After how many decimal places does the decimal expansion of 51/1500 terminate?

After how many decimal places does the decimal expansion of $\frac{51}{1500}$ terminate?

Class 10Real NumbersNumber Systems

Reduce to lowest terms FIRST:

\frac{51}{1500} = \frac{17}{500}

(divide both by 3). Now

500 = 2^{2} \times 5^{3}

is of the form 2ᵐ × 5ⁿ, so it terminates, after as many places as the larger exponent, 3. Indeed

\frac{17}{500} = \frac{34}{1000} = 0.034

. Testing the un-reduced

1500 = 2^{2} \cdot 3 \cdot 5^{3}

is exactly the trap.

Answer: A) 3

**Theorem**: A rational number

\frac{p}{q}

(in lowest terms) has a terminating decimal expansion if and only if

q

has the form **2ᵐ × 5ⁿ** (m, n ≥ 0).

**Examples**:
-

\frac{3}{8} : q = 8 = 2^{3}

→ terminates

. (\frac{3}{8} = 0.375.) - \frac{7}{40} : q = 40 = 2^{3} \times 5

→ terminates

. (\frac{7}{40} = 0.175.) - \frac{1}{3} : q = 3

→ does NOT terminate

. (\frac{1}{3} = 0.333

… recurring.)
-

\frac{4}{15} : q = 15 = 3 \times 5

→ does NOT terminate (because of the 3).

**Why**: any decimal expansion is a fraction with denominator 10ⁿ = 2ⁿ × 5ⁿ. To convert

\frac{p}{q}

to decimal form, you need

q

to ONLY have 2s and 5s as primes — otherwise you can't multiply numerator/denominator into a power of 10.

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