Grade 7 Mathematics cheat sheet

A ratio compares two quantities by division; the ratio 3 to 4 can be written 3:4, 3/4, or 0.75, and order matters.

A rate is a ratio comparing quantities with different units; a unit rate rewrites it so the second quantity is 1, e.g. 120 miles in 3 hours is 40 miles per hour.

A proportion states two ratios are equal: a/b = c/d. Solve by cross-multiplying, ad = bc, then dividing by the remaining factor.

Two quantities are proportional only if the graph is a straight line through the origin (0, 0); a line through the origin with slope k means y = kx.

The constant of proportionality k equals y/x for every matching pair; in a table, divide each y by its x and confirm the quotient is always the same.

Percent problems are proportions: part/whole = percent/100. Find the part with part = percent x whole (using the decimal form of the percent).

Percent change = (new value - original value) / original value, expressed as a percent; a positive result is an increase, a negative result a decrease.

Unit price comparison: divide total cost by quantity to get cost per unit, then compare; the lower unit price is the better buy only when units match.

Scale drawings: the scale is a ratio of drawing length to actual length, so actual length = drawing length / scale factor when the scale is written as a fraction.

Adding rational numbers with the same sign: add absolute values and keep the sign. With opposite signs: subtract the smaller absolute value from the larger and keep the sign of the larger.

Subtracting a number equals adding its opposite: a - b = a + (-b). This converts every subtraction to an addition problem.

Multiplication and division sign rules: same signs give a positive product or quotient; opposite signs give a negative one.

A rational number can be written as a fraction p/q with integers p and q and q not 0; every terminating or repeating decimal is rational.

Converting a fraction to a decimal: divide numerator by denominator. The decimal terminates only if the reduced denominator has no prime factors other than 2 and 5.

To add or subtract fractions, rewrite with a common denominator first; to multiply, multiply numerators and denominators directly; to divide, multiply by the reciprocal.

Order of operations applies to rational numbers too: parentheses, exponents, multiply/divide left to right, then add/subtract left to right.

Absolute value is distance from 0 on the number line, so |-7| = 7 and |7| = 7; opposites have equal absolute values and sum to 0.

Combine like terms only when variable parts match exactly: 5x + 3x = 8x, but 5x + 3y cannot be combined.

Distributive property: a(b + c) = ab + ac; it also expands negatives, as in -3(x - 4) = -3x + 12.

Solve a linear equation by undoing operations in reverse order: first add or subtract, then multiply or divide, applying the same operation to both sides.

An inequality reverses its direction when both sides are multiplied or divided by a negative number: -2x > 6 becomes x < -3.

Graph an inequality on a number line with an open circle for < or > and a closed circle for <= or >=; shade in the direction of the solution.

A proportional relationship has the form y = kx with no added constant; any equation with a nonzero added term is not proportional.

Word problems: define a variable for the unknown, translate each phrase into an operation, then solve and check by substituting back into the original statement.

Factoring out the greatest common factor reverses distribution: 12x + 18 = 6(2x + 3).

Scale factor k changes length by k, area by k squared, and volume by k cubed; doubling the sides of a figure multiplies area by 4.

Circumference of a circle: C = 2 x pi x r, or C = pi x d. Area of a circle: A = pi x r squared.

Area of a triangle: A = (1/2) x base x height, where the height is perpendicular to the chosen base.

Area of a parallelogram: A = base x height; area of a trapezoid: A = (1/2) x (base 1 + base 2) x height.

Volume of a right prism or cylinder: V = area of the base x height; for a cylinder the base area is pi x r squared.

Surface area is the sum of the areas of all faces; for a rectangular prism, SA = 2(lw + lh + wh).

Angles in a triangle sum to 180 degrees; supplementary angles sum to 180 degrees and complementary angles sum to 90 degrees.

Vertical angles are equal, and a transversal cutting parallel lines creates equal alternate interior and corresponding angles.

Mean = sum of the data values divided by the number of values; it is pulled toward outliers. Median is the middle value of ordered data, or the mean of the two middle values.

Mode is the most frequent value, and range is maximum minus minimum; neither is affected by how far the extremes lie from the center.

Mean absolute deviation (MAD) measures spread: find the mean, take each value's distance from the mean, then average those distances.

A random sample must give every member of the population an equal chance of selection; biased samples over- or under-represent a group.

Probability of an event = number of favorable outcomes / total number of equally likely outcomes, so every probability lies from 0 to 1.

The complement rule: P(not A) = 1 - P(A); probabilities of all outcomes in a sample space sum to 1.

For two independent events, P(A and B) = P(A) x P(B); this multiplication rule does not apply to dependent events.

Theoretical probability comes from counting outcomes, while experimental probability comes from actual trials; more trials generally bring the experimental value closer to the theoretical one.