1000 meters.
Statistics and Probability
How is theoretical probability calculated?
Theoretical probability is the ratio of the number of successful outcomes to the number of possible
outcomes.
example: What is the probability of picking a prime number given the numbers {1, 2, 3, 4, 5, 6, 7, 8, 9}?
The prime numbers are {2, 3, 5, 7} so the probability would be 4 9 .
How do you find the mean (average) of a set of numbers?
Add up all the numbers and then divide by the number of numbers.
2 + 6 + 13
=7
example: The mean of 2, 6, and 13 is
3
How do you find the median of a set of numbers?
Order the numbers from small to large. The median is the middle number. If there is no middle
number, the median is the average of the two numbers in the middle.
example: The median of 2, 8, and 1 is 2. (2 is the middle number for 1, 2, 8)
What is the mode of a set of numbers?
The most frequently occurring number. If there is a tie, all the tied numbers are the mode.
example: For the numbers 2, 4, 3, 4, 4, 4, 8, 1, the mode is 4.
What is the minimum of a set of numbers? What is the maximum of a set of numbers?
The minimum is the smallest number in the set and the maximum is the largest number in the set.
example: The minimum of 2, 5, 10, 1, 8 is 1. The maximum of 2, 5, 10, 1, 8 is 10.
What is the range of a set of numbers?
The difference between the maximum number and the minimum number.
example: The range of 2, 5, 10, 1, 8 is 10 − 1 = 9 .
Geometry
What is perimeter? (Give some example units for perimeter.)
The distance around a two dimensional shape. (feet, inches, meters, miles, centimeters, …)
Pre-Algebra (page 7/9)
How do you find the perimeter of a two dimensional shape?
Add up all the sides.
example: The perimeter of the shape below is 14 meters.
5m
2m
2m
5m
What is area? (Give some example units for area.)
The amount of surface covered by a two dimensional shape. (ft2, in2, m2, km2, cm2, …)
What is the formula for the area of a rectangle?
Length times width. (Area of a rectangle is also written as base times height.)
example: The area of the rectangle below is 10 in2.
5 in
2 in
2 in
5 in
What is the formula for the area of a triangle?
One half base times height. ( A = 12 bh )
example: The area of the triangle below is 12 ft2.
3 ft
8 ft
What is the formula for the area of a circle?
Pi times radius squared. A = π r 2
(
)
example: The area of the circle below is 9π cm2.
3 cm
What is the formula for the circumference of a circle?
Two times pi times radius. ( A = 2π r )
example: The circumference of the circle below is 6π yards.
3 yd
Pre-Algebra (page 8/9)
What is volume? (Give some example units for volume.)
The amount of three dimensional space enclosed by an object. (ft3, in3, m3, km3, cm3, …)
What is the formula for the volume of a box?
Length times width times height.
example: The volume of the box below is 60 mm3.
4 mm
3 mm
5 mm
What is the Pythagorean Theorem and when can it be used?
a squared plus b squared equals c squared a 2 + b2 = c 2 . It can only be used on right triangles to find a
(
)
missing side when two sides are already known.
example: The missing side in the following triangle is 12.
52 + b2 = 132 → 25 + b2 = 169 → b2 = 144 → b = 12
(
5
)
13
b
Graphical
How do you graph inequalities on a number line?
Less than " < " and greater than " > " are open circles. Less than or equal " ≤ " and greater than or equal
" ≥ " are closed circles. Greater " > or ≥ " is shaded to the right. Less " < or ≤ " is shaded to the left.
examples: x ≥ 10
x < 10
6
8
10
12
14
6
8
10
12
14
How is a point (x, y) plotted in a coordinate plane?
The x coordinate is plotted left and right. Positive numbers are to the
right and negative numbers are to the left. The y coordinate is
plotted up and down. Positive numbers are up and negative numbers
are down.
examples: Plot the points A (2, −4) and B ( −1,3) .
5
B
3
1
-5
-3
-1-1
1
3
5
-3
A
-5
Pre-Algebra (page 9/9)