Ratio, Proportion, Indices, and Logarithms Study Guide

RATIO AND PROPORTION, INDICES, LOGARITHMS

• Conceptual Overview: Ratios are ubiquitous in practical fields. They allow for the calculation of unknown parts from given totals or other parts. Examples include dividing money into three parts in a specific ratio or determining the count of girls in a school if the total student count and the boys-to-girls ratio are known.
• Fundamental Definition of Ratio: A ratio is a comparison of the sizes of two or more quantities of the same kind by division. If $a$ and $b$ are two quantities of the same kind (expressed in the same units), the fraction $\frac{a}{b}$ is the ratio of $a$ to $b$. This is written as $a : b$.
• Terms of a Ratio: In the ratio $a : b$, the quantities $a$ and $b$ are called the terms.     * $a$ is the first term or antecedent.     * $b$ is the second term or consequent.     * Example: In the ratio $5 : 6$, $5$ is the antecedent and $6$ is the consequent.
• Remarks on Ratios:     * Scaling: Terms of a ratio remain unchanged in value if both are multiplied or divided by the same non-zero number. Ratios are typically expressed in their lowest terms (simplest form). For example: $12 : 16 = \frac{12}{16} = \frac{3 \times 4}{4 \times 4} = 3 : 4$.     * Order Importance: The order of terms is significant ($3 : 4$ is not equal to $4 : 3$).     * Same Kind Constraint: Ratios only exist between quantities of the same kind. There is no ratio between the number of students and a teacher's salary, or between a child's weight and another child's age.     * Unit Consistency: Quantities must be in the same units prior to comparison.         * Illustration: Ratio between $150\,g$ and $2\,kg = 150 : 2000 = 3 : 40$.         * Illustration: Ratio between $25\,min$ and $45\,sec = (25 \times 60) : 45 = 1500 : 45 = 100 : 3$.
• Comparing Ratios: To compare two ratios, they must be converted into equivalent like fractions with a common denominator.     * Example: Compare $2\frac{1}{3} : 3\frac{1}{3}$ and $3.6 : 4.8$.     * $7/3 : 10/3 = 7 : 10 = 14/20$.     * $3.6 : 4.8 = 36/48 = 3/4 = 15/20$.     * Since $15/20 > 14/20$, the ratio $3.6 : 4.8$ is greater.
• Quantity Changes: If a quantity increases or decreases in the ratio $a : b$, then the:     $\text{New Quantity} = \frac{b}{a} \times \text{Original Quantity}$     * The fraction $\frac{b}{a}$ is known as the factor multiplying ratio.     * Illustration: Rounaq weighs $56.7\,kg$. He reduces his weight in the ratio $7 : 6$.     $\text{New Weight} = \frac{6}{7} \times 56.7\,kg = 48.6\,kg$

TYPES OF RATIOS AND PROPERTIES

• Inverse Ratio: One ratio is the inverse of another if their product is $1$. The inverse of $a : b$ is $b : a$.
• Inequality of Ratios:     * Greater Inequality: If antecedent $a > b$.     * Lesser Inequality: If antecedent $a < b$.
• Compound Ratio: The ratio compounded of $a : b$ and $c : d$ is calculated by multiplying the antecedents and the consequents: $ac : bd$.     * Example: Compound ratio of $3 : 4$ and $5 : 7$ is $15 : 28$.
• Duplicate and Triplicate Ratios:     * Duplicate Ratio: A ratio compounded of itself ($a^2 : b^2$). Example: Duplicate of $2 : 3$ is $4 : 9$.     * Triplicate Ratio: $a^3 : b^3$. Example: Triplicate of $2 : 3$ is $8 : 27$.
• Sub-Duplicate and Sub-Triplicate Ratios:     * Sub-duplicate ratio: $\sqrt{a} : \sqrt{b}$. Example: Sub-duplicate of $4 : 9$ is $2 : 3$.     * Sub-triplicate ratio: $\sqrt[3]{a} : \sqrt[3]{b}$. Example: Sub-triplicate of $8 : 27$ is $2 : 3$.
• Commensurability: Quantities are commensurable if their ratio can be expressed as a rational number (ratio of two integers). Otherwise, they are incommensurable (e.g., $\sqrt{3} : \sqrt{2}$).
• Continued Ratio: A relation between three or more quantities of the same kind, written as $a : b : c$.     * Illustration: The continued ratio of $\$200$, $\$400$, and $\$600$ is $1 : 2 : 3$.

PROPORTIONS

• Definition: An equality of two ratios is called a proportion. Four quantities $a, b, c, d$ are in proportion if $a : b = c : d$ ($a/b = c/d$). This is also written as $a : b :: c : d$.
• Cross Product Rule: For quantities in proportion, $ad = bc$, meaning the product of extremes equals the product of means.     * Extremes: First term ($a$) and fourth term ($d$).     * Means: Second term ($b$) and third term ($c$).
• Fourth Proportional: If $a : b = c : d$, then $d$ is the fourth proportional to $a, b, c$.
• Continued Proportion: Three quantities $a, b, c$ are in continued proportion if $a : b = b : c$. Here, $b^2 = ac$.     * Mean Proportional: The middle term $b$ is the mean proportional between $a$ and $c$ ($b = \sqrt{ac}$).     * Third Proportional: The term $c$ is called the third proportional.
• Proportion Consistency: While ratio terms must all be the same kind, in proportion ($a : b = c : d$), only the first two terms ($a, b$) must be of the same kind, and the last two terms ($c, d$) must be of the same kind.     * Example: $\$6 : \$8 = 12\,toffees : 16\,toffees$.

PROPERTIES OF PROPORTION

If $\frac{a}{b} = \frac{c}{d}$, then:

• Invertendo: $\frac{b}{a} = \frac{d}{c}$.
• Alternendo: $\frac{a}{c} = \frac{b}{d}$.
• Componendo: $\frac{a + b}{b} = \frac{c + d}{d}$.
• Dividendo: $\frac{a - b}{b} = \frac{c - d}{d}$.
• Componendo and Dividendo: $\frac{a + b}{a - b} = \frac{c + d}{c - d}$.
• Addendo: If $\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \dots$, then each ratio is equal to $\frac{a + c + e + \dots}{b + d + f + \dots}$.
• Subtrahendo: If $\frac{a}{b} = \frac{c}{d} = \dots$, then each ratio is equal to $\frac{a - c - e - \dots}{b - d - f - \dots}$.

INDICES

• Definition: Indices represent repeated multiplication. In $a^n$, $a$ is the base and $n$ is the index (power/exponent).
• Zero Power: Any base raised to the power zero is $1$: $a^0 = 1$.
• Laws of Indices:     * Law 1 (Product Rule): $a^m \times a^n = a^{m+n}$.     * Law 2 (Quotient Rule): $\frac{a^m}{a^n} = a^{m-n}$.     * Law 3 (Power of a Power): $(a^m)^n = a^{mn}$.     * Law 4 (Product Power): $(ab)^n = a^n b^n$.
• Fractional and Negative Indices:     * $a^{-n} = \frac{1}{a^n}$.     * $a^{\frac{1}{m}} = \sqrt[m]{a}$.     * Example: $3^{-5} = \frac{1}{3^5} = \frac{1}{243}$.
• Equations with Indices:     * If $a^x = a^y$, then $x = y$ ($a \neq 0, 1, -1$).     * If $x^a = y^a$, then $x = y$.

LOGARITHMS

• Definition: If $a^x = n$ ($n > 0, a > 0, a \neq 1$), then $x$ is the logarithm of $n$ to the base $a$, written as $\log_{a}(n) = x$.     * Example: $2^4 = 16 \Rightarrow \log_{2}(16) = 4$.     * Example: $10^3 = 1000 \Rightarrow \log_{10}(1000) = 3$.
• Fundamental Laws of Logarithm:     1. Product Law: $\log_{a}(mn) = \log_{a}(m) + \log_{a}(n)$.     2. Quotient Law: $\log_{a}\left(\frac{m}{n}\right) = \log_{a}(m) - \log_{a}(n)$.     3. Power Law: $\log_{a}(m^n) = n \log_{a}(m)$.
• Base Change Property:     * $\log_{a}(m) = \log_{b}(m) \times \log_{a}(b) = \frac{\log_{b}(m)}{\log_{b}(a)}$.     * $\log_{b}(a) \times \log_{a}(b) = 1$, meaning $\log_{b}(a) = \frac{1}{\log_{a}(b)}$.
• Common and Natural Logarithms:     * Common Logarithms: Use base $10$. (e.g., $\log_{10}(10) = 1$, $\log_{10}(1) = 0$).     * Natural Logarithms: Use base $e$ ($e \approx 2.33$ in context, traditionally $2.71828$).
• Logarithm Tables - Characteristic and Mantissa: A common logarithm has two parts: the integral part (characteristic) and the fractional part (mantissa).     * Characteristic:         * For numbers $> 1$: It is (number of digits to the left of the decimal point $- 1$). Example: log 75 has characteristic $1$. log 1.76 has characteristic $0$.         * For numbers $< 1$: It is negative and numerically (number of zeros immediately after the decimal point $+ 1$). Example: $.07$ has characteristic $-2$. This is often written with a bar, e.g., $\bar{2}$.     * Mantissa: Always a positive quantity obtained from log tables. For the same sequence of figures, the mantissa remains constant regardless of the decimal placement.
• Antilogarithms: If $\log_{a}(n) = x$, then $n = \text{antilog}(x)$.     * Example: If $\log(61720) = 4.7904$, then $61720 = \text{antilog}(4.7904)$.

PRACTICAL BUSINESS APPLICATIONS

• Partnership and Profit Sharing: Profits are typically divided according to the ratio of investments and the duration for which the capital was used.     * Example: P and Q invest $\$85000$ and $\$15000$ respectively. Ratio of profit is $85000 : 15000 = 17 : 3$.     * Compound sharing Example: Reena invests $\$35000$ for $8\,months$ and Shaloo invests $\$42000$ for $10\,months$. The profit is shared in ratio $(35000 \times 8) : (42000 \times 10) = 280000 : 420000 = 2 : 3$.
• Mix Proportion Problems: Determining mixture ratios to achieve a target cost price.     * Illustration: Dealer mixes tea at $\$6.92/kg$ and $\$7.77/kg$. Selling price is $\$8.80$ with a $17\frac{1}{2}\%$ profit on sale price.     * Profit = $\frac{165}{200} \times 8.80 = \$1.452$.     * $\text{Cost Price of mixture} = 8.80 - 1.54 = \$7.26$.     * Required ratio is found via the difference method: $(7.77 - 7.26) : (7.26 - 6.92) = 0.51 : 0.34 = 3 : 2$.
• Ages and Work: Proportion is used to find ages when comparisons are given at different times, or to determine work efficiency (e.g., if A and B together take 8 days, B and C take 12 days, and all three take 6 days).