Comprehensive Study Guide for Mathematics: Fractions, Decimals, Percentages, and Algebraic Techniques

Dividing Quantities into Ratios

To divide a specific quantity into a given ratio, a three-step process is followed to ensure the proportions are mathematically accurate. First, calculate the total number of parts by summing the components of the ratio. For instance, in a ratio of $3:4$, the total number of parts is $3 + 4 = 7$. Second, determine the proportion of the desired quantity by expressing the relevant part as a fraction of the total parts. Using the previous example, the first component's proportion would be $\frac{3}{7}$. Finally, use this fraction to calculate the actual amount by multiplying it by the total quantity. If the total quantity were $210$, the calculation would be $\frac{3}{7} \times 210 = 90$.

A practical application of this is seen in mixing liquids. If a cordial mix is made of $1$ part syrup to $9$ parts water, the ratio is written as $1:9$. In this scenario, the total number of parts is $10$. The proportion of syrup in the mix is $\frac{1}{10}$, while the proportion of water is $\frac{9}{10}$. If a student prepares $400\,ml$ of this cordial mix, the amount of syrup required would be calculated as $\frac{1}{10} \times 400 = 40\,ml$.

Introduction to Formal Algebra and Terminology

Algebra serves as a framework to describe the rules and conventions governing numbers and arithmetic. Central to algebra is the equals symbol ($=$), which indicates that two or more expressions have exactly the same numerical value. Such a statement is known as an identity because the sides are identical in value. For example, since $3 + 5 = 8$ and $5 + 3 = 8$, it can be stated as an identity that $3 + 5 = 5 + 3$. This principle extends to variables: $a + b = b + a$ and $b + c + a = a + b + c$, demonstrating that numbers can be added in any order.

Several fundamental algebra facts define common operations. The expression $-3 + a$ is equivalent to $a - 3$ because the negative sign "belongs" to the number $3$. Multiplication is commutative, meaning $a \times 3 = 3 \times a$, and it is standard convention to make the multiplication symbol "invisible," resulting in $3a$. Repeated addition is expressed as multiplication; for example, $a + a + a = 3a$, which signifies three lots of $a$. Division and fractions are intrinsically related, where the first number in a division operation serves as the numerator of the fraction: $a \div 8 = \frac{a}{8}$.

The vocabulary of algebra includes specific terms to describe the components of expressions. A pronumeral is a letter or symbol used to represent one or more numerical values, while a variable specifically refers to a pronumeral that represents more than one value. An algebraic expression is a statement containing numbers and pronumerals connected by mathematical operations but containing no equals sign, such as $3a + 5$. A term is a component of an expression; for instance, the expression $3a + 5$ contains two terms. Like terms are those that contain exactly the same pronumerals, such as $3a$ and $5a$, whereas $3a$ and $5a^2$ are not like terms. A constant term is the part of an expression that contains no pronumerals, such as the number $5$ in $3a + 5$.

Algebraic Techniques and Evaluation

A coefficient is a numeral placed before a pronumeral to indicate that the pronumeral is multiplied by that factor. In the expression $3a + b + 5$, the coefficient of $a$ is $3$, while the coefficient of $b$ is implicitly $1$. Equivalent expressions are those that always yield the same numerical value regardless of what numbers are substituted for the pronumerals, such as $3a + 5$ and $5 + 3a$. Simplification involves finding the simplest possible equivalent version of an expression, such as reducing $3a + 5a$ to $8a$. Note that expressions like $3a + 5$ cannot be simplified further because they do not contain like terms.

Evaluating an expression involves substituting specific numerical values for variables to calculate a final result. If $x = 3$ and $y = 8$, evaluating $x + 2y$ requires replacing the pronumerals to get $3 + 2 \times 8 = 19$. It is critical to remember that term notation like $4a$ represents multiplication ($4 \times a$) and is distinct from the placement of digits in whole numbers ($42$ vs $4 \times 2$). Once substitution is complete, the standard order of operations applies: calculate brackets first, then multiplication and division from left to right, followed by addition and subtraction from left to right. For example, $(4 + 3) \times 2 - 20 \div 4 + 2$ simplifies to $7 \times 2 - 5 + 2 = 14 - 5 + 2 = 11$.

Equivalent Algebraic Expressions and Like Terms

Algebraic rules determine equivalence, such as $x + 12 = 12 + x$ (the commutative property of addition) and $3k = k + k + k$ (repeated addition as multiplication). Tables of values can be used to test for equivalence by substituting various integers ($x = 0, 1, 2, 3$) into expressions like $2x + 2$ and $(x + 1) \times 2$ to see if results consistently match. Like terms are defined by containing identical pronumerals; for example, $5ab$ and $6ba$ are like terms because the order of multiplication ($a \times b$ vs $b \times a$) does not change the value. However, $8ab$ and $5a$ are not like terms because the set of pronumerals is different.

Like terms can be combined (collected) to simplify expressions. For example, $5ab + 3ab$ simplifies to $8ab$. Terms without the same pronumerals, such as $4x$ and $5y$, can be added to form the expression $4x + 5y$, but this cannot be reduced effectively. In complex strings such as $1a + 2b + 3a + 4b + ... + 19a + 20b$, only the $a$ terms can be combined with other $a$ terms, and $b$ terms with other $b$ terms.

Multiplication and Division in Algebra

In algebraic shorthand, multiplication signs are omitted, and numbers are written before pronumerals, which are usually listed in alphabetical order. Thus, $b \times 2 \times a$ is written as $2ab$. Because of the associative property of multiplication, brackets are unneccessary when only multiplication is involved, so $3 \times (a \times b)$ is simply $3ab$. Repeated multiplication of the same pronumeral is written using indices, such as $a \times a = a^2$.

Division in algebra is typically represented as a fraction, where $7xy \div 3xz$ is written as $\frac{7xy}{3xz}$. Simplifying these fractions involves dividing by common factors. Just as numerical fractions like $\frac{10}{15}$ are simplified to $\frac{2}{3}$ by dividing by the common factor of $5$, algebraic fractions can be simplified by cancelling common pronumerals. In $\frac{7xy}{3xz}$, the common factor $x$ can be cancelled from the numerator and denominator, leaving $\frac{7y}{3z}$. Another example is $\frac{24ab}{12bc}$, where dividing by the common factor of $12b$ results in $\frac{2a}{c}$.

Expansion and the Distributive Law

Expanding or eliminating brackets involves creating an equivalent expression without brackets. This is achieved using the distributive law, which states that $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$. This requires multiplying every term inside the brackets by the term outside. For example, expanding $3(a + 2)$ results in $3 \times a + 3 \times 2 = 3a + 6$. This can also be conceptualized as repeated addition: $3(a + 2) = (a + 2) + (a + 2) + (a + 2) = 3a + 6$. The distributive law is also utilized in mental arithmetic to solve problems like $5 \times 27$, which can be broken down into $5(20 + 7) = 100 + 35 = 135$.

A real-world application involves calculating areas. If a house is $15\,m$ long and $10\,m$ wide, and a deck of variable width $x$ is added, the total floor area $A$ can be written as $A = 15(10 + x)$. Using expansion, this becomes $A = 150 + 15x$. Similarly, the area of a rectangle with height $4$ and width $x + 3$ is expressed as $4(x + 3)$, which expands to $4x + 12$.

Decimal Place Value and Ordering

The decimal point is a structural tool used to separate whole numbers from the fractional or decimal part. Place value tables must be extended to include columns for tenths, hundredths, thousandths, and so on. For instance, the number $428.357$ is decomposed as $4$ hundreds ($4 \times 100$), $2$ tens ($2 \times 10$), $8$ ones ($8 \times 1$), $3$ tenths ($\frac{3}{10}$), $5$ hundredths ($\frac{5}{100}$), and $7$ thousandths ($\frac{7}{1000}$). Every integer can be represented as a decimal; for example, the integer $345$ is equivalent to $345.0$. Adding extra zeros to the right of the decimal part, such as $12.5 = 12.50 = 12.500$, does not change the value of the number.

Rounding Decimals

Rounding involves approximating a decimal to a lower number of decimal places. The first step is to "cut" the number after the specified decimal place. The digit immediately to the right of that place is the critical digit. If the critical digit is less than $5$ ($0, 1, 2, 3, 4$), the number is rounded down, meaning the digits after the cut are simply removed. If the critical digit is $5$ or more ($5, 6, 7, 8, 9$), the number is rounded up, increasing the final digit by $1$. For the number $25.8174$: rounding to $1$ decimal place uses $1$ as the critical digit, resulting in $25.8$; rounding to $2$ places uses $7$ as the critical digit, resulting in $25.82$; rounding to $3$ places uses $4$ as the critical digit, resulting in $25.817$.

Operations with Decimals

When adding or subtracting decimals, the decimal points must be aligned vertically, ensuring that corresponding place values are in the same column. The decimal point in the final answer is placed directly in line with these points. It is often helpful to append zeros so that all numbers have the same count of decimal places; for example, adding $7.12$, $8.5$, and $13.032$ is best handled by writing them as $7.120$, $8.500$, and $13.032$.

Multiplying and dividing by powers of $10$ involves shifting the decimal point. Multiplying by $10, 100, 1000$ moves the point to the right by the same number of places as there are zeros in the multiplier (e.g., $5.7839 \times 1000 = 5783.9$). Dividing by these powers moves the point to the left (e.g., $2975.6 \div 100 = 29.756$).

General decimal multiplication is performed by initially ignoring the decimal points and multiplying the numbers as integers. The point is سپس reintroduced into the answer so that the total number of decimal places in the final result equals the sum of the decimal places in the numbers being multiplied. For example, in $5.34 \times 1.2$, the problem is calculated as $534 \times 12 = 6408$. Since the factors have a total of $3$ decimal places ($2$ from $5.34$ and $1$ from $1.2$), the answer is $6.408$. Dividing decimals by whole numbers follows general long division, with the decimal point in the quotient placed directly above the point in the dividend. To divide by another decimal, the divisor must be changed into a whole number by shifting the decimal point; the dividend's decimal point must be shifted by the exact same number of places before dividing.

Fractions, Decimals, and Percentages Interconnectivity

Decimals are converted to fractions using place value knowledge; for example, $0.25$ becomes $\frac{25}{100}$, which simplifies to $\frac{1}{4}$. Fractions are converted to decimals by either finding an equivalent fraction with a denominator of $10, 100, 1000$, or by dividing the numerator by the denominator (e.g., $\frac{5}{8}$ is solved by $5.000 \div 8 = 0.625$). Recurring decimals have repeating patterns, denoted by a dot or bar over the repeating digits, such as $\frac{1}{3} = 0.\dot{3}$.

The term "per cent" originates from the Latin "per centum," meaning "out of $100$." Any percentage can be written as a fraction with a denominator of $100$, such as $37\% = \frac{37}{100}$. To convert a percentage to a decimal, divide by $100$ by moving the point two places left ($42\% = 0.42$). Conversely, to convert a decimal to a percentage, multiply by $100$ by moving the point two places right ($0.654 = 65.4\%$).

Comparing proportions is easier when the values are all expressed as the same type of number (e.g., all as percentages). For instance, comparing scores of $17$ out of $20$, $0.65$, and $82\%$ involves converting all to percentages: $85\%$, $65\%$, and $82\%$. To find a percentage of a quantity, express the percentage as a fraction, change "of" to a multiplication sign, and multiply (e.g., $25\%$ of $60 = \frac{25}{100} \times \frac{60}{1} = 15$). Useful mental shortcuts include finding $10\%$ by dividing by $10$, $50\%$ by dividing by $2$, and $25\%$ by dividing by $4$.

Advanced Fraction Operations

Fractions with the same denominator are called "like" fractions and are added or subtracted by operating on the numerators while keeping the denominator constant. If denominators are different, they must be converted to equivalent fractions with a Lowest Common Denominator (LCD). For subtraction involving mixed numerals, if the fraction part of the first numeral is smaller than the second, one must regroup a whole number (e.g., converting $7\frac{1}{4}$ to $6\frac{5}{4}$) or convert both to improper fractions.

To multiply fractions, multiply the numerators and multiply the denominators: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$. Fractions should be simplified by cancelling common factors vertically or diagonally before the final multiplication. Division requires the use of a reciprocal. The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Dividing by a fraction is equivalent to multiplying by its reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. Before multiplying or dividing, mixed numerals must always be converted to improper fractions.