to calculate
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to calculate
To determine a result through a mathematical calculation; to calculate something.
Können Sie mir bitte die genaue Summe aller Kosten errechnen?
Could you please calculate the exact total of all the costs for me?
'Errechnen' (to calculate) has the inseparable prefix 'er-'. This prefix often means that you achieve a result through an action. So 'errechnen' means to arrive at a result by calculating.
Der Computer hat den schnellsten Weg für uns errechnet.
The computer calculated the fastest route for us.
The verb 'errechnen' (to calculate) is transitive and always needs an accusative object: the thing being calculated. Optionally, a person in the dative can be added, for whom something is being calculated.
Ich habe den Gesamtbetrag errechnet.
I calculated the total amount.
Common objects for 'errechnen' (to calculate) are financial or mathematical values. Typical combinations are 'die Kosten errechnen' (to calculate the costs), 'den Gewinn errechnen' (to calculate the profit), 'den Durchschnitt errechnen' (to calculate the average), or 'den Wert errechnen' (to calculate the value).
Das Unternehmen muss den zu erwartenden Gewinn für das nächste Quartal errechnen.
The company has to calculate the expected profit for the next quarter.
to determine / investigate
'ermitteln' and 'errechnen' can both refer to finding a result or a value. 'errechnen' clearly says that the result comes from a calculation. 'ermitteln' is more general: a value can be determined by calculating, measuring, comparing, or checking. So if it is only about mathematics or a calculation, 'errechnen' fits very well. If the method is not only calculation, or if you do not want to name it exactly, 'ermitteln' is more natural.
Wenn ihr die Strecke und die Zeit kennt, könnt ihr die Geschwindigkeit errechnen.
If you know the distance and the time, you can calculate the speed.
Nach der Arbeit ermittelt Lisa den Bedarf an Material für die nächste Woche.
After work, Lisa determines how much material will be needed for the next week.