A tree is at present 9 feet tall. If every year it grows 1/9th of its height, what will be the height of the tree after three years?

Correct — B, 12·34 feet. Growing by one-ninth of its own height each year means the height is multiplied by 1 + 1/9 = 10/9 every year, exactly as compound interest multiplies a principal by 1 + r. After three years the height is 9 × (10/9)³. The 9 in front cancels one factor of 9 in the denominator, leaving 1000/81. Dividing, 81 × 12 = 972, so 1000/81 = 12·345…, which the paper prints as 12·34 feet. The trap is to compute one year's growth, 9 × 1/9 = 1 foot, and then add a foot three times to reach 12 — that treats the growth as a fixed quantity when the stem ties it to the current height, which rises each year.

• (a)12 feet — The simple-interest answer. It adds one foot a year, the growth of the first year only, and ignores that in the second year one-ninth is taken of 10 feet and in the third of 11·11 feet.
• (c)13 feet — Above the true value. Even a fourth year of growth would only take the tree to about 13·7 feet, so 13 exactly corresponds to no whole number of years.
• (d)13·10 feet — Overshoots by about three-quarters of a foot. It is roughly what you get by adding a further ninth after the correct three-year figure, that is by counting four years of growth.

Any quantity that grows by a fixed fraction of its current size is a geometric progression, and the same formula covers compound interest, population growth, depreciation and this tree: final = initial × (1 + r)ⁿ. Growth by a fixed amount each period is arithmetic instead, and the two diverge quickly — after three years at one-ninth, the tree stands about a third of a foot taller than steady one-foot growth would leave it.

The arithmetic is friendlier than it looks if you keep the fraction rather than converting to decimals. 9 × (10/9)³ = 10³/9² = 1000/81, and 1000/81 is a value worth recognising: 81 × 12 = 972 with 28 left over, so the answer is a little over 12·34. Year by year the heights run 9, 10, 11·11 and 12·35, and writing those four numbers down takes about as long as the formula does. A note on the printed options — this paper writes decimal points as raised dots, so 12·34 means twelve point three four feet.

• Growth by 1/9 of the current height multiplies the height by 10/9 each year.
• After n years the height is 9 × (10/9)ⁿ; for n = 3 this is 1000/81 feet.
• 1000/81 = 12·345…, printed in the paper as 12·34 feet.
• The year-by-year heights are 9, 10, 11·11 and 12·35 feet.
• Adding a fixed one foot a year would give 12 feet, which is the arithmetic-growth answer offered as a distractor.

• Adding the first year's growth three times over.
• Taking one-ninth of the original height in every year instead of the current height.
• Rounding 1000/81 down to 12·3 and then failing to separate the two nearby options.

Compound growth wearing a different costume. Any stem that says a quantity grows by a fraction of itself is asking for the multiplier form.

• practice — not a real PYQ

  A plant 8 cm tall grows by one-quarter of its height each month. Its height after two months is

  • (a)10 cm
  • (b)12 cm
  • (c)12·5 cm
  • (d)16 cm

  Answer(c) 12·5 cm — 8 × (5/4)² = 8 × 25/16 = 12·5.

• practice — not a real PYQ

  A machine loses one-tenth of its value every year. If it is worth ₹1,00,000 today, its value after two years is

  • (a)₹80,000
  • (b)₹81,000
  • (c)₹82,000
  • (d)₹90,000

  Answer(b) ₹81,000 — the multiplier is 9/10 each year, so 1,00,000 × 81/100.