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Percent changes one after another

Ratio, proportion and percent, grade 8 Lesson 10 of 12Learning

Percent changes one after another

A rise and a fall by the same percent

A shirt cost 100 ₪. Its price rose by 10%, and then fell by 10%.
The price did not go back to 100 ₪:

  • After the rise: 1.1×100=110\displaystyle {1.1\times100}={110} ₪.
  • The fall is 10% of 110, that is 11 ₪: 0.9×110=99\displaystyle {0.9\times110}={99} ₪.

The rise was 10% of 100, and the fall 10% of a bigger amount, so the fall is bigger than the rise.

Several changes: one multiplication after another

Every percent change is a multiplication, so several changes one after another are one multiplication after another:

100×1.1×0.9=100×0.99=99100\times1.1\times0.9=100\times0.99=99
  • 1.1×0.9=0.99\displaystyle {1.1\times0.9}={0.99}: a 10% rise followed by a 10% fall are together a 1% fall.
  • In multiplication the order does not matter: a 10% fall followed by a 10% rise gives the same price, 99 ₪.
Exercise 1

Work out the final price.

a
200 ₪, a 10% rise followed by a 10% fall

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b
    500 ₪, a 20% rise followed by a 5% rise

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      A discount on a discount

      At the end of the season a shop gives 20% off everything, and at the till another 10% is taken off the discounted price.
      Is that a 30% discount?

      0.8×0.9=0.720.8\times0.9=0.72

      You pay 72% of the price, so the discount is 28% – not 30%. The second 10% is taken off a price that has already fallen.

      Exercise 2

      In shop A there is a 40% discount on a coat that costs 400 ₪.
      In shop B the same coat has a 30% discount, followed by another 10% discount.
      In which shop is the coat cheaper, and by how many shekels?

      Shop A

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        Shop B

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Difference

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            A change repeated every year

            A town had 50,000 residents, and every year the number of residents grows by 10%.

            • After a year: 1.1×50,000=55,000\displaystyle {1.1\times50{,}000}={55{,}000}.
            • After two years: 1.1×55,000=60,500\displaystyle {1.1\times55{,}000}={60{,}500} – that is, 1.1×1.1×50,000\displaystyle 1.1\times1.1\times50{,}000.

            1.1×1.1=1.21\displaystyle {1.1\times1.1}={1.21}: in two years the number of residents grew by 21%, not 20% – in the second year the 10% is of 55,000.

            Exercise 3

            A savings account holds 2000 ₪, and every year the amount grows by 5%.
            How much money will be in the account after three years?

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              Exercise 4
              Try it with a price of 100 ₪.

              The price of a phone rose by 25%.
              Some time later its price fell, back to exactly the price before the rise.
              By what percent did the price fall?

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power