Simulic
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Percentages with a bar model – parts, wholes, multipliers and compound interest

MathematicsNumbers & ArithmeticAges 11–12

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On the Percentage bar screen, drag along a bar split into equal parts or across a 10 × 10 grid to find p% of an amount, find the whole from a part, or find what percentage one amount is of another, including percentages over 100%, with the unitary method shown. The Change screen shows a percentage increase or decrease as a multiplier 1 ± r/100 for discounts, tax, profit and loss, solves reverse percentage problems and finds the percentage change. The Successive screen shows why +20% then −20% does not return to the start. The Interest screen compares simple and compound interest (and straight-line with reducing-balance depreciation) on a graph and a year-by-year table.

Lesson: Percentages, percentage change and interest

What it shows

A percentage is a number of hundredths, so p% of an amount W is p/100 × W, and the whole is always 100%. Increasing by r% multiplies by 1 + r/100 and decreasing by r% multiplies by 1 − r/100, so a reverse percentage problem divides by the multiplier. Successive changes multiply their multipliers: +20% then −20% gives 1.2 × 0.8 = 0.96, an overall fall of 4%. Simple interest adds the same amount every year, A = P(1 + nr/100), while compound interest multiplies by the same factor every year, A = P(1 + r/100)ⁿ, so it grows faster and faster.

How to use

On Percentage bar, choose what to Find, switch between Bar and 10 × 10 grid, type the Whole or Part and drag the orange handle or the Percentage slider. On Change, pick a Situation, choose what to Find, set the Direction, the amount and the Rate. On Successive, set Change 1 and Change 2 or press a preset. On Interest, set Principal P, Rate per year and Years, tick Depreciation and press Play by year.

Parameters you can change

  • Screen Percentage bar, Change, Successive, Interest
  • Quantity to find (Percentage bar) Part, Whole (100%), Percentage
  • Representation (Percentage bar) Bar, 10 × 10 grid
  • Starting whole (Percentage bar) 1–10000
  • Starting percentage (Percentage bar) 0–200 %
  • Quantity to find (Change) New value, Original value, Percentage change
  • Situation (Change) General, Discount, Adding tax, Profit and loss
  • Direction (Change) Increase, Decrease
  • Rate of change (Change) 0–100 %
  • Change 1 (Successive) -50–50 %
  • Change 2 (Successive) -50–50 %
  • Principal (Interest) 1–100000
  • Interest rate per year (Interest) 0.5–20 %
  • Number of years (Interest) 1–30

Questions to explore

  1. Why is 25% off followed by 25% on not the same as the original price?
  2. A price after a 20% increase is 96. Why is the original price 80 and not 76.8?
  3. Why does compound interest grow faster than simple interest over many years?

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