Mathematical Operations, Fraction Conversions, and Rounding Precision

Percentage Calculations and Conversions

A percentage represents a dimensionless ratio or fraction expressed with a denominator of $100$, designated by the percent symbol $\%$. The foundational mathematical procedure for converting a numerical quantity or rational ratio into a percentage involves scaling the proportion against a standard base of one hundred units. In problem $Sa$, this relationship is stated through the expression:

$\frac{13 \times 100}{100} = 13\%$

The arithmetic structure of $\frac{13 \times 100}{100}$ illustrates the direct correspondence between division by $100$ and the percentage scale. When a base quantity such as $13$ is scaled by a factor of $100$ and subsequently divided by $100$, the numerator multiplier and denominator divisor cancel each other out. This operation preserves the underlying scalar value of $13$ while formally translating it into percentage notation as $13\%$. For any arbitrary rational value $\frac{a}{b}$, conversion to a percentage is achieved by evaluating $\frac{a}{b} \times 100\%$, where the percentage sign functions algebraically as multiplication by $\frac{1}{100}$.

Conversion of Common Fractions to Decimal Fractions

A rational number written as a fraction of two integers can be transformed into a decimal traction or decimal fraction through factorization, simplification, and division. Problem $b$ introduces the fraction:

$\frac{52}{65}$

To represent $\frac{52}{65}$ as a decimal traction, common divisors shared by the numerator $52$ and denominator $65$ are factored out. The integer $52$ factors into $4 \times 13$, while the integer $65$ factors into $5 \times 13$. Dividing both the numerator and the denominator by their greatest common factor of $13$ reduces the fraction to its simplified form:

$\frac{4}{5}$

Evaluating the irreducible fraction $\frac{4}{5}$ through standard division yields the decimal value $0.8$. The associated working values $655$ and $5$ demonstrate division procedures where dividends are partitioned by a divisor of $5$. Performing division by $5$ corresponds mathematically to scaling by $\frac{2}{10}$, which doubles the numerator value and adjusts the decimal position by one place to the left.

Numerical Rounding to Decimal Places

Rounding constitutes an approximation technique that replaces a numerical value with an adjacent representation defined by a prescribed precision threshold. Problem $6$ demonstrates rounding values to specific decimal place limits. In the first instance, the quantity $78, 96$ is evaluated to one decimal place:

$78, 96$

The evaluated result of this rounding operation is recorded as:

$781$

Rounding a quantity to one decimal place necessitates inspecting the digit positioned in the second decimal position to the right of the radix point. Standard mathematical conventions dictate that if the subsequent digit equals or exceeds $5$, the target digit increments upward by $1$. In the case of $78, 96$, the digit following the first position triggers an upward adjustment, resulting in $781$ or $78.1$ depending on whether decimal or separator punctuation is applied.

Problem $b$ requires the rounding of the quantity $78,096$ to two decimal place specifications:

$78,096$

Evaluating a number to two decimal places establishes the truncation boundary after the second digit following the decimal point. The digit in the third decimal place dictates whether the hundredths digit remains static or rounds up. For $78,096$, the digit $6$ exceeds the threshold of $5$, necessitating that the preceding hundredths place increments upward.

Approximations by Significant Figures

Significant figures, recorded under the terminology are bignificant figure, denote the meaningful digits in a number that contribute to its overall precision and scale. Unlike decimal place rounding, which fixes precision strictly relative to the decimal mark, significant figures prioritize leading non-zero digits regardless of decimal point location. Problem $c$ presents the values $78,$ and $€ 78 096$ evaluated to are bignificant figure, yielding:

$78,010$

In the expression $€ 78 096$, the symbol $€$ establishes the value as a monetary currency quantity. Rounding $€ 78 096$ to significant figures requires retaining the leading critical digits that define the order of magnitude. Approximating the quantity yields $78,010$, wherein the terminal zero acts as a positional place-value holder to preserve the magnitude of tens of thousands while matching the precision criteria specified by are bignificant figure.