The Ultimate PSAT 10 Math Formula Cheat Sheet

The PSAT 10 Math covers many topics, from as early as elementary school all the way to high school.

While you have probably learned many of these formulas at some point, it may have been a long time since you’ve actually used them. This is where most test takers have a hard time preparing for the test.

So, what formulas do you need to have memorized for the PSAT 10 Math before the test day?

Following is a quick formula reference sheet that lists all important PSAT 10 Math formulas you MUST know before you sit down for the test.

If you learn every formula in this PSAT 10 Math Formula Cheat Sheet, you will save yourself valuable time on the test and probably get a few extra questions correct.

Looking for a comprehensive and complete list of all PSAT 10 Math formulas? Please have a look at the PSAT 10 Math Formulas

Decimals

Is a fraction written in a special form? For example, instead of writing \(\frac{1}{2}\) you can write \(0.5\).

Mixed Numbers

A number is composed of a whole number and a fraction. Example: \(2 \frac{2}{ 3}\) Converting between improper fractions and mixed numbers: \(a \frac{c}{b}=a+\frac{c}{b}= \frac{ab+ c}{b}\)

Factoring Numbers

Factor a number means breaking it up into numbers that can be multiplied together to get the original number. Example:\(12=2×2×3\)

Integers

\( \{…,-3,-2,-1,0,1,2,3,…\} \)
Includes: zero, counting numbers, and the negative of the counting numbers

Real Numbers

All numbers that are on a number line. Integers plus fractions, decimals, and irrationals, etc.) (\(\sqrt{2},\sqrt{3},π\), etc.)

Order of Operations

PEMDAS
(parentheses/ exponents/ multiply/ divide/ add/ subtract)

Absolute Value

Refers to the distance of a number from, the distances are positive as the absolute value of a number cannot be negative. \(|-22|=22\)

Ratios

A ratio is a comparison of two numbers by division. Example: \(3: 5\), or \(\frac{3}{5}\)

Percentages

Use the following formula to find part, whole, or percent
part \(=\frac{percent}{100}×whole\)

Proportional Ratios

A proportion means that two ratios are equal. It can be written in two ways:
\(\frac{a}{b}=\frac{c}{d}\), \(a: b = c: d  \)

Percent of Change

\(\frac{New \ Value \ – \ Old \ Value}{Old Value}×100\%\)

Expressions and Variables

A variable is a letter that represents unspecified numbers. One may use a variable in the same manner as all other numbers: Addition: \(2+a\): \(2\) plus a
Subtraction: \(y-3\): \(y\) minus \(3\)
Division: \(\frac{4}{x}\): 4 divided by x
Multiplication: \(5a\): \(5\) times a

Distributive Property

\(a(b+c)=ab+ac\)

Equations

The values of the two mathematical expressions are equal.
\(ax+b=c\)

Distance from A to B:

\(\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2 }\)

Parallel and Perpendicular lines:

Parallel lines have equal slopes. Perpendicular lines (i.e., those that make a \(90^° \) angle where they intersect) have negative reciprocal slopes: \(m_{1}\).\(m_{2}=-1\).
Parallel Lines (l \(\parallel\) m)

Mid-point of the segment AB:

M (\(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\))

Slope of the line:

\(\frac{y_{2}- y_{1}}{x_{2} – x_{1} }=\frac{rise}{run}\)

Point-slope form:

Given the slope m and a point \((x_{1},y_{1})\) on the line, the equation of the line is
\((y-y_{1})=m \ (x-x_{1})\).

Slope-intercept form:

given the slope m and the y-intercept b, then the equation of the line is:
\(y=mx+b\).

Factoring:

“FOIL”
\((x+a)(x+b)\)
\(=x^2+(b+a)x +ab\) “Difference of Squares”
\(a^2-b^2= (a+b)(a-b)\)
\(a^2+2ab+b^2=(a+b)(a+b) \)
\(a^2-2ab+b^2=(a-b)(a-b)\) “Reverse FOIL”
\(x^2+(b+a)x+ab=\) \((x+a)(x+b)\)

Exponents:

Refers to the number of times a number is multiplied by itself.
\(8 = 2 × 2 × 2 = 2^3\)

Scientific Notation:

It is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.
In scientific notation all numbers are written in this form: \(m \times 10^n\)
Scientific notation:
\(5×10^0\)
\(-2.5×10^4\)
\(5×10^{-1}\)
(2.122456×10^3)

Square:

The number we get after multiplying an integer (not a fraction) by itself. Example: \(2×2=4,2^2=4\)

Square Roots:

A square root of \(x\) is a number r whose square is \(x: r^2=x\)
\(r\) is a square root of \(x\)

Pythagorean Theorem:

\(a^2+b^2=c^2\)

Triangles

Area: \(A=\frac{1}{2}bh\) where \(b\) is the base and \(h\) is the height.
Perimeter: \(P=a+b+c\) (sum of all three sides).
Pythagorean Theorem (right triangles): \(a^2+b^2=c^2\) where \(c\) is the hypotenuse.
Sum of interior angles: \(180°\)

All triangles:

Area \(=\frac{1}{2}\) b. H
Angles on the inside of any triangle add up to \(180^\circ\).

Equilateral:

These triangles have three equal sides, and all three angles are \(60^\circ\).

Isosceles:

An isosceles triangle has two equal sides. The “base” angles (the ones opposite the two sides) are equal (see the \(45^\circ\) triangle above).

Circles

Area \(=πr^2\)
Circumference \(=2πr\)
Full circle \(=360^\circ\)

Rectangles

(Square if l=w)
Area=lw

Parallelogram

(Rhombus if l=w)
Area=lh
Regular polygons are n-sided figures with all sides equal and all angles equal.
The sum of the inside angles of an n-sided regular polygon is
\((n-2).180^\circ\).

Area of a trapezoid:

\(A =\frac{1}{2} h (b_{1}+b_{2})\)

Surface Area and Volume of a Rectangular/right prism:

\(SA=ph+2B\)
\(V=Bh\)

Surface Area and Volume of a Cylinder:

\(SA =2πrh+2πr^2\)
\(V =πr^2 h  \)

Surface Area and Volume of a Cone

\(SA =πrs+πr^2\)
\(V=\frac{1}{3} \ πr^2 \ h\)

Surface Area and Volume of a Sphere

\(SA =4πr^2\)
\(V =\frac{4}{3} \ πr^3\)
(p \(=\) perimeter of base B; \(π ~ 3.14 \))

Simple interest:

\(I=prt\)
(I = interest, p = principal, r = rate, t = time)

mean:

mean: \(\frac{sum \ of \ the \ data}{of \ data \ entires}\)

mode:

value in the list that appears most often

range:

largest value \(-\) smallest value

Median

The middle value in the list (which must be sorted)
Example: median of
\( \{3,10,9,27,50\} = 10\)
Example: median of
\( \{3,9,10,27\}=\frac{(9+10)}{2}=9.5 \)

Average

\( \frac{sum \ of \ terms}{number \ of \ terms}\)

Average speed

\(\frac{total \ distance}{total \ time}\)

Probability

\(\frac{number \ of \ desired \ outcomes}{number \ of \ total \ outcomes}\)
The probability of two different events A and B both happening is:
P(A and B)=p(A).p(B)
as long as the events are independent, meaning one does not change the chance of the other. (For mutually exclusive events, P(A and B) = 0.)

Powers, Exponents, Roots

\(x^a.x^b=x^{a+b}\)
\(\frac{x^a}{x^b} = x^{a-b}\)
\(\frac{1}{x^b }= x^{-b}\)
\((x^a)^b=x^{a.b}\)
\((xy)^a= x^a.y^a\)
\(x^0=1\)
\(\sqrt{xy}=\sqrt{x}.\sqrt{y}\)
\((-1)^n=-1\), if n is odd.
\((-1)^n=+1\), if n is even.
If 0 < x < 1, then
0 < x³ < x² < x < √x < ∛x < 1.

Simple Interest

The charge for borrowing money or the return for lending it.
Interest = principal \(×\) rate \(×\) time
OR
\(I=prt\)

Powers/ Exponents

\(x^a×x^b=x^{a+b}\)
\(\frac{x^a}{x^b}=x^{a-b}\)
\((x^a)^b=x^{ab}\)
\(x^0=1\)
\(x^{-a}=\frac{1}{x^a}\)
\(x^{\frac{1}{n}}=\sqrt[n]{x}\)

Positive Exponents

An exponent is simply shorthand for multiplying that number of identical factors. So \(4^3\) is the same as (4)(4)(4), three identical factors of 4. And \(x^3\) is just three factors of x, \((x)(x)(x)\).

Negative Exponents

A negative exponent means to divide by that number of factors instead of multiplying.
So \(4^{-3}\) is the same as \( \frac{1}{4^3}\) and
\(x^{-3}=\frac{1}{x^3}\)

Factorials

Factorial- the product of a number and all counting numbers below it.
8 factorial \(=8!=\)
\(8×7×6×5×4×3×2×1=40,320\)
5 factorial \(=5!=\)
\(5×4×3×2×1=120\)
2 factorial \(=2!=2× 1=2\)

Multiplying Two Powers of the SAME Base

When the bases are the same, you find the new power by just adding the exponents
\(x^a.x^b=x^{a+b }\)

Powers of Powers

For the power of power: you multiply the exponents.
\((x^a)^b=x^{(ab)}\)

Dividing Powers

\(\frac{x^a}{x^b} =x^a x^{-b}= x^{a-b}\)

The Zero Exponent

Anything to the 0 power is 1.
\(x^0= 1\)