数える and 計算: counting items and calculating
Distinguish counting actual items from calculating with given numbers. A request to count or a calculated total does not establish that every item was physically checked.
数える counts individual objects
数える, read かぞえる, means to count. 箱の中の鉛筆を一本ずつ数えるよう頼まれました。 says that someone was asked to count the pencils in the box one by one.
The completed event in this sentence is receiving the request. 数えるよう頼まれました alone does not establish that all the pencils have already been counted.
計算 works with numbers
計算, read けいさん, means calculation. 買い物の合計金額を暗算で計算した。 says that someone calculated the shopping total mentally.
計算の途中で符号を間違えた。 says that someone made a sign error during the calculation. Performing a calculation and obtaining a correct result are separate matters to check.
Separate given numbers from a physical check
If there are 8 boxes with 5 pencils in each, you can calculate 8 times 5 to obtain 40 pencils. This calculation uses the given numbers.
Obtaining 40 does not establish that someone opened every box and counted each pencil. Look for an actual checking action before concluding that the objects were physically counted.
Does calculating 40 pencils from 8 boxes with 5 pencils each mean all the pencils were directly counted?
Show the answer and why
No. That is 計算 using given numbers. Check separately whether someone completed the action of 数える the actual items one by one.