To find X% of Y, multiply: X% × Y. To add X% to Y: Y × (1 + X/100). To subtract X%: Y × (1 − X/100). Example: 20% off $80 = $80 × 0.80 = $64.
To multiply a two-digit number by 11, add the digits and place the sum between them: 35 × 11 → 3(3+5)5 = 385. If the sum exceeds 9, carry over: 78 × 11 → 7(7+8)8 → 7(15)8 → 858.
When multiplying/dividing, the result should have the same number of significant figures as the least precise input. When adding/subtracting, match the least number of decimal places.
For ax² + bx + c, find two numbers that multiply to ac and add to b. Then split the middle term and factor by grouping. If b²−4ac is a perfect square, it factors nicely over integers.
To square a number ending in 5: multiply the remaining digits by (itself + 1), then append 25. Example: 85² = 8×9 = 72, so 85² = 7225. 35² = 3×4=12, so 35² = 1225.
In a normal distribution: ~68% of data falls within 1σ, ~95% within 2σ, and ~99.7% within 3σ of the mean. Useful for quick probability estimates in statistics.
1/7 = 0.142857..., 1/6 = 0.1666..., 1/8 = 0.125, 1/9 = 0.111..., 1/12 = 0.0833... Memorizing common fractions speeds up mental math significantly.
To solve proportions a/b = c/d, cross-multiply: a×d = b×c. This works for any proportion and is the fastest way to find a missing value. Example: 3/4 = x/20 → 3×20 = 4x → x = 15.
For ax² + bx + c = 0: x = (−b ± √(b²−4ac)) / 2a. The discriminant Δ = b²−4ac tells you: Δ > 0 = two real roots, Δ = 0 = one repeated root, Δ < 0 = two complex roots.
a² − b² = (a+b)(a−b). This is one of the most useful factoring patterns. Use it backwards to quickly multiply: 97 × 103 = (100−3)(100+3) = 10000 − 9 = 9991.
P(A or B) = P(A) + P(B) − P(A and B). For independent events: P(A and B) = P(A) × P(B). Complement rule: P(not A) = 1 − P(A). Total probability always equals 1.
To check if a number is divisible: by 2 (even), by 3 (sum of digits divisible by 3), by 4 (last two digits divisible by 4), by 9 (sum of digits divisible by 9), by 5 (ends in 0 or 5).
On the number line, the number to the right is always greater. Positive numbers are greater than zero, and zero is greater than negative numbers. For any two real numbers a and b:
if a − b > 0, then a > b; if a − b < 0, then a < b; if a − b = 0, then a = b.
When comparing negative numbers, the one with the smaller absolute value is greater: −3 > −7 because |−3| < |−7|. To compare fractions, convert them to a common denominator or to decimal form.
A common fraction a/b (where b ≠ 0) represents parts of a whole. Equivalent fractions represent the same value: a/b = (a·k)/(b·k) for k ≠ 0. Reducing a fraction means dividing numerator and denominator by their greatest common divisor (GCD).
Decimal fractions use place values: 0.75 = 75/100 = 3/4. To compare decimal fractions, align decimal points and compare digit by digit.
To add or subtract fractions, find the least common denominator (LCD). To multiply: (a/b)·(c/d) = (ac)/(bd). To divide: (a/b) ÷ (c/d) = (ad)/(bc).
Two quantities x and y are inversely proportional if their product is constant: x · y = k, or equivalently y = k/x, where k ≠ 0. The graph is a hyperbola with two branches.
Key properties: (1) If x is multiplied by a factor n, then y is divided by n. (2) The product of any pair of corresponding values is always k. (3) The function never reaches zero.
Typical problems: If 6 workers complete a job in 10 days, how many days for 15 workers? 6 × 10 = 15 × d, giving d = 4 days.
A linear inequality with a parameter has the form ax + b > 0 (or <, ≤, ≥). The solution depends on the parameter a: if a > 0, then x > −b/a; if a < 0, the inequality sign reverses.
For quadratic inequalities ax² + bx + c > 0, first find the discriminant D = b² − 4ac. Consider cases: D > 0 (two roots), D = 0 (one double root), D < 0 (no real roots).
The graphical method is powerful: sketch the parabola and identify where it is above or below the x-axis. When parameters are involved, consider all cases for the sign of the leading coefficient.
A rational inequality involves a fraction with polynomials, such as P(x)/Q(x) > 0. The interval method is the standard approach: (1) Find all zeros of numerator and denominator. (2) Plot on number line. (3) Determine sign in each interval. (4) Select matching intervals.
Signs alternate when crossing a root of odd multiplicity but stay the same at even multiplicity. Zeros of the denominator are never included.
Common mistakes: Multiplying both sides by a denominator without knowing its sign; forgetting that zeros of the denominator are excluded; canceling factors that change the domain.
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