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Percent change

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

Percent change

A rise in percent

A concert ticket cost 80 ₪, and its price rose by 15%.

  • In two steps: the rise is 15% of 80, 0.15×80=12\displaystyle {0.15\times80}={12} ₪, and the new price is 80+12=92\displaystyle {80+12}={92} ₪.
  • In one step: the old price is 100%, and the new one is 100%+15%=115%\displaystyle {100\%+15\%}={115\%} of it – 1.15×80=92\displaystyle {1.15\times80}={92} ₪.

A 15% rise is multiplying by 1.15.

A fall in percent

In the same way, a coat cost 300 ₪ and got a 20% discount.

  • After the discount 100%−20%=80%\displaystyle {100\%-20\%}={80\%} of the price is left: 0.8×300=240\displaystyle {0.8\times300}={240} ₪.
  • A 20% fall is multiplying by 0.8.
  • A 5% fall is multiplying by 0.95, and a 5% rise – by 1.05.
Exercise 1

Work out the amount after the change – in one multiplication.

a
240 ₪, a 15% 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

    b
    60 ₪, a 35% discount

      / – 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

      c
      1200 residents, an 8% growth

        / – 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

        d
        500 kg, a 2% 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

          The percent change
          Check: 1.15×80=92\displaystyle {1.15\times80}={92}.

          A ticket once cost 80 ₪, and now it costs 92 ₪.
          By what percent did the price rise?

          • The change: 92−80=12\displaystyle {92-80}={12} ₪.
          • What part it is of the price before the change: 1280=15100=15%\displaystyle {\frac{12}{80}}={\frac{15}{100}}={15\%}.

          When the amount before the change is a\displaystyle a, and the amount after it is b\displaystyle b:

          p%=b−aa\displaystyle p\%=\frac{b-a}{a}
          The change out of the amount before it

          The percent change is always worked out of the amount before the change.

          • From 80 to 100: a rise of 20, and 2080=25%\displaystyle {\frac{20}{80}}={25\%}.
          • From 100 to 80: a fall of 20, and 20100=20%\displaystyle {\frac{20}{100}}={20\%}.

          The same 20 ₪, but a different percent: each time it is part of a different amount.

          Exercise 2

          The price of a bag fell from 250 ₪ to 200 ₪.
          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

            Exercise 3

            Last year the robotics club had 24 children, and this year it has 30.
            By what percent did the number of children grow?

              / – 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

              A worker’s salary rose from 8000 ₪ to 8400 ₪, and another worker’s from 5000 ₪ to 5300 ₪.
              Whose salary rose by more shekels?
              And whose by a bigger percent?