Q1(vi):

Write the following in decimal form and say what kind of decimal expansion each has : (vi) $\frac{329}{400}$

Initial Setup & Theoretical Foundation

We are tasked with converting the rational number $\frac{329}{400}$ into its decimal form and determining the nature of its decimal expansion.

[Per the Rational Number Decimal Expansion Theorem]: Let $x = \frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is of the form $2^n \times 5^m$, where $n$ and $m$ are non-negative integers. Then $x$ has a decimal expansion which terminates. If the prime factorization of $q$ contains any prime other than $2$ or $5$, the decimal expansion is non-terminating repeating.

Step 1: Prime Factorization of the Denominator

To predict the type of decimal expansion, we first analyze the denominator, $q = 400$.

• We decompose $400$ into its prime factors:
• $400 = 4 \times 100$
• $400 = 2^2 \times (10^2)$
• $400 = 2^2 \times (2 \times 5)^2$
• $400 = 2^2 \times 2^2 \times 5^2 = 2^4 \times 5^2$

Since the prime factors of the denominator consist exclusively of the primes $2$ and $5$ (where $n=4$ and $m=2$), we can definitively conclude that the fraction will yield a terminating decimal expansion.

Step 2: Algebraic Manipulation to Base 10

To convert the fraction to a decimal without long division, we can manipulate the denominator to become a power of $10$. A power of $10$ requires equal powers of $2$ and $5$ ($10^k = 2^k \times 5^k$).

Currently, the denominator is $2^4 \times 5^2$. To make the powers of $2$ and $5$ equal, we must multiply both the numerator and the denominator by $5^2$ (which is $25$).

$\frac{329}{400} = \frac{329}{2^4 \times 5^2}$

Multiplying numerator and denominator by $5^2$:

$= \frac{329 \times 5^2}{2^4 \times 5^2 \times 5^2}$

$= \frac{329 \times 25}{2^4 \times 5^4}$

$= \frac{8225}{(2 \times 5)^4}$

$= \frac{8225}{10^4}$

$= \frac{8225}{10000}$

Step 3: Decimal Conversion

Dividing by $10000$ (which has four zeros) requires shifting the decimal point of the numerator four places to the left.

$\frac{8225}{10000} = 0.8225$

Visual Representation of the Transformation

Step 4: Verification via Long Division

Alternatively, we can verify this result using the standard long division algorithm:

• $329 \div 400 = 0$ with a remainder of $329$. (Add a decimal point and a zero).
• $3290 \div 400 = 8$ ($400 \times 8 = 3200$), remainder is $90$. (Bring down a zero).
• $900 \div 400 = 2$ ($400 \times 2 = 800$), remainder is $100$. (Bring down a zero).
• $1000 \div 400 = 2$ ($400 \times 2 = 800$), remainder is $200$. (Bring down a zero).
• $2000 \div 400 = 5$ ($400 \times 5 = 2000$), remainder is $0$.

The division process ends with a remainder of $0$, confirming that the decimal expansion terminates.

Final Solution: The decimal form of $\frac{329}{400}$ is $0.8225$, and it has a terminating decimal expansion.