Transcript Presentation

Survival Guide
Number Sense and
Percents
Contents
Divisibility Rules
Factors and Factor Pairs
Multiples
Primes and Composites
Prime Factorization
GCF and GCF Word Problems
LCM and LCM Word Problems
Contents
Adding Mixed Numbers
Subtracting Mixed Numbers
Multiplying Mixed Numbers
Dividing Mixed Numbers
Adding & Subtracting Decimals
Multiplying Decimals
Dividing Decimals
Contents
Changing Fractions to Decimals
Changing Decimals to Fractions
Changing Fractions to Percents
Changing Percents to Fractions
Changing Decimals to Percents
Changing Percents to Decimals
The Percent Proportion
Divisibility Rules
You can always divide. These rules are shortcuts
to determine if one number divides evenly into
another number.
1: All whole numbers are divisible by 1
2: All even numbers (numbers ending in 0, 2, 4, 6, or 8)
3: The sum of the digits is divisible by 3
4: The number formed by the last 2 digits are divisible by 4
5: The number in the one’s place is a 0 or 5
6: The number is divisible by 2 AND 3
8: The number formed by the last 3 digits is divisible by 8
9: The sum of the digits is divisible by 9
10: The number in the one’s place is a 0
Divisibility Rules
EXAMPLES:
162,720 is divisible by… 1, 2, 3, 4, 5, 6, 8, 9, and 10
1
2 (ends in a 0)
3 (the sum is 1 + 6 + 2 + 7 + 2 + 0 = 18 and 18 ÷ 3 = 6)
4 (20 ÷ 4 = 5)
5 (ends in a 0)
6 (the number is divisible by 2 and 3)
8 (720 ÷ 8 = 90)
9 (sum is 18 and 18 ÷ 9 = 2)
10 (ends in a 0)
Divisibility Rules
EXAMPLES:
32,592,165 is divisible by… 1, 3, and 5
1
not 2 (ends in a 5)
3 (the sum is 3+2+5+9+2+1+6+5 = 33 and 33 ÷ 3 = 11)
not 4 (4 does not divide evenly into an odd number)
5 (ends in a 5)
not 6 (the number is divisible by 3 but not 2)
not 8 (8 does not divide evenly into an odd number)
not 9 (sum is 33 and 33 ÷ 9 is not a whole number)
not 10 (does not end in a 0)
Divisibility Rules
EXAMPLES:
745,168 is divisible by… 1, 2, 3, 4, and 8
1
2 (ends in a 8)
3 (the sum is 7+4+5+1+6+8 = 31 and 31 ÷ 3 does not
equal a whole number)
4 (68 ÷ 4 = 17)
not 5 (does not end in a 0 or 5)
not 6 (the number is divisible by 2 but not 3)
8 (168 ÷ 8 = 21)
not 9 (sum is 31 and 31 ÷ 9 is not a whole number)
not 10 (does not end in a 0)
Factors and Factor Pairs
Factors are numbers you multiply together to get a
product. 2 and 5 are factors of 10 because 2 x 5 = 10
A number can have many factors. The factors of 12 are 1, 2,
3, 4, 6, and 12 because 1 x 12 = 12, 2 x 6 = 12, and 3 x 4 = 12.
Factor pairs are the two numbers used to get the product.
There are 3 factor pairs for 18: 1 and 18, 2 and 9, 3 and 6.
There are 4 factor pairs for 40: 1 and 40, 2 and 20, 4 and 10,
5 and 8.
There is one factor pair for 11: 1 and 11
Multiples
The multiples of a number are the product of the number
and any other number.
If you list the multiples of a number, the list is endless.
The first multiple of 3 is 3 because 3 x 1 = 3.
The second multiple of 3 is 6 because 3 x 2 = 6.
The third multiple of 3 is 9 because 3 x 3 = 9.
The fourth multiple of 3 is 12 because 3 x 4 = 12.
The tenth multiple of 3 is 30 because 3 x 10 = 30.
The seventy-fifth multiple of 3 is 225 because 3 x 75 = 225.
Primes and Composites
Prime numbers have exactly two factors (1 and itself).
Composite numbers have three or more factors.
Number
1
2
3
4
5
6
7
8
9
10
11
12
Factors
1
1, 2
1, 3
1, 2, 4
1, 5
1, 2, 3, 6
1, 7
1, 2, 4, 8
1, 3, 9
1, 2, 5, 10
1, 11
1, 2, 3, 4, 6, 12
Prime or Composite
Neither
Prime
Prime
Composite
Prime
Composite
Prime
Composite
Composite
Composite
Prime
Composite
Primes and Composites
Prime numbers are white.
Composite numbers are yellow.
One is neither prime nor composite!!
Prime Factorization
Every number has a unique prime factorization. When a
number is written as a product of prime numbers, it is
called the prime factorization for that number.
The prime factorization for 12 is 2 x 2 x 3 and the prime
factorization for 50 is 2 x 5 x 5. Always write the prime
numbers from least to greatest.
Two common methods for finding the prime factorization is
a factor tree
and prime division.
These two diagrams
show how to find the
prime factorization for
80 using both common
methods.
Greatest Common Factor
All natural numbers share a common factor of 1. Many
numbers share other factors in common.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The common factors are 1, 2, 3, 4, 6 and 12. The greatest
common factor (GCF) is 12. It is the largest factor the
numbers have in common.
The list method, used above, works well for determining
the GCF of smaller numbers. For larger numbers, other
methods, such as “the slide” and prime factorization are
useful.
Greatest Common Factor
The SLIDE Method
The GCF can be found
by multiplying the 2 and
the 4. The GCF of 40 and
72 is 8.
Prime Factorization Method
The numbers in the prime
factorization that 40 and 72
have in common are 2 x 2
x 2. The GCF is 8.
Greatest Common Factor
The SLIDE Method
The GCF can be found
by multiplying 4 x 3 x 10.
The GCF of 120 and 360 is
120.
Prime Factorization Method
The numbers in the prime
factorization that 120 and 360
have in common are 2 x 2 x 2
x 3 x 5 = 120. The GCF is 120.
GCF Word Problems
Greatest common factor word problems usually contain
one of the following words in the question…greatest, most,
largest, maximum, highest.
1. Juliana is putting together first-aid kits. She has 20 large
bandages and 8 small bandages, and she wants each kit to be
identical, with no bandages left over. What is the greatest
number of first-aid kits Juliana could put together?
2. A florist has 5 tulips and 15 carnations. If the florist wants to
create identical bouquets without any leftover flowers, what is
the maximum number of bouquets the florist can make?
Least Common Multiple
All numbers have common multiples. It is often helpful to
find the least common multiple (also known as the least
common denominator) when adding and subtracting
fractions.
The first 10 multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.
The first 10 multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80.
The common multiples in the list are 24 and 48. The least
common multiple (LMC) of 6 and 8 is 24. It is the smallest
multiple the two numbers have in common.
The list method, used above, works well for determining
the LCM of smaller numbers. For larger numbers, other
methods, such as “the slide” and prime factorization are
useful.
Least Common Multiple
The SLIDE Method
The LCM can be found
by multiplying the 2, 4,
5 and 9. The LCM of 40
and 72 is 360.
Prime Factorization Method
To find the LCM, multiply the
numbers they have in common
(the 2 x 2 x 2) with the numbers
not in common (3 x 3 x 5). The
LCM of 40 and 72 is 360.
Least Common Multiple
The SLIDE Method
The LCM can be found
by multiplying the 2, 7,
and 8. The LCM of 14
and 16 is 2 x 7 x 8 = 112.
Prime Factorization Method
To find the LCM, multiply the
number they have in common
(the 2) with the numbers not
in common (2 x 2 x 2 x 7). The
LCM of 14 and 16 is 112.
LCM Word Problems
Least common multiple word problems usually contain one
of the following words in the question…least, smallest,
minimum.
1. Sara's Bath Shop sells bars of soap in boxes of 2 bars and bottles
of soap in boxes of 19 bottles. An employee is surprised to
discover that the shop sold the same number of bars and bottles
last week. What is the smallest number of each type of soap that
the shop could have sold?
2. Butler Stationery sells cards in packs of 18 and envelopes in packs
of 16. If Evan wants the same number of each, what is the least
number of cards that he will have to buy?
LCM Word Problems
A second type of least common multiple word problem will
have two or more things happening over and over again.
The question will want to know when those things will
happen again at the same time.
1. Jordan rides his bike every 3 days, swims every 2 days, and runs
every 4 days. If he did all three exercises today, in how many
days will he do all three again on the same day?
2. Maria cleans her over every 40 days and washes windows every
50 days. If she did both chores today, in how many days will she
do both chores again on the same day?
Adding Mixed Numbers
1. Find the least common denominator (LCD). For this
problem, the LCD is 20.
2. Rewrite the problem with equivalent fractions using the
LCD.
3. Add the numerators.
4. Keep the denominator.
5. Add the whole numbers.
6. Finalize the problem by
making sure the fraction
is not improper.
Always simplify!
Subtracting Mixed Numbers
1. Find the least common denominator (LCD). For this
problem, the LCD is 20.
2. Rewrite the problem with
equivalent fractions using
the LCD.
3. Subtract the numerators.
4. Keep the denominator.
5. Subtract the whole numbers.
6. Always simplify!
Subtracting Mixed Numbers
In this problem, you must rename (also known as borrow)
before you can subtract the numerators.
To rename a mixed number, subtract 1 from the whole
number and add it to the fractional part. Change the
fractional part to an improper fraction.
Multiplying Mixed Numbers
1. Rewrite every mixed number as an improper fraction.
2. Optional: Cross-cancel and reduce fractions to make the
next steps easier!
3. Multiply the numerators straight across.
4. Multiply the denominators straight across.
5. If the answer is improper, change it to a mixed number.
6. Always simplify!
Dividing Mixed Numbers
1. Rewrite every mixed number as an improper fraction.
2. Multiply by the reciprocal. This means change the divide
sign to multiply and “flip” the fraction that came after
divide sign.
3. Optional: Cross-cancel and reduce fractions to make the
next steps easier!
4. Multiply the numerators straight across.
5. Multiply the denominators straight across.
6. If the answer is improper, change it to a mixed number.
7. Always simplify!
Adding & Subtracting Decimals
1.
2.
3.
4.
Line up the decimal points!
Add zeros after the decimal points, if needed.
Bring the decimal point straight down.
Add or subtract.
Multiplying Decimals
1. Line up the last digit of each number. There is no need
to line up the decimals.
2. Multiply the numbers as if they are whole numbers.
Ignore the decimals for now.
3. After multiplying, determine where the decimal should
be placed. Count the number of digits after the decimal
point for both factors. Your answer needs
to have the same number of digits after the
decimal point.
In this first example, there are three
digits after the decimal point (3, 2, 7).
As a result, the answer has three
digits after the decimal point (0, 2, 4).
Dividing by Whole Numbers
When dividing by a whole number,
remember to repeat the following
steps: divide, multiply, subtract, and
bring down the next number.
When you reach the end of the whole
number, you are no longer to write the
leftover number as a remainder!
Instead, add a decimal and bring it
up to the answer. Then add as many
zeros as you need to keep dividing.
In some cases, as the second example,
you will begin to notice a repeating
pattern. Use a bar over the repeating
numbers.
REMEMBER TO KEEP ALL YOUR NUMBERS AND DECIMALS LINED UP!!
Dividing by Decimals
When dividing by a decimal, you must FIRST make the divisor into a whole
number. This is done by moving the decimal to the right. Then count the
number of places you moved the decimal point and do the exact same thing to
the dividend. The dividend does NOT need to be a whole number!
Notice in the first example, when the decimal is moved once in the divisor, it is
moved once in the dividend. In the second example, when the decimal is
moved twice in the divisor, it is moved twice in the dividend. Zeros are added
as needed.
REMEMBER TO KEEP ALL YOUR NUMBERS AND DECIMALS LINED UP!!
Changing Fractions to Decimals
Many fractions can be changed to decimals by
using place value. Here are a few examples:
•
3
5
•
7
25
is equal to
6
.
10
is equal to
Six tenths is 0.6 as a decimal.
28
100
which is 0.28 as a decimal.
When this does not work, remember that
a fraction is just another way to write a
division problem.
1
•
is equal to 1 ÷ 8 which equals 0.125
8
•
7
12
is equal to 7 ÷ 12 which equals 0.58𝟑
Changing Decimals to Fractions
Changing decimals to fractions is as easy as
reading the number! Remember to simplify
the fraction.
32
8
• 0.32 is thirty-two hundredths =
=
• 0.9 is nine tenths =
9
10
100
25
12
1000
3
250
145
1000
29
200
• 0.012 is twelve thousandths =
=
• 0.145 is one hundred forty-five thousandths
=
=
Changing Decimals to Percents
To change a decimal to a percent,
move the decimal two places to the
right and add a percent sign.
0.57 = 57%
0.29 = 29%
0.5 = 50%
0.06 = 6%
0.002 = 0.2%
1.05 = 105%
Changing Percents to Decimals
To change a percent to a decimal,
move the decimal two places to the
left and remove the percent sign.
36% = 0.36
45% = 0.45
70% = 0.70 = 0.7
8% = 0.8
1.9% = 0.019
435% = 4.35
Changing Percents to Fractions
Percent means per hundred. Every percent
can be changed into a fraction by placing
the percent over 100. Remember to simplify.
Examples:
68% =
68
100
=
17
25
115% =
115
100
=
15
1
100
=1
3
20
If the percent contains a decimal, a little
more work is required. Change the percent
to a decimal and then to a fraction.
Examples:
37.5% =
Examples:
5.25% =
375
3
0.375 =
=
1000
8
525
21
0.0525 =
=
10000
400
Changing Fractions to Percents
Percent means per hundred. If you can write an
equivalent fraction with a denominator of 100, the
numerator is the percent.
Examples:
12
48
=
25
100
36
12
=
300
100
so
so
12
= 48%
25
36
= 12%
300
When this doesn’t work, change the fraction to a
decimal by dividing and then change the decimal
to a percent.
19
Examples:
=
32
19 ÷ 32 = 0.59375 = 59.375%
Percent Problems: The % Proportion
The Percent Proportion is
is
of
=
%
.
100
Examples:
30
%
What percent of 50 is 30?
=
50
100
32
40
32 is 40% of what number? =
x
100
x
75
75% of 420 is what number?
=
420
100
805
115
115% of what number is 805?
=
x
100
Answer: 60%
Answer: 80
Answer: 315
Answer: 700
Percent Problems: The % Proportion
For story problems, this percent
proportion works better:
part
whole
=
%
100
Example 1: 56% of the students in a class wear flip-flops. If
there are 32 students, how many students were flip-flops?
Since 32 students represents the whole class, the proportion
would be
x
32
=
56
100
.
Solve to find18 students wear flip-flops.
Example 2: On a quiz, Marcy got 38 questions correct out of
40. What percent did Marcy get on the test? The proportion
38
40
=
𝑥
100
would give you an answer of 95%.