ICAS Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Order of operations

Evaluate brackets first, then indices, then multiplication and division from left to right, and finally addition and subtraction from left to right.

Divisibility by 3 test

A whole number is divisible by 3 exactly when the sum of its digits is divisible by 3; for example 471 gives 4 + 7 + 1 = 12.

Divisibility by 4 test

A number is divisible by 4 when its final two digits form a multiple of 4; 1,536 ends in 36, so it is divisible by 4.

HCF and LCM relationship

For two positive whole numbers, the highest common factor multiplied by the lowest common multiple equals the product of the two numbers.

Percentage change formula

Percentage change = (new value − original value) ÷ original value × 100%; a positive answer is an increase and a negative answer a decrease.

Dividing by a fraction

To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c, so 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.

Prime number definition

A prime has exactly two factors, 1 and itself; 1 is not prime, 2 is the only even prime, and 9 = 3 × 3 is composite.

Rounding to a place value

Locate the required place value, then check the digit to its right: 5 or more rounds the place value up by one, 4 or less leaves it unchanged.

nth term of arithmetic sequence

A sequence with first term a and constant difference d has nth term a + (n − 1)d; the sequence 5, 8, 11, 14 has rule 3n + 2.

Gradient-intercept form

In the rule y = mx + c, m is the gradient measuring steepness and c is the y-intercept, the value of y when x equals zero.

Expanding double brackets

(x + a)(x + b) = x² + (a + b)x + ab; multiply every term in the first bracket by every term in the second, then collect like terms.

Difference of two squares

(a − b)(a + b) = a² − b²; this identity speeds up mental arithmetic, for example 51 × 49 = 50² − 1 = 2499.

Solving a linear equation

Apply the same operation to both sides to keep the equation balanced, undoing operations in reverse order until the variable stands alone.

Combining like terms

Terms with identical variable parts can be added or subtracted, so 3x + 5x = 8x, but 3x and 3x² cannot be combined.

Second difference test

If the first differences between consecutive terms are constant the rule is linear; if the second differences are constant the rule is quadratic.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses its direction, so 2 < 5 becomes −2 > −5.

Area of a triangle

Area = ½ × base × perpendicular height; the height must be measured at right angles to the base, not along a sloping side.

Circumference of a circle

The distance around a circle is C = 2πr or C = πd, where r is the radius, d is the diameter and π is about 3.14159.

Area of a trapezium

Area = ½(a + b)h, where a and b are the lengths of the two parallel sides and h is the perpendicular distance between them.

Pythagoras' theorem

In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².

Angle sum of a triangle

The three interior angles of any triangle add to 180°, and each exterior angle equals the sum of the two interior opposite angles.

Volume of a prism

Volume = area of the uniform cross-section × length of the prism; for a cylinder this becomes V = πr²h.

Converting area units

Because each length unit is squared, 1 m² = 10 000 cm² and 1 cm² = 100 mm²; area conversion factors are the squares of length factors.

Similar figures scale factors

Similar figures have equal matching angles and sides in a common ratio k; their areas scale by k² and their volumes by k³.

Mean of a data set

Mean = sum of all values ÷ number of values; it uses every data point, so a single outlier can pull it up or down.

Median of a data set

The median is the middle value once the data are ordered; with an even number of values it is the average of the two middle values.

Range as a measure of spread

Range = largest value − smallest value; it describes spread using only two data points and ignores every value in between.

Probability complement rule

The probability an event does not occur is 1 minus the probability it does: P(not A) = 1 − P(A).

Independent events multiplication rule

Events are independent when one outcome does not affect the other; then P(A and B) = P(A) × P(B), as with repeated coin tosses.

Mutually exclusive events

Two events that cannot happen at the same time satisfy P(A or B) = P(A) + P(B), because their outcomes never overlap.

Pie chart sector angle

Sector angle = (category frequency ÷ total frequency) × 360°, so a category holding 25% of the data takes a 90° sector.

Adding a constant to data

Adding the same value k to every observation adds k to the mean and median, but the range and standard deviation are unchanged.