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Buying and selling: discounts and VAT

Finance and economics, grade 10 (3 units) Lesson 7 of 15Learning

Buying and selling: discounts and VAT

A discount in one multiplication

A p%\displaystyle p\% discount leaves 100%−p%\displaystyle 100\%-p\% of the price, so the price after the discount is one multiplication.

A pair of shoes costs 350 ₪, with a 20% discount.

  • 100%−20%=80%\displaystyle {100\%-20\%}={80\%} of the price is left.
  • The price after the discount: 0.8×350=280\displaystyle {0.8\times350}={280} ₪.
  • The discount itself: 350−280=70\displaystyle {350-280}={70} ₪, which is 20%\displaystyle 20\% of 350.

When a\displaystyle a is the price before the discount and b\displaystyle b the price after it:

b\displaystyle b =\displaystyle = (100%−p%)⋅a\displaystyle (100\%-p\%)\cdot a
Exercise 1

Work out the price after the discount – in one multiplication.

a
240 ₪, a 15% 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

    b
    1250 ₪, a 30% 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
      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

        VAT

        VAT (value added tax) is a tax added to the price of most goods and services, which the business passes on to the state. In Israel VAT is 18% of the price.

        • A price in a shop already includes VAT.
        • A business’s quote to another business, or a tradesperson’s, is often written “before VAT” – and 18% is added to it.
        • The price with VAT is 100%+18%=118%\displaystyle {100\%+18\%}={118\%} of the price without VAT: multiplying by 1.18.

        Renovating a kitchen costs 2500 ₪ before VAT.
        With VAT: 1.18×2500=2950\displaystyle {1.18\times2500}={2950} ₪, and of that the VAT is 2950−2500=450\displaystyle {2950-2500}={450} ₪.

        b\displaystyle b =\displaystyle = 1.18⋅a\displaystyle 1.18\cdot a
        Exercise 2

        Work out.

        a
        The price with VAT of a computer that costs 1800 ₪ before VAT

          / – 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
          The price with VAT of a repair that costs 650 ₪ before VAT

            / – 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
            How many shekels of VAT you pay on a job that costs 3000 ₪ before VAT

              / – 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 price before the change

              When you know the price after the change, the price before it is the unknown of an equation.

              Headphones cost 590 ₪ in a shop, with VAT. What is their price without VAT?

              • Call the price without VAT x\displaystyle x. With VAT it is 1.18x\displaystyle 1.18x, so 1.18x=590\displaystyle {1.18x}={590}.
              • Divide both sides by 1.18: x=500\displaystyle {x}={500} ₪.
              • Check: 1.18×500=590\displaystyle {1.18\times500}={590}.

              Careful: the price without VAT is not 590 minus 18% of 590 – the 18% is out of the price before VAT, not out of 590.

              In the same way, after a 20% discount a bag costs 160 ₪: 0.8x=160\displaystyle {0.8x}={160}, and x=200\displaystyle {x}={200} ₪.

              Exercise 3

              Write an equation and solve it.

              a
              A fridge costs 944 ₪ with VAT. What is its price without VAT?
                b
                A garage’s bill is 1416 ₪ with VAT. What is the amount without VAT?
                c
                After a 25% discount, a coat costs 270 ₪. How much did it cost before the discount?
                Two discounts one after another

                Two discounts one after another make a discount smaller than the sum of their percents, because the second is taken from a price that has already gone down.

                A clothes shop has a 20% discount, and at the till you get another 10% off the price after the first discount.
                In the example below we work out the price of a dress that costs 500 ₪, one discount after the other:

                • After the first discount: 0.8×500=400\displaystyle {0.8\times500}={400} ₪.
                • The second discount is 10% of 400, not of 500: 0.9×400=360\displaystyle {0.9\times400}={360} ₪.
                • In one multiplication: 0.8×0.9=0.72\displaystyle {0.8\times0.9}={0.72} – you pay 72% of the price, and the whole discount is 28%, not 30%.
                Offers in percent

                An offer can be written as a discount in percent on everything you buy in it.

                “The second at half price”, on two items of 100 ₪:

                • You pay 100+50=150\displaystyle {100+50}={150} ₪ instead of 200 ₪.
                • The discount: 50 ₪ out of 200 ₪, 50200=25%\displaystyle {\frac{50}{200}}={25\%}.

                “Buy one, get one free” – you pay for one item and get two: a 50% discount on both.
                But the discount is worth something only to someone who needs the second item.

                Exercise 4

                A coat costs 400 ₪ in two shops.
                Shop A has a 30% discount.
                Shop B has a 20% discount, and at the till another 10% off the price after the discount.
                Work out the price in each shop.
                In which shop is the coat cheaper, and by how much?

                a
                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

                  b
                  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

                      / – 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 5

                      A shirt costs 120 ₪, and Dan wants to buy three shirts.
                      One shop has a “3 for 2” offer: you buy three and pay for two.
                      Another shop has a 30% discount on every shirt.
                      In which shop will Dan pay less?
                      What percent discount is the “3 for 2” offer?

                        / – 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