S1 Math Assessment: Decimals, Sequences & More

Within the domain of decimal arithmetic, the cumulative sum of $142.7$ and $88.4$ manifests as ______, thus exemplifying the additive properties inherent to real number manipulation.

Given a scalar quantity of $8.5$ undergoing multiplicative transformation by a factor of $9$, the resultant product, indicative of scaled magnitude, is precisely ______.

In the context of metric differential analysis, the disparity in stature between Margaret, quantified at $1.41$ meters, and Kevin, registering at $1.23$ meters, reveals a height differential of ______ meters favoring Margaret.

Considering the economic transactional framework wherein a digital album procurement incurs a cost of $£8.49$ per unit, the aggregate financial burden for the acquisition of three such albums escalates to ______, thereby delineating the principles of scalar multiplication within a monetary context.

Within an educational procurement scenario, given that the collective expenditure for nine mathematical textbooks amounts to $£51.12$, the unitary cost attributable to each individual textbook, reflective of equitable cost distribution, is delineated as ______.

In the expression $\frac{a}{b} + \frac{c}{d}$, where $a, b, c, d \in \mathbb{Z}$ and $b, d \neq 0$, the resultant fraction can be simplified to its lowest terms if the greatest common divisor of the numerator and ______ is greater than 1.

Given two mixed fractions $x \frac{y}{z}$ and $p \frac{q}{r}$, where $x, y, z, p, q, r \in \mathbb{N}$, the most efficient method to compute their sum, while minimizing the risk of arithmetic errors especially with large numbers, typically involves converting them into ______ before performing the addition.

When calculating the perimeter of a rectangle with fractional side lengths $l$ and $w$, expressing the result as a mixed number $a \frac{b}{c}$ provides insight into the whole number of units ($a$) and the remaining ______ part ($\frac{b}{c}$) of the perimeter measured beyond that whole number.

Consider a scenario where a vehicle travels for $x \frac{y}{z}$ hours and then continues for $p \frac{q}{r}$ hours, with $x, y, z, p, q, r \in \mathbb{Q}^+$. Determining the total travel time necessitates the consolidation of these time intervals, which relies on the fundamental arithmetic operation of ______ applied to mixed numbers.

Suppose a rectangular plot has a length specified as $a\frac{b}{c}$ units and a width of $d\frac{e}{f}$ units, where $a, b, c, d, e, f \in \mathbb{N}$. To determine the effect on the perimeter if both dimensions are increased by a factor of $k$, the initial and final perimeters must be calculated using ______ arithmetic operations with fractions before comparing the results.

Given the product $\frac{1}{4} \times \frac{4}{7}$, the resultant fraction, when expressed in its simplest form requires the identification of the ______ between the numerator of the first fraction and the denominator of the second.

In evaluating the expression $3 \frac{1}{4} \times 1 \frac{1}{3}$, one must first convert the mixed numbers to improper fractions, perform the ______ of the resulting numerators and denominators, and then simplify the result back into a mixed number if necessary.

When dividing $\frac{5}{12} \div \frac{5}{3}$, the operation is transformed into multiplication by the ______ of the second fraction, allowing for simplification before or after multiplication.

To solve $5 \div 1 \frac{1}{4}$, convert the mixed number to an ______ fraction before inverting and multiplying, which enables the calculation of the quotient.

In simplifying $\frac{2}{5} \div \frac{9}{10}$, multiplying by the reciprocal uncovers opportunities for ______ between the numerator of the first fraction and the denominator of the second and vice versa, facilitating the reduction to the simplest form.

Evaluate $2 \frac{7}{8} \div 1 \frac{3}{9}$. Upon converting to improper fractions and transforming division into multiplication by the reciprocal, watch for ______ factors to simplify and find the answer more efficiently.

The phrase 'expressed as a fraction in its simplest form' implies that the numerator and the denominator of the resultant fraction must be ______, sharing no common factors other than 1.

The reciprocal of a fraction $\frac{a}{b}$, where $a$ and $b$ are non-zero integers, is denoted as $\frac{b}{a}$, illustrating the ______ property in the context of multiplicative inverses.

Given the recurrence relation $a_{n+2} = p a_{n+1} + q a_n$, where $a_0 = 1$, $a_1 = 2$, $a_2 = 5$, and $a_3 = 12$, the values of $p$ and $q$ are such that $p + q = $ ______.

In the context of continued fractions, the convergents of a real number provide increasingly accurate rational approximations. If the continued fraction representation of $\pi$ is given by $[3; 7, 15, 1, 292, ...]$, the third convergent, obtained by truncating the continued fraction after the third term, expressed as a simplified fraction, has a denominator equal to ______.

Consider the equation $ax + b = c$, where $a$, $b$, and $c$ are real numbers. Assuming $a \neq 0$, if the solution for $x$ is such that $x = \frac{c-b}{a}$, and we know that $a$, $b$, and $c$ form an arithmetic sequence in that order with a common difference of $d$, then $x$ is equal to ______.

Given the geometric sequence defined by $a_n = a_1 r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio, if the sum of the first two terms ($a_1 + a_2$) is 15 and the sum of the next two terms ($a_3 + a_4$) is 60, then the common ratio $r$ is equal to ______.

Suppose you have the infinite series $\sum_{n=1}^{\infty} ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio. If $a = \frac{1}{2}$ and the sum of the series converges to $\frac{3}{4}$, then the value of $r$ is ______.

Consider a recursive sequence defined by $a_{n+1} = \frac{1}{2} \left( a_n + \frac{A}{a_n} \right)$, where $A > 0$. This sequence converges to $\sqrt{A}$. If $a_0 = 1$ and $a_2 = \frac{17}{12}$, then $A$ is ______.

The expression $\frac{x^3 - 8}{x - 2}$ can be simplified to a quadratic expression. When $x \neq 2$, the value of this simplified quadratic expression when $x = 2$ is ______.

If $\frac{a}{b} = \frac{3}{4}$ and $\frac{c}{d} = \frac{5}{6}$, the value of the expression $\frac{ac}{bd} + \frac{9}{16}$ is ______.

Given the equation $2^{2x} - 3 \cdot 2^{x+1} + 8 = 0$, the sum of the distinct real solutions for $x$ is ______.

Express the repeating decimal $0.1\overline{6}$ as a simplified fraction. The denominator of the fraction in its reduced form is ______.

Decimal Addition

Decimal Subtraction

Decimal Multiplication

Decimal Division

Perimeter

Adding fractions (same denominator)

Subtracting fractions (same denominator)

Simplest form of a fraction

Adding mixed numbers

Perimeter of a rectangle

Sequence

Formula (for a sequence)

Percentage (%)

Decimal

Fraction

Simplify

Equation

Solve (an equation)

Improper Fraction

Mixed Number

Multiplying Fractions

Simplifying Fractions

Multiplying Mixed Numbers

Dividing Fractions

Factor

Greatest Common Factor (GCF)