Question

Fill in the blanks with the correct Roman numerals.
\[\overline{XXXVII}CCXIX\div \_\_\_\_\_\_=CDIX\]

Answer 1

Hint: To solve calculations involving Roman numbers, we have to follow some steps as following:
Step 1: Convert all the Roman numerals to the Hindu-Arabic numbers, that we are familiar with.
Step 2: Perform the given operations/ solve the given equation, and find the solution to the question in Hindu-Arabic numbers.
Step 3: Convert the solution from step 2 from Hindu-Arabic numbers to the Roman numerals.
We will follow these steps to solve the given question.

Complete step by step answer:
We are given the Roman numerals calculation/ equation as \[\overline{XXXVII}CCXIX\div \_\_\_\_\_\_=CDIX\]. In the first step, we will convert the Roman numerals to the Hindu-Arabic numbers.
The first Roman number is \[\overline{XXXVII}CCXIX\], we will evaluate it as two parts and then add them. The first part is \[\overline{XXXVII}\], we know the \[X,V,I\] are \[10,5,1\] respectively in Hindu-Arabic system hence this numeral can be written as, \[\left( 10+10+10+5+1+1 \right)\times 1000=37,000\]. The second part is \[CCXIX\], we know that the \[C\] is \[100\] in Hindu- Arabic system. Hence, this numeral can be written as, \[100+100+10-1+10=219\]. Adding the values of the two parts, we get \[\overline{XXXVII}CCXIX=37,219\] in the Hindu-Arabic system.
The second numeral is \[CDIX\], we know that the \[D\] is \[500\] in the Hindu-Arabic system. Hence, the numeral can be written as, \[-100+500-1+10=409\].
Replacing the Roman numerals with the Hindu-Arabic numbers, we get \[37,219\div \_\_\_\_\_\_=409\]. Let the number to be fill in the blank is \[y\]. Hence, we get
\[\begin{align}
  & \Rightarrow 37,219\div y=409 \\
 & \Rightarrow \dfrac{37,219}{y}=409 \\
\end{align}\]
Solving the above equation, we get \[y=91\]. Now, we have to convert this to the Roman system. We can do this as follows,
\[91=-10+100+1\]
Hence, in Roman system \[91\] is written as \[XCI\].
The solution of the given fill in the blanks is \[XCI\].

Note:
One should know how to convert the Hindu-Arabic numbers to Roman, and Roman numerals to the Hindu-Arabic system. Also, the different conventions in which Roman numbers are expressed. In this example, the numeral \[\overline{XXXVII}\] has a line above it, which means to multiply the value of the numeral by \[1000\].