Mathematics, three units · Grade 11

Why does a 10 percent rise followed by a 10 percent fall not return the original price?

Because the two percentages are taken from different bases. The rise is computed on the original price, but the fall is computed on the higher price already reached, so in shekels it is the larger of the two. A hundred becomes 110 and then 99. The result is always below the original, in any order and at any percentage.

Learning objectives

Multiply, do not add

Every percentage change is a multiplication by a single number. A rise of 20 percent is a multiplication by 1.2. A fall of 20 percent is a multiplication by 0.8. That is the whole tool.

Successive changes are handled by multiplying the multipliers together, never by adding the percentages. A rise of 20 followed by a rise of 30 is 1.2 then 1.3, which is 1.56 — a rise of 56 percent, not 50.

That also answers the question in the title. 1.1 times 0.9 is 0.99, always less than one. The larger the percentage the wider the gap: 1.5 times 0.5 is 0.75, a quarter below where you started.

Compound interest

When a sum earns interest that stays in the account, next year's interest is calculated on that too. It is the same repeated multiplication: the starting sum times the multiplier raised to the number of periods.

1,000 shekels at 5 percent a year becomes, after 3 years, 1,000 times 1.05 cubed, which is about 1,157.6. Not 1,150 — the difference is the interest on the interest.

Watch the period. An annual rate of 12 percent compounded monthly is not 12 percent a month but one percent, and the number of periods is the number of months, not of years.

Decay and half-life

Decay is the same model with a multiplier below one. A population falling by 8 percent a year is multiplied by 0.92 each year, so after t years it is the starting figure times 0.92 to the power t.

A quantity that halves — a drug clearing the blood, a radioactive substance — is described by a multiplier of one half per half-life. After three half-lives an eighth remains, not a third and not nothing.

The question "after how long" needs a logarithm, so on a three-unit paper it is phrased differently: usually you are asked for the value after a given number of periods, or to test how many periods are needed. If you find yourself needing a logarithm, you have probably picked up a paper from another level.

Worked examples

  1. A price of 800 rose by 25 percent and then fell by 20 percent. What is the final price?

    1. Rise: multiply by 1.25, giving 1,000
    2. Fall: multiply by 0.8
    3. 1,000 times 0.8

    Answer: 800 — here exactly the original, because 1.25 times 0.8 equals 1

  2. 2,000 shekels are deposited at 4 percent a year. What is the balance after 5 years?

    1. The annual multiplier is 1.04
    2. The number of periods is 5
    3. 2,000 times 1.04 to the power 5

    Answer: About 2,433 shekels

  3. There are 400 mg of a drug in the blood and its half-life is 6 hours. How much is left after 18 hours?

    1. 18 over 6, which is three half-lives
    2. After one: 200
    3. After two: 100
    4. After three: 50

    Answer: 50 mg, which is one eighth of the original amount

Common mistakes

Adding the percentages of successive changes
Percentages do not add because each is computed on a different base. 20 and 30 make 56, not 50, and the gap widens as the percentages and the number of steps grow.
Computing simple interest instead of compound
With compound interest the interest stays in the account and earns interest itself. Multiplying simply by the number of years understates the total, and the shortfall grows with the term.
Assuming two half-lives clear everything
Each half-life halves what is currently there, not what was there originally. After two a quarter remains and after three an eighth; the amount approaches zero but never reaches it.

What to remember

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