Full Grade 8 test

A trapezoid midline is m = (a + b)/2. To find base b, rearrange to b = 2m − a. The midline is the average of the base lengths.

First calculate IQR = Q3 − Q1. The upper outlier fence is Q3 + 1.5 × IQR. A value is an upper outlier if it is strictly greater than this fence.

Find the lower bound from the first inequality and the upper bound from the second. Both conditions must hold. If the lower bound exceeds the upper bound, the intersection is empty.

A cone radius and height are perpendicular legs; its slant height is the hypotenuse. Thus l = √(r² + h²).

One-year total repayment = loan + loan × p/100 + fee. Compare both total repayments; a lower interest rate does not necessarily mean a lower total cost.

The lower endpoint is excluded, so the first integer solution is one greater than that endpoint. The upper endpoint is included. Express the number of consecutive integers in terms of the two endpoints and isolate the unknown endpoint.

Find vector AB by subtracting each coordinate of A from the corresponding coordinate of B. Apply the same displacement to C: add the x and y changes separately, keeping their signs. Check that subtracting C from your chosen D gives the same vector.

Find a factor common to all terms. Divide each term by it and write the quotients in parentheses: ax + ab = a(x + b). Check by expanding.

In axⁿ, a is the coefficient and n is the exponent of x. The degree of this nonzero one-variable monomial is n; the coefficient size does not change it.

Multiply each term in the first parentheses by every term in the second. (x + a)(x + b) = x² + ax + bx + ab = x² + (a + b)x + ab.

Triangle angles total 180°. Isosceles base angles are equal, so each is (180° − vertex angle)/2. In an equilateral triangle all three are equal: 180°/3 = 60°.

The opposite vector has the same magnitude and reversed direction. Both coordinates change sign: (a, b) → (−a, −b).

The magnitude of (a, b) is √(a² + b²), by Pythagoras. Magnitude is always nonnegative.

When the rate is USD per euro, multiply euros by the rate to obtain dollars. Divide dollars by the rate to obtain euros. For the same euro amount without fees, the larger USD/Eur rate gives more dollars.

The cube root ∛n is the number whose cube equals n. For example, ∛27 = 3 because 3 × 3 × 3 = 27.

Multiply the coefficients and add exponents of the shared base x: axᵐ × bxⁿ = abxᵐ⁺ⁿ. Check the coefficient sign separately.

Expand the brackets and collect like terms. Solve both parts and check whether the endpoints are included. List all integer solutions and add them; the number of solutions is not their sum.

If b is nonnegative, compare n with b²: the same comparison holds between √n and b. For example, 20 < 25 = 5², so √20 < 5.

Evaluate the roots and divide the first by the second. For a ≥ 0 and b > 0, √a ÷ √b = √(a ÷ b). For example, √36 ÷ √4 = 6 ÷ 2 = 3.

Find two numbers whose sum is s and product is p. If they are a and b, then x² + sx + p = (x + a)(x + b). Check both conditions.

Subtract vectors coordinate by coordinate: (u₁, u₂) − (v₁, v₂) = (u₁ − v₁, u₂ − v₂).

1. Subtract the first coordinate of the second vector from the first coordinate of the first vector.
2. Do the same with the second coordinates.
3. Write the two results as a new ordered pair.

Take particular care with signs: subtracting a negative coordinate becomes addition.

Sort the data. Q2 is the overall median; Q1 is the median of the lower half and Q3 the median of the upper half. In these tasks, an even-length list is split into two equal halves; the median of an even-sized group is the average of its two middle values.

Substitute the radius and π = 3.14 into V = 4πr³/3. Cube r, multiply by 4 and 3.14, then divide by 3. Use cm³.

Compound interest is calculated on the current balance each period, including earlier interest. Final amount = initial deposit × (1 + p/100)ⁿ. With a positive rate and more than one period, this exceeds the amount from simple interest.

From a² + b² = c², the missing leg is a = √(c² − b²). Subtract the known leg squared from the hypotenuse squared and take the square root.

Multiply every coordinate by k: k(a, b) = (ka, kb). A negative k also reverses the direction.

Rational numbers Q are all numbers that can be written as a ratio p / q of two integers, where q ≠ 0.

1. Check whether the number is written as a fraction of integers.
2. Make sure that the denominator is not zero.
3. If the question asks for the smallest set, compare it with the chain N ⊂ Z ⊂ Q ⊂ R.

A fraction may have a terminating or repeating decimal form; in both cases it is rational. An irrational number cannot be written in this form.

Perpendicular lengths form the legs of a right triangle. By Pythagoras, hypotenuse c = √(a² + b²). Square the lengths, add and take the square root; do not simply add the lengths.

Like terms have the same variables raised to the same powers. Add their coefficients and keep the variable part: 3x² + 5x² = 8x².

A triangle midline joins the midpoints of two sides. It is parallel to the third side and half as long. Divide the side by 2 for the midline; multiply the midline by 2 for the side.

Subtract the constant from all three parts, then divide all parts by the positive coefficient. Exclude a strict boundary and include a non-strict one.

Sort the data. Q2 is the overall median; Q1 is the median of the lower half and Q3 the median of the upper half. In these tasks, an even-length list is split into two equal halves; the median of an even-sized group is the average of its two middle values.