Summary of Ratios and Direct Proportion

## Introduction Ratios and proportions are ways to compare quantities. A **ratio** compares two or more amounts by division, while a **proportion** states that two ratios are equal. These tools help solve sharing problems, scale drawings, mixtures, and many real-world situations. > **Definition:** A ratio is a comparison of two quantities using division, written as $a:b$ or $\frac{a}{b}$. A proportion is an equation of the form $\frac{a}{b} = \frac{c}{d}$. ## Basic Concepts ### What is a ratio? - A ratio compares two quantities: $a:b$ means $a$ parts to $b$ parts. - Ratios can be simplified by dividing both terms by their greatest common divisor (GCD). - You can write ratios as fractions $\frac{a}{b}$ or using a colon $a:b$. Example: Compare 160 cm and 4 m. - Convert units first: $4\text{ m} = 400\text{ cm}$. - Ratio: $160:400$. - Simplify by GCD $=40$: $\frac{160}{400} = \frac{4}{10} = \frac{2}{5}$, so the simplest ratio is $2:5$. > **Definition:** The simplest form of a ratio means both terms have no common factor greater than 1. ### Writing and simplifying ratios (step-by-step) 1. Put both quantities in the same units, if necessary. 2. Write the ratio $a:b$ or $\frac{a}{b}$. 3. Divide both terms by their GCD to simplify. ### Ratios with money and sharing When two people pay different amounts and want to split items in proportion to what they paid, use their payments to form a ratio. Example: Andrews paid $7.50 and Marc paid $3.50$ and they bought 48 chocolates together. - Form the ratio using payments: $7.50:3.50$. - Convert to cents or decimals and simplify: $750:350$ dividing by 50 gives $15:7$. - Total parts: $15 + 7 = 22$ parts. - Each part = $\frac{48}{22} = \frac{24}{11}$ chocolates (this may be a fractional result; usually we expect whole chocolates, so check context). If whole chocolates are required, adjust or allocate nearest whole numbers while keeping fairness. - Andrews gets $15$ parts: $15 \times \frac{24}{11} = \frac{360}{11} \approx 32\text{ chocolates}$ (exactly $\frac{360}{11}$). Marc gets $7$ parts: $7 \times \frac{24}{11} = \frac{168}{11} \approx 15\text{ chocolates}$. ### Ratios in mixtures and colors Problem: Two shades of pink are made from red and white in different ratios. - Perfect Pink: $3:4$ red:white. Fraction red = $\frac{3}{3+4} = \frac{3}{7}$. - Rose Pink: $2:3$ red:white. Fraction red = $\frac{2}{2+3} = \frac{2}{5}$. To compare fairly, convert to a common denominator: $\frac{3}{7} = \frac{15}{35}$ and $\frac{2}{5} = \frac{14}{35}$. Perfect Pink has more red, so it is darker. ### Using ratios to find angles Given a circle divided into angles in ratio $3:2:4:6$, the total parts are $3+2+4+6=15$. - Each part measures $\frac{360^\circ}{15} = 24^\circ$. - Then angles are $24^\circ\times 3 = 72^\circ$, $24^\circ\times 2 = 48^\circ$, $24^\circ\times 4 = 96^\circ$, $24^\circ\times 6 = 144^\circ$. ## Proportions and Solving Problems ### What is a proportion? - A proportion states $\frac{a}{b} = \frac{c}{d}$. - To solve for an unknown, use cross-multiplication: $a\cdot d = b\cdot c$. Example: If $\frac{2}{5} = \frac{x}{30}$ then cross-multiply: $2\cdot 30 = 5\cdot x$ so $x = \frac{60}{5} = 12$. ### Direct proportion - Two quantities are directly proportional if increasing one increases the other at the same rate: $y = kx$ where $k$ is the constant of proportionality. - To find $k$, divide $y$ by $x$: $k = \frac{y}{x}$. ## Comparison Table | Concept | Symbol | Example | How to simplify or use | |---|---:|---|---| | Ratio | $a:b$ or $\frac{a}{b}$ | $160:400$ | Convert units, divide by GCD to get $2:5$ | | Fraction form | $\frac{a}{b}$ | $\frac{3}{7}$ | Compare by common denominator | | Proportion | $\frac{a}{b} = \frac{c}{d}$ | $\frac{2}{5} = \frac{x}{30}$ | Use cross-multiplication | | Direct proportion | $y = kx$ | $y$ proportional to $x$ | Find $k = \frac{y}{x}$ | ## Practical Tips and Common Mistakes - Always convert to the same unit before forming a ratio. Example: meters to centimeter