⚹   Three Variables to Find

Every simple interest problem has three quantities hidden in the wording. Before computing anything, identify them:

· P — the principal (the amount borrowed, loaned, or invested)
· r — the rate, as a decimal (e.g. 4% = 0.04)
· t — the time, in years (months must be converted)

For each scenario, enter P, r (as a decimal), and t (in years).

⚹   The Simple Interest Formula

With P, r, and t identified, the interest is their product:  I = P × r × t. We'll build the whole toolkit: converting units, computing interest, finding the total to repay, and solving the reverse problems.

⚹   Compute the Interest

Given P, r (as a percent), and t (in years), compute the interest. Use I = P × r × t (remember to convert the percent to a decimal).

⚹   Forward Interest — Find I, Find Total

Forward: given P (principal), r (rate), t (time) → find interest and total. Formula: I = P × r × t. Total to repay = P + I. Two conversions before computing: percent → decimal (4% → 0.04) and months → years (18 months → 1.5 years). Skipping either conversion is the most common error.

⚹   Find the Hypotenuse

Every right triangle has three sides and one special side — the hypotenuse. It's the side opposite the right angle, and always the longest.

Click the side that is the hypotenuse in each triangle below.

⚹   Pythagoras’ Theorem

In any right triangle, a² + b² = c², where c is the hypotenuse. We'll build the skills — squaring, square roots, the "add then root" case, and the "subtract then root" case — one atom at a time.

⚹   Find the Unknown Side

Some triangles ask you to find the hypotenuse (add the squares). Others ask you to find a shorter side (subtract). The diagram tells you which.

⚹   Real Problems

Ladders, ropes, gardens, and triangles that can’t exist. Read carefully, draw if it helps.

⚹   Label the Sides

Trigonometry always starts from one angle of the triangle — usually called θ (theta). Once you pick an angle, the three sides get names relative to it:

· hypotenuse — the longest side, opposite the right angle (always the same)
· opposite — the side directly across from θ
· adjacent — the side next to θ (but NOT the hypotenuse)

Below, the angle θ is marked. Click the side that matches the label shown.

⚹   SOH · CAH · TOA

Three ratios connect an angle θ to pairs of sides:  sin = opp/hyp · cos = adj/hyp · tan = opp/adj. We'll build the whole toolkit: picking the ratio, finding a side, and finding the angle.

⚹   Find an Unknown Side

Given an angle and one side, choose the right ratio to find another side. Work to 1 decimal place.

⚹   Trigonometry In Context

Word problems. Draw a triangle first — identify θ, hypotenuse, opposite, adjacent — then pick the ratio and compute. Work to 1 decimal place.

⚹   Read the Coordinates

Every point on the Cartesian plane has two numbers: (x, y). The x comes first — how far right (positive) or left (negative) from the origin. Then y — how far up (positive) or down (negative).

A point is plotted below. Enter its coordinates as x,y — for example 3,-4.

⚹   The Coordinate Toolkit

For any two points A(x₁, y₁) and B(x₂, y₂) there are three things to calculate — the midpoint, the gradient, and the distance — all built on differences of coordinates. We'll build them one at a time, starting with the sign-handling skill that underlies all three.

⚹   Apply the Three Formulas

Six questions. Each asks for the midpoint, gradient, or distance between two points. Work carefully with negatives.

⚹   Coordinates In Context

Four word problems. Read carefully, pick the right formula, then compute. For coordinate answers, enter as x,y.

⚹   Collect Like Terms · Expand Brackets

Two moves happen before solving: collect like terms (add the coefficients of terms with the same variable) and expand brackets (multiply the outside number over every term inside).

· 3x + 5x = 8x   —  like terms add together
· 2(x + 3) = 2x + 6   —  the 2 multiplies both x and 3

Simplify each expression below. Enter without spaces (for example 5x+4, 8x-20, or -2x+6).

⚹   The Balance Method

An equation is a scale — whatever you do to one side, you must do to the other. Solve by unwrapping the variable with inverse operations. We'll build every move, one atom at a time.

⚹   Solve the Equation

Six equations, increasing in difficulty. Find the value of the pronumeral. Enter just the number (e.g. 5 or -3).

⚹   Powers & Real Problems

Four final problems: three index equations (variable to a power) and one word problem. For impossible equations (no real solution), enter none.

⚹   Rounding to Decimal Places

"Round to n decimal places" means: keep n digits after the decimal point, then look at the NEXT digit. If it's 5 or more, round up. If it's less than 5, round down.

· 8.47 rounded to 1 d.p. → look at 7 → round up → 8.5
· 8.42 rounded to 1 d.p. → look at 2 → round down → 8.4
· 8.95 rounded to 1 d.p. → 5 rounds up → 9.0 (watch for propagation!)

Round each number below to the specified decimal places.

⚹   The Foundations

Number foundations covers five distinct skills: recognising primes, following the order of operations, converting fractions, finding common factors, and solving sum-and-product puzzles. We'll drill each one before the mixed-test stage.

⚹   Multiple Choice Practice

Six mixed MCQs. Tap the option you think is correct.

⚹   Number Puzzles

Four final problems mixing sum-product puzzles, fractions, and factors.