Aspire Faculty ID #4195 · Topic: NIMCET 2019 · Just now
NIMCET 2019

The base ( or radix) of the number system such that the following equation holds $\dfrac{312}{20}=13.1$ is

Solution

Base (Radix) Problem Solution

Question: Find the base r such that

(312)r ÷ (20)r = (13.1)r

Expansion in base 10

  • (312)r = 3r² + 1r + 2
  • (20)r = 2r
  • (13.1)r = r + 3 + 1/r

Equation

(3r² + r + 2) / (2r) = r + 3 + 1/r

Multiply both sides by 2r

3r² + r + 2 = 2r² + 6r + 2

Simplification

r² - 5r = 0
⇒ r(r - 5) = 0

✅ Final Answer

Since base must be > 1, the valid solution is:
Base (radix) = 5

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Paramendra Vikram Singh
Paramendra Vikram Singh , Nimcet Aspirants
Commented Oct 17, 2023
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Shwetank Kumar , Batch two
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