The Mathematics 11
Amount, Base, and Rate
Sorting out a percent problem always involves correctly attaching numbers to three fundamental
quantities: the percent (rate), the base value to which the percent rate is applied, and the
resulting amount of material or amount of change. Here we will use the full words to represent
each quantity, but when you’ve developed some skill in algebra, you could use shorter symbols
(such as A for “amount,” b for “base,” and r for “rate”).
The percent rate is just the percent value with which we are dealing. So when we say
“the metal alloy is 10% zinc,” the percent rate of zinc in the alloy is equal to 10%
“Hank has received a 5% salary increase,” the percent rate of change in Hank’s salary is
equal to 5%.
When it comes time to do calculations involving the percent rate, we always use the fractional or
decimal value corresponding to the percent rate.
The base value is the value of the quantity to which the percent rate is applied. Every percent
rate is always associated with a very specific base – apart from this base value, a percent rate
has no meaning. The easiest way to identify the base quantity is to ask “percent of what?” The
“what” here must be a numerical quantity. So when we say
“the metal alloy is 10% zinc,” the base that the percent rate refers to is the total mass of
metal alloy. (Ask: Zinc is 10% of what? Answer: It is the total mass of the metal alloy
that zinc makes up 10%.)
“Hank has received a 5% salary increase” the base to which the percent rate is applied is
his previous salary. (Ask: The increase is 5% of what?” Answer: It is Hank’s previous
salary that the increase is 5% of.)
Sometimes when people are speaking of percent change, they use the term original value
instead of base. The two terms mean exactly the same thing.
The amount is the actual quantity of stuff or quantity of change that is represented by the stated
percent of the base. The amount and the base, as these words are used here, always have the
same units (and in fact, are almost always quantities that have units of measurement). So, when
“the metal alloy is 10% zinc,” the amount of zinc is the actual number of kilograms (say)
of zinc in the kilograms of alloy that is present.
“Hank has received a 5% salary increase,” the amount of this increase is the additional
dollars and cents he receives for each hour, say, of work.
The relationship between these three fundamental characteristics of a percent problem is very
Amount = base × rate
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If you know a bit of algebra already, you know that this formula can easily be rearranged to give
base = and rate =
Thus, if we know any two of these three quantities, it is easy to calculate the third one. In all of
these formulas, “rate” stands for the fractional or decimal form of the percent value.
Example 1: Calculate 67% of 586.
Clearly 67% is the rate here. Since the value 586 follows the word “of” in “67% of 586,” 586 must
be the base to which the percent refers. Then, the question is asking for the amount
corresponding to 67% of 586. So
Amount = base x rate
= 586 x 0.67
Note that we used rate = 0.67, the decimal equivalent of 67%. So, to respond to the instruction,
67% of 586 is 392.62.
You might get a better sense of the way language is being used here if you note that 67% is
equivalent to 67/100 or about two-thirds as a fraction. When we say, “calculate two-thirds of
586,” we mean multiply 586 by two-thirds. In a similar way, since 67% is equivalent to the
decimal value 0.67, to “calculate 67% of 586” is to “calculate 0.67 of 586,” and indicates the result
of multiplying 586 by 0.67.
Example 2: Hank buys a DVD player on sale. Before tax, he is given a 30% discount which
reduces the price by $68.99. What was the original price of the DVD player?
The rate here is 30% or 0.30. This 30% discount is applied to the original price of the DVD player
(so the original price must be the base corresponding to the rate of 30% -- this means that the
question is asking us to find a base value). The $68.99 is the amount of dollars that the 30% of
the original (base) price “amounted” to. So, we are told the rate and the amount, and we are
asked to find the base. Using
= 229.96666 $
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Now, prices of DVD players do not normally include fractions of cents. So, at the least we should
round our final answer off to the nearest cent. To respond to the question, then we can state that
it appears that the original price of the DVD player must have been $229.97.
(Note that you can check this answer. Just calculate how much money that 30% of $229.97
amounts to. You should get $68.99 if we haven’t made a mistake here.)
Example 3: Hank buys a house for $227,000. One month later he sells the house for an
additional $15,500 more than he paid for it. By what percent did the price of the house increase?
The question asks us for a percent, which means we are being asked to calculate a rate. The
formula we have for a rate is
So, to be able to calculate a rate, we need to identify the values of the relevant base and amount.
Recalling that the base is the original value of whatever is changing, we conclude that the
appropriate base value here must be the original price of the house, $227,000. The “Amount” is
the amount by which the base values changes. Here the original price of the house increased by
$15,500, so this is the “Amount” in the formula. Substituting these values into the formula gives
rate = = ≅ 0.06828
rounded to five decimal places. To get the percent equivalent of this decimal value, we need to
multiply by 100:
0.06828 x 100 ⇒ 6.828%
Thus, the price of the house increased by 6.828%.
Example 4: If the price of a new car is $27,545, and tax is payable on this car at a rate of 14.5%,
how much tax must the purchaser actually pay?
The rate of tax is 14.5%. This rate of taxation is based on the actual price of the car. The tax
paid, the amount of tax paid in dollars is then
Amount = base x rate
= $27,545 x 0.145
Example 5: One day Hank spent $42.50 on gasoline and $123.75 on a textbook. What percent
of his purchases that day was for gasoline?
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There’s a bit of a twist here, but it’s not too hard to sort things out. Clearly, we are asked for a
percent, so we need to calculate a rate, and the formula for that is
The part of his expenses for which a rate is requested is the part Hank spent on gasoline. So
Amount = $42.50
The base value to which this percent applies must then be the total amount of money Hank spent
base = $42.50 + $123.75 = $166.25
rounded to four decimal places.
This decimal fraction corresponds to 25.56%. So, 25.56% of Hank’s expenses for that day were
Example 6: The label on a 1.5 litre bottle of wine that Hank buys states that it is 11.7% alcohol
by volume. How many millilitres of alcohol are present in that bottle of wine.
Here we have a straightforward request to calculate the Amount. When we are told the alcohol is
11.7% of the total of 1.5 litres of wine in the bottle, we get that the amount of alcohol present must
Amount = base x rate
= (1.5 litres)(0.117)
= 0.1755 litres
Notice that we carried units along where appropriate (the base quantity here was a number of
litres of wine, and hence the Amount we calculated carries with it units of litres). But the question
asked for the answer as millilitres of alcohol, so we must do a unit conversion here:
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Amount = 0.1755 litres × = 175.5 millilitres
Thus, this bottle of wine contains 175.5 millilitres of alcohol.
NOTE: A method for carrying out unit conversions correctly and reliably is described in detail in
other documents on this site. If you haven’t studied that method, you can still easily calculate the
answer above because if involves a very simple unit conversion.
1 litre = 1000 millilitres (by definition)
0.1755 litres = 0.1755 x 1000 millilitres
= 175.5 millilitres
This rather informal approach will do here – for more involved unit conversions you’ll definitely
want to use the more systematic method described in the documents on unit conversion
elsewhere in these notes.
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