Add the 10% GST to a pre-GST price, find the GST contained in a GST-inclusive total and the pre-GST price, calculate percentage increases and decreases, and combine successive percentage changes using multipliers

What this dot point is asking

NESA wants two closely linked skills here. The first is the Goods and Services
Tax (GST): a flat $10\%$ tax the Australian Government adds to most goods and
services. You must be able to add the GST to a price. Just as often, you must
work backwards from a GST-inclusive total (a price that already has the tax in
it). From that total you find either the GST it contains or the price before GST.
The second skill is percentage change: increasing or decreasing a quantity by
a percentage, and combining several such changes one after another. One idea ties
both topics together and makes them click. That idea is the multiplier: a
single number you multiply by. Every percentage change is one of these multipliers,
and that is also exactly how GST works.

This is strictly Year 11 Money Matters (MS-F1). The arithmetic is percentages you
already have; what is examined is choosing the right operation, especially the
direction of a GST calculation and what to do when changes are stacked.

The answer

Everything on this page reduces to one move: turn the percentage into a
multiplier and multiply.

• Adding the GST. The GST is $10\%$ of the pre-GST price. To add it, find
  $10\%$ and add it on, or in one step multiply the pre-GST price by $1.1$ (which
  is $100\% + 10\% = 110\%$):

• Finding the GST inside an inclusive total. Because the price was multiplied
  by $1.1$, the GST is one eleventh of the inclusive total, so divide by $11$:

• Finding the pre-GST price from an inclusive total. Undo the $\times 1.1$ by
  dividing by $1.1$:

• A percentage increase or decrease of $r$ (written as a decimal) multiplies
  the amount by $(1 + r)$ for an increase or $(1 - r)$ for a decrease:

• Successive percentage changes multiply their multipliers. Apply each change
  in turn, or just multiply the multipliers together once:

Adding the GST: multiply by 1.1

To add the $10\%$ GST you can find $10\%$ and add it on. It is faster and safer,
though, to multiply the pre-GST price by $1.1$ in a single step. This works because
$100\%$ of the price plus another $10\%$ is $110\% = 1.1$. A plumber who quotes
$680 excluding GST adds $0.10 \times 680 = 68$, i.e. $68.00 of GST, for a
total of $680 \times 1.1 = 748$, i.e. $748.00. The two routes agree because
$680 + 68 = 748$. Quotes from tradespeople are very often given "plus GST", so
adding the GST is the everyday calculation.

Working backwards: the divide-by-11 method

The trap that catches the most students is going backwards from a price that
already includes GST. Suppose a receipt total is $88.00, GST included. The GST
is not $10\%$ of $88; that would give $8.80, which is too much. The
$88 is $110\%$ of the pre-GST price, so the pre-GST price is
$88 \div 1.1 = 80$, i.e. $80.00, and the GST is the remaining
$88 - 80 = 8$, i.e. $8.00. The shortcut is to divide the inclusive total by
$11$ directly: $88 \div 11 = 8$.

Here is why $\div 11$ works. The inclusive total is the pre-GST price times
$1.1 = \tfrac{11}{10}$, and the GST is the pre-GST price times
$0.1 = \tfrac{1}{10}$. So the GST as a fraction of the inclusive total is

i.e. the GST is exactly $\tfrac{1}{11}$ of any GST-inclusive total, about
$9.09\%$ of it, never $10\%$ of it. The bar below splits a GST-inclusive grocery
and homewares receipt of $132.00 into its pre-GST price and its GST.

For the $132.00 receipt, the GST is $132 \div 11 = 12$, i.e. $12.00, and the
pre-GST price is $132 \div 1.1 = 120$, i.e. $120.00. The bar shows the GST as a
single skinny eleventh sitting beside the ten parts of pre-GST price: that picture
is the whole reason the GST is the total divided by $11$, not by $10$.

Percentage change as a multiplier

A percentage increase or decrease is the same idea as adding GST, just with
a different rate. To change an amount by $r$ (the rate as a decimal), multiply by
$(1 + r)$ to increase or $(1 - r)$ to decrease. The number you multiply by is the
multiplier. A $1450 laptop discounted by $30\%$ keeps
$100\% - 30\% = 70\%$ of its price, so the sale price is
$1450 \times 0.70 = 1015$, i.e. $1015.00, and the saving is the rest,
$1450 - 1015 = 435$, i.e. $435.00. An hourly rate of $24.80 raised by
$3.5\%$ becomes $24.80 \times 1.035 = 25.67$, i.e. $25.67. Going through the
multiplier is one calculation; finding the change and then adding or subtracting it
is two, with two chances to slip.

The multiplier also lets you find a percentage change from two prices: divide
the new by the old to get the multiplier, then read off the rate. If rent rises
from $540 to $580.50 a week, the multiplier is
$580.50 \div 540 = 1.075$, so the increase is $0.075 = 7.5\%$.

Successive percentage changes: multiply the multipliers

When two or more percentage changes happen one after another, you apply each to the
result of the previous one, never to the original. The clean way is to multiply
the multipliers together. This is where students lose marks by adding
percentages, which is always wrong. You see this stacking everywhere. A shop marks
an item up then runs a sale on the new price. A price rises one year and falls the
next. A sale takes a discount and then a further discount at the register. The
diagram traces a sound system originally $480.00 through a $20\%$ discount and
then a further $10\%$ discount.

After the first discount the price is $480 \times 0.80 = 384$, i.e. $384.00.
The further $10\%$ comes off that figure: $384 \times 0.90 = 345.60$, i.e.
$345.60. Combining the multipliers gives $0.80 \times 0.90 = 0.72$, so the two
discounts together are a single $\times 0.72$, which keeps $72\%$ of the price and
so is a $28\%$ discount overall, not the $30\%$ you would get by adding
$20\% + 10\%$. Two discounts never add, because the second one acts on an
already-reduced price.

The same logic explains why a price that goes up then down by the same percentage
does not return to where it started. Raise $200 by $10\%$ to get
$200 \times 1.1 = 220$, i.e. $220.00, then drop it by $10\%$:
$220 \times 0.9 = 198$, i.e. $198.00. The combined multiplier is
$1.1 \times 0.9 = 0.99$, a net fall of $1\%$, because the $10\%$ fall is taken
from the larger $220 and so removes more than the $10\%$ rise added.

How exam questions ask about this

Each wording points to one of the moves above. Learn the translation:

• "A $10\%$ GST is added. Find the GST / the total price." Multiply the pre-GST
  price by $0.10$ for the GST, or by $1.1$ for the total.
• "This price includes GST. How much GST is included?" Divide the inclusive
  total by $11$.
• "This price includes GST. What was the price before GST?" Divide the
  inclusive total by $1.1$.
• "Increase / decrease ... by ... %." Multiply by $(1 + r)$ to increase or
  $(1 - r)$ to decrease.
• "Find the percentage increase / decrease" (given two prices). Divide the new
  amount by the old to get the multiplier, subtract $1$, and convert to a percent;
  or divide the change by the original amount.
• "... and then ..." (one change after another). Apply each change to the
  previous result, or multiply the multipliers together.
• "What single discount has the same effect?" Multiply the multipliers, then
  the single discount is $100\%$ minus the combined multiplier as a percent.

Edge case: GST-free items and "plus GST" quotes

Two practical points sit just inside the syllabus. First, not everything carries
GST. Basic foods (plain bread, milk, fresh fruit and vegetables) and some health
and education items are GST-free, so a receipt can mix taxed and untaxed lines
and the GST is $\tfrac{1}{11}$ only of the taxable part, not of the whole
total. An exam will tell you which items are taxable. Second, watch the wording of
a price. "$680 plus GST" means $680 is the pre-GST price and you add GST to
get $680 \times 1.1 = 748$, i.e. $748.00; "$680 including GST" means the
$680 is already the inclusive total and the GST inside it is
$680 \div 11 \approx 61.82$, i.e. about $61.82. Reading which one a question
gives you decides whether you multiply by $1.1$ or divide by $11$.