Fractions, Decimals, Percents Flashcards
For adding & subtracting fractions, first find a common denominator (Least Common Multiple of 3 and 5)
Multiply horizontally:
Dividing fractions: Flip the 2nd fraction and multiply
When we take the RECIPROCAL, we flip the fraction
This is the same as 1 divided by the fraction.
Write “z is x percent of y” as an equation
Percent means “divide by 100”
“of” means “multiply”
“is” means =
Write “x is 20% greater than y” as an equation
Write “x is 15% less than y” as an equation,
a) write using a decimal
b) convert to a fraction
Write “x is n percent greater than y” as an equation
A bike was $50, and is now on sale for $32. What is the percent change in price?
Distribute:
-3(x-7)
= -3*x + -3*-7
= -3x + 21
Be careful with your SIGNS! Double negative turns into a positive
Factor out the Common Factor:
4x + 4y =7
Both terms have “4” as a factor.
So, we can factor it out (this is the inverse process to “distribution”)
4(x+y) = 7
Factor out the Common Factor:
3x + 6y + 15z = 7
Factor out the Common Factor:
The common factor between 3, 6, and 15 is 3.
(we can prove this by writing the prime factorization: 6=2*3, 15=5*3)
We factor out 3 from each “term”.
3(x + 2y + 5z)
Multiplying Exponents:
What is 23 * 24 ?
27
When we multiply exponents with the same bases, we add the powers
ab * ac = ab+c
What is 2 * x3 * y2 * 3 * y4 * x ?
We match up what can be simplified:
2*3 = 6
x3 * x = x4
y2 * y4 = y6
6x4y6
Distribute: -2x(-3x + 5)
- 2x * -3x = 6x2
- 2x * 5 = -10x
6x2 - 10x
Factor out the Common Factor:
15x2 - 10x
This time, we can factor out both a number and a variable:
The common prime factor of 15 and 10 is 5.
The common variable is x
So, we factor out 5x:
5x(3x-2)
Scientific Notation:
Write out as a number: 2.3 * 107 = ?
23,000,000
* 107 is like multiplying by 10, 7 times. This moves the decimal point 7 places to the right.
What is the units digit, tens digit, and hundreds digit of the number 5318?
units: 8
tens: 1
hundreds: 3
What is the units digit of 3219 ?
We need to find the pattern.
3219 will have the same units digit as 219
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
The pattern repeats for every 4 numbers in the pattern: {2, 4, 8, 6}
Every 4th power will have a units digit of 6: 24, 28, 212, 216
So, 217 ends in 2, 218 ends in 4, 219 ends in 8.
What is the units digit of 714 ?
71 =7
72 = 49
Only the units digit will affect the next power: so we can do 9*7 =63 –> 73 ends with 3
3*7 =21 –> 74 ends with 1
1*7 = 7 –>75 ends with 7.
So, the pattern repeats for every 4 numbers in the pattern: {7, 9, 3, 1}
So, the 4th, 8th, and 12th powers would have units digit of 1.
713 has units digit of 7
714 has units digit of 9
If x and y are integers, with y > 1, how do we determine whether the fraction x/y is a repeating decimal (for example, 1/3 = .33333333… repeating), versus a “Terminating Decimal” with a finite number of digits (example: 1/40 = 0.025)?
The numerator doesn’t matter.
If the denominator only includes the prime factors 2 and/or 5, it is not repeating.
Example: 1/200 = ?
200 = 2*2*2*5*5 —> so it will be a “Terminating Decimal”
1/200 = 0.005
1/60= 3*2*2*5 = .01666666…… —> “Repeating Decimal”
For every 5 men, there are 3 women.
Express the above ratio in 2 ways
The ratio of men to women is:
5 : 3 or
5/3
Note: The order of wording matters! the ratio of women to men is 3:5
Be mindful to prevent this “Backwards” error.
If there are 25% more women than men, what is the ratio of men to women?
If the ratio of men to women is 5:7, what fraction of the total are women?
If the ratio of men to women is 3:4, which of the following could NOT be the total number of men and women combined?
A) 35
B) 72
C) 49
D) 21
E) 280
We can’t have a fraction of a person.
Therefore, the total must be a multiple of 7. (3+4)
7, 14, 21, 28, etc.
Only B is not a multiple of 7. (we might notice that 70 is a multiple of 7, so 72 isn’t)
35= 5*7
49=7*7
21= 3*7
280= 40*7
If the ratio of flour to water in a recipe is 3:2, how many cups of flour would we need for 7 cups of water?
Unlike for people, We CAN have fractions of units.
We can use the Unknown Multiplier method:
The ratio of flour to water is 3x : 2x, with x as our unknown multiplier.
We know the amount of water, 7, so 2x=7.
Solve for x: x=3.5
Now plug in x=3.5 and find the flour:
3x = 3*3.5 = 10.5 cups of flour
If the ratio of dogs to cats to birds is 2:3:5, and there are 24 cats, how many total animals are there?
We can use the unknown multiplier:
2x:3x:5x
3x represents cats. 3x=24
x=8
So, our multiplier is 8.
Counting all 3 animals, we have 2x+3x+5x = 10x
So, 10x = 10*8 = 80
There are 3x for every 2y.
There are 3z for every 5x
What is the ratio of y to z?
ratio 1) 3x:2y
ratio 2) 3z:5x
We use a similar process to finding a common denominator with fractions:
What is the common factor between 3x and 5x? 15x
Multiply the 1st ratio by 5: 15x:10y
Multiply the 2nd ratio by 3: 9z:15x
So, for every 15x there is 10y, and 9z
Therefore the ratio of y to z is 10:9
What is the definition of “directly proportional”?
When 2 quantites vary by a constant ratio.
It is a linear relationship: y = mx, where m is the slope or constant ratio The graph would start at 0,0
examples: someone earns $20 per hour, with no other payments.
