Simplify calculation of a deductible rate

To deduce a percentage, I often see students or salespeople tapping away at their calculators, then getting bogged down trying to string together three consecutive arithmetic operations. With a little oral preparation, it would be simpler for them to perform just one multiplication on their machine.

Value of a percentage, usual calculation

To take, for example, the 5% of $87, we can

  • multiply $87 by 5, i.e. $87×5 = $435
  • then divide by 100, i.e. $435/100 = $4.35

The calculation required two arithmetic operations: multiplication and division.

Value of a percentage, simplified calculation

To take, for example, the 5% of $87, we can

  • Instead of multiplying by 5, then dividing by 100, it's simpler to multiply by 5/100 = 0.05, an operation that can be performed mentally. Note that 0.05 can be read as five hundredths. We can also say that 5 per cent is equivalent to 0.05 per one.
  • The calculation is now reduced to a single arithmetic operation, i.e. $87×0.05 = $4.35

To summarize :

\[5 \% = \frac{5}{100} = 0.05 \]

and taking the 5% from 87 is achieved by multiplying

\[5 \% \times 87 = 0.05 \times 87 = 4.35 \]

Deduction of a percentage, usual calculation

To deduct the 5% of $87, for example, you can

  • multiply $87 by 5, i.e. $87×5 = $435
  • then divide by 100, i.e. $435/100 = $4.35
  • finally, we perform the discount: $87 - $4.35 = $82.65

This required three arithmetic operations: multiplication, division and subtraction.

If it's not the final value that needs to be calculated, but the percentage increase or decrease, the Percentage Increase Calculator will be appropriate.

Deduction of a percentage, simplified calculation

To deduct the 5% of $87, for example, we can

  • start with an oral calculation: when you deduct the 5%, you are left with the 95%.
  • The result can then be calculated directly:
\[ 87 \times 0.95 = 82.65 \]

With this approach, the calculator was used for just one arithmetic operation: multiplication, which increases speed and reduces the risk of making a mistake.

To remember

  • Take the 5% of a quantity means
    multiply the quantity by 5/100, i.e.
    multiply the quantity by 0.05.
  • Deduct the 5% from a quantity means
    take the 95% from the quantity, i.e.
    multiply the quantity by 0.95.

Mathematical formulation

The simplified calculation is based on the mathematical concept of the variation factor, also known as the growth or decrease factor, depending on the context.

When a quantity changes from an initial value to a final value, the variation factor is the number by which the initial value must be multiplied to obtain the final value.

\[ \begin{equation*} \begin{aligned} t &= \text{percentage} \\ i &= \frac{ t }{ 100 } = \text{rate} \\ r &= 1 + i = \text{variation factor} \\ (\text{final value}) &= r \times (\text{initial value}) \end{aligned} \end{equation*} \]

These formulas show, for example, that a 100% increase corresponds to a doubling in value : t = 100, i = 1 and r = 2.

Numerical example with a negative rate expressing a decrease :

\[ \begin{equation*} \begin{aligned} t &= -5 = \text{percentage}\\ i &= -0.05 = \text{rate} \\ r &= 0.95 = \text{variation factor} \\ 82.65 &= 0.95 \times 87 \end{aligned} \end{equation*} \]
  • When the rate is positive, the variation factor is greater than 1. In this case, it's also called the growth factor.
  • When the rate is negative, the variation factor is between 0 and 1. In this case, it is also called the decrease factor.
Contact  |  Home