Percentage operation gives different results with two scientific calculator

Question

The operation is: $21.99 + 9.75$$\%$

On the Texas Instruments model TI36X-Pro the answer is: $22.0875$

But when I tried on the Casio FX-991ES Plus the result is: $24.134025$

Which is the correct answer?

And how these two calculators reach different results?

Answer 1

It depends on how we 'read' $21.99 + 9.75\%$

(1) As $\%$ is derived from the latin 'per centum' (which means 'out of one hundred'), we have $$ 9.75\% = \frac{9.75}{100} = 0.00975 $$ giving answer one.

(2) Many calculators allow the input 21.99 + 9.75% being a shortcut for the operation 'add 9.75% of 21.99 to the given 21.99', which is $$ 9.75\% \cdot 21.99 = 2.144025 $$ giving the second answer.

If I should choose the 'correct' one, this would be (1), but the better way is not to use notation like that.

Answer 2

The Casio one adds $9.75\%$ of $21.99$, which is $2.144025$.

The Texas one adds $9.75\% = 0.0975$ as a plain number.

As to which one is correct, that depends on the specific question.

Answer 3

Since $9.75\% = 0.0975$, the correct answer is $22.0875$.

The number you got with the second calculator is actually $21.99 \cdot 1.0974$, which is the number $21.99$, increased by $9.75$ percent.

With more simple numbers:

If I add $\frac12$, which is $50\% $ to the number $10$, I will get $10+\frac12 = \frac{21}{2}=10.5$. However, if I increase $10$ by $50\%$, I will increase it by $5$ (because $5=\frac12 10$) and get $15$.

Answer 4

Other calculators give yet more answers:

• Simply divide argument by 100:
  ${\bf 22.0875} = 21.99 + (9.75 / 100)$
  e.g. Texas Instruments TI36X-Pro
• Use argument to add mark up¹:
  ${\bf 24.134025} = 21.99 \times (1 + 9.75 / 100)$
  e.g. Casio FX-991ES Plus, MS Windows Calculator
• Use argument to add margin:
  ${\bf 24.36565} = 21.99 \div (1 - 9.75 / 100)$
  e.g. Casio DS-20
• Total as percent of second argument:
  ${\bf 325.53846} = (21.99 + 9.75) \div 9.75 \times 100$
  e.g. Casio fx-3400P, Casio fx-115W, Casio fx-260
• Some calculators have no percentage button.
  e.g. Texas Instruments TI84 Plus

There is no consensus for the calculator keystrokes "21.99+9.75%=", and probably not² for the notation "$21.99+9.75\%$", so there isn't a right answer.

And that's just for addition³. So don't rely on the percent button without knowing what that particular model of calculator does!

<sub>1. I have found the mark up behaviour to be the most common for no-frills office calculators.
2. What would "$1.23 + 9.8\% + 3.32 + 66.4\%$" mean?
3. See also Calculators are needlessly bad by Harold Thimbleby (Int. J. Human-Computer Studies (2000) 52, 1031-1069).</sub>