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Pre-Algebra formula sheet

45 formulas · what each one means · a free lesson for each

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Integers and signs

Add a negative
a + (−b) = a − b
Adding a negative number is the same as taking it away.Lesson: Integers →
Subtract a negative
a − (−b) = a + b
Taking away a negative number is the same as adding it.Lesson: Integers →
Multiply and divide signs
(−) × (−) = + (−) × (+) = −
Same signs give a plus answer; different signs give a minus answer. ÷ works the same way.Lesson: Integers →
Absolute value
|−a| = a |a| = a
How far a number is from 0. It is never negative.Lesson: Powers, square roots, absolute value →
Distance between two numbers
distance = |a − b|
Take one number from the other and drop the sign.Lesson: Negative numbers in real life →

Order of operations

The order
( ) → an → × ÷ → + −
Brackets first, then powers, then × and ÷ from left to right, then + and − from left to right.Lesson: Order of operations →
Change the order (+)
a + b = b + a
You can add in any order.Lesson: Number properties for mental math →
Change the order (×)
a × b = b × a
You can multiply in any order.Lesson: Number properties for mental math →
Change the grouping
(a + b) + c = a + (b + c)
With only + (or only ×), the brackets can move.Lesson: Number properties for mental math →
Distributive property
a(b + c) = ab + ac
The number outside multiplies each part inside.Lesson: Number properties for mental math →

Fractions and decimals

Simplify
a × cb × c = ab
Divide the top and the bottom by the same number.Lesson: Fractions →
Multiply
ab × cd = a × cb × d
Top times top, bottom times bottom.Lesson: Fractions →
Divide
ab ÷ cd = ab × dc
Flip the second fraction, then multiply.Lesson: Fractions →
Add with the same bottom
ac + bc = a + bc
Add the tops; the bottom stays.Lesson: Fractions →
Add with different bottoms
ab + cd = a × db × d + c × bd × b
First make the bottoms the same, then add the tops.Lesson: Fractions →
Mixed number
abc = a × c + bc
Whole part times the bottom, plus the top, over the same bottom.Lesson: Fractions →
Fraction to decimal
ab = a ÷ b
The fraction bar means divide.Lesson: Fractions and decimals →
Decimal to fraction
0.75 = 75100 = 34
The digits over 10, 100 or 1000 (one 0 for each decimal place), then simplify.Lesson: Fractions and decimals →

Factors and multiples

Prime number
2, 3, 5, 7, 11, 13, …
A whole number above 1 whose only factors are 1 and itself.Lesson: Factors and multiples →
Greatest common factor
GCF(12, 18) = 6
The biggest number that goes into both.Lesson: Factors and multiples →
Least common multiple
LCM(4, 6) = 12
The smallest number both go into.Lesson: Factors and multiples →
Divisible by 3 (or 9)
123 → 1 + 2 + 3 = 6
Add the digits: if 3 (or 9) goes into that sum, it goes into the number.Lesson: Divisibility tests →
Divisible by 2, 5, 10
last digit for 2: 0, 2, 4, 6, 8 · for 5: 0, 5 · for 10: 0
Only the last digit matters.Lesson: Divisibility tests →

Percent

Percent means out of 100
p% = p100
35% is 35 out of 100.Lesson: Percents, decimals and fractions →
Percent of a number
part = p100 × whole
“Of” means times.Lesson: Percent →
Find the percent
percent = partwhole × 100%
Divide the part by the whole, then × 100.Lesson: Percent →
Find the whole
whole = part ÷ p100
Divide the part by the percent as a decimal.Lesson: Percent →
Percent change
new − oldold × 100%
Plus = an increase, minus = a decrease. Always divide by the old value.Lesson: Percent →
Percent error
|measured − actual|actual × 100%
How far off the measurement is, as a part of the real value.Lesson: Percent error →
Price after a discount
new price = price × (1 − r)
r is the discount as a decimal: 20% off means × 0.80.Lesson: Percents and money →
Price with tax or markup
new price = price × (1 + r)
r is the tax or markup as a decimal: 8% tax means × 1.08.Lesson: Percents and money →

Ratios and rates

Unit rate
unit rate = total amountnumber of units
How much for one: $12 for 4 pounds is $3 per pound.Lesson: Ratios, unit rates, compare deals →
Proportional relationship
y = kx k = yx
y ÷ x is the same every time; k is the constant of proportionality.Lesson: Proportional relationships →
Share in a ratio a : b
one part = totala + b
Add the parts of the ratio, divide the total by that, then multiply.Lesson: Sharing in a ratio →
Speed
speed = distancetime
How far for each hour (or second).Lesson: Ratios, unit rates, compare deals →
Density
density = massvolume
How much mass in each cm³ (or m³).Lesson: Density, pressure, population density →

Powers and roots

Power
an = a × a × … × a
a multiplied by itself n times.Lesson: Powers, square roots, absolute value →
Square root
√a = b means b2 = a
The number that times itself gives a.Lesson: Powers, square roots, absolute value →
x² = k
x = √k or x = −√k
Two answers: a square of a negative is positive too.Lesson: Solve x² = 49 and x³ = 27 →
x³ = k
x = 3√k
One answer: the cube root keeps the sign.Lesson: Solve x² = 49 and x³ = 27 →

Data

Mean
x = sum of the valuesnumber of values
Add them all, divide by how many.Lesson: Mean, median, mode, range →
Median
the middle value in order
Put the values in order; with two in the middle, take their mean.Lesson: Mean, median, mode, range →
Mode and range
range = largest − smallest
The mode is the value that comes up most often.Lesson: Mean, median, mode, range →
Mean absolute deviation
MAD = Σ|x − x|n
The mean distance of the values from their mean.Lesson: Mean absolute deviation →
Circle graph angle
angle = partwhole × 360°
Each part gets its share of the full turn.Lesson: Circle graphs →

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