Year 8 Unit 8 Ratio, Proportion and Compound Measures Notes

Learning Outcomes

• Support Objectives:     - Describe the proportion of something using words, fractions, decimals, or percentages.     - To change freely between standard metric units of length, mass, or time (Sparx: M530, M515, M772; Corbett: 349a, 349b).     - To change freely between standard metric units of volume (Sparx: M761, M774; Corbett: 349c, 322).     - Exchange between units of money (Corbett: 351).     - Express a quantity as a ratio and as a proportion (Sparx: M901).

• Core Objectives:     - Use percentages, decimals, or fractions to calculate proportions (Sparx: M885; Corbett: 267, 478).     - Simplify ratios to their simplest form $a:b$ where $a$ and $b$ are integers (Sparx: M885; Corbett: 269).     - Write a ratio in the form $1:n$ or $n:1$ (Sparx: M543; Corbett: 271c).     - Divide an amount into a given ratio (Sparx: M525; Corbett: 270).     - Make comparisons between two quantities and represent them as a ratio (Sparx: M801, M885; Corbett: 271b).     - Apply ratio to real contexts and problems (Sparx: M478).

• Extension (Higher Level) Objectives:     - Solve best-buy problems using informal strategies or using the unitary method of solution (Sparx: M681; Corbett: 210).     - Use equality of ratios to solve problems (Sparx: M478; Corbett: 271d).     - Understand and use compound measures such as speed, rates of pay, unit pricing, density, and pressure — without unit changes (Sparx: U151, U256, U527, U842, U910; Corbett: 299, 384, 385).     - Use equations that describe direct and inverse proportion (Sparx: M472, M665, M478; Corbett: 254, 255).     - Solve problems involving direct and inverse proportion algebraically, specifically for $y = kx$ and $y = \frac{1}{x}$ (Sparx: M472, M665, M478; Corbett: 254, 255).

Conversion of Metric Units

• Key Concepts and Units:     - Length: Millimetres ($mm$), Centimetres ($cm$), Metres ($m$), Kilometres ($km$).     - Weight: Grams ($g$), Kilograms ($kg$).     - Capacity: Millilitres ($ml$), Litres ($l$).     - Metric units always utilize conversions of multiples of $10$, such as $10$, $100$, or $1000$.

• Metric Conversions for Length, Weight, and Capacity:     - $1\,m = 100\,cm$.     - $1\,m = 1000\,mm$.     - $1\,km = 1000\,m$.     - $1\,kg = 1000\,g$.     - $1\,l = 1000\,ml$.

• Converting Areas:     - To convert between area units, the linear conversion factor is squared.     - Example: Area = $1\,m^2$ vs. Area = $10000\,cm^2$.     - Conversion formula: ($\times 100^2$) or ($\div 100^2$).

• Converting Volumes:     - To convert between volume units, the linear conversion factor is cubed.     - Example: Volume = $1\,m^3$ vs. Volume = $1000000\,cm^3$.     - Conversion formula: ($\times 100^3$) or ($\div 100^3$).

• Worked Examples:     - a) Convert $12\,cm$ into $mm$: $12 \times 10 = 120\,mm$.     - b) Convert $1783\,g$ into $kg$: $1783 \div 1000 = 1.783\,kg$.     - c) Convert $2.5\,litres$ into $ml$: $2.5 \times 1000 = 2500\,ml$.     - d) Convert $6.8\,m$ into $mm$: $6.8 \times 1000 = 6800\,mm$.     - e) Convert $5000000\,cm^3$ into $m^3$: $5000000 \div 100^3 = 5\,m^3$.     - f) Convert $2\,m^2$ into $cm^2$: $2 \times 100^2 = 20000\,cm^2$.

Ratio Principles

• Key Definitions:     - Ratio: A relationship between two numbers.     - Part: Denotes the numeric value of "1" equivalent unit in a ratio.     - Simplify: To divide both parts of a ratio by their greatest common divisor.     - Equivalent: Values that are equal in value albeit represented differently.     - Convert: To change values from one mathematical form to another (e.g., ratio to fraction).

• Methodological Tips:     - When working with ratios, write letters above the numbers to maintain the correct order (e.g., for specific people or items).

• Simplification Examples:     - Simplify $60:40:100$         - Divide by $10 \rightarrow 6:4:10$         - Divide by $2 \rightarrow 3:2:5$         - Alternative: Divide by $20$ in one step.

• Writing Ratios in the Form $1:n$:     - Example: Write $2:5$ in the form $1:n$.     - Divide both sides by $2$: $2 \div 2 = 1$ and $5 \div 2 = 2.5$.     - Result: $1:2.5$.

• Ratio and Fractions:     - A ratio such as $1:4$ can be converted into a fraction. The total parts are $1 + 4 = 5$. Therefore, the fraction is $\frac{1}{5}$.

Dividing Amounts into Ratios

• Unitary Method for Ratio Division:     - To share an amount, find the total number of parts by adding the numbers in the ratio.     - Divide the total amount by the total number of parts to find the value of one "box" (one part).

• Examples of Sharing Ratios:     - Case 1: Share $\pounds 45$ in the ratio $2:7$.         - Total parts: $2 + 7 = 9$.         - Value per part: $\pounds 45 \div 9 = \pounds 5$.         - Distribution: $2 \times 5 = \pounds 10$ and $7 \times 5 = \pounds 35$.         - Final Answer: $\pounds 10 : \pounds 35$.     - Case 2: Share $\pounds 400$ into a ratio of $2:3$.         - Total parts: $2 + 3 = 5$.         - Value per part: $400 \div 5 = 80$.         - Child 1 receives: $2 \times 80 = \pounds 160$.         - Child 2 receives: $3 \times 80 = \pounds 240$.

• Solving Ratios via Differences:     - Example: Joy and Martin share money in the ratio $2:5$. Martin gets $\pounds 18$ more than Joy. How much do they each get?         - Difference in parts: $5 - 2 = 3$ parts.         - Difference in value: $\pounds 18$.         - Value of one part: $18 \div 3 = 6$.         - Joy gets: $2 \times 6 = \pounds 12$.         - Martin gets: $5 \times 6 = \pounds 30$.     - Example: Party attendees (boys and girls) are in the ratio $5:2$. There are $15$ more boys than girls. Calculate total people.         - Extra parts: $5 - 2 = 3$.         - Value of one part: $15 \div 3 = 5$.         - Total parts: $5 + 2 = 7$.         - Total people: $7 \times 5 = 35$ people.

Ratio and Direct Proportion (Unitary Method)

• Key Words:     - Unitary Method: Finding the value of one single item.     - Best Value: Comparing prices to find the most economical option.     - Proportion: The relative relation between quantities.

• Weight/Unitary Calculation Example:     - If $20$ apples weigh $600\,g$, how much do $28$ apples weigh?         - Weight of $4$ apples: $600 \div 5 = 120\,g$.         - Weight of $28$ apples: $7 \times 120 = 840\,g$.

• Best Value Calculations:     - Formula for money: $\text{Price} \div \text{Quantity}$.     - Box A: $8$ fish fingers for $\pounds 1.40 \rightarrow 1.40 \div 8 = \pounds 0.175$ per unit.     - Box B: $20$ fish fingers for $\pounds 3.40 \rightarrow 3.40 \div 20 = \pounds 0.17$ per unit.     - Conclusion: Box B is better value as each unit costs less.

• Recipe Scaling:     - Scaling for $25$ flapjacks from a baseline of $10$.     - Method 1 (Unitary): Divide baseline amount by $10$ (to find amount for 1) then multiply by $25$.     - Method 2 (Proportional): Find the amount for $5$ (baseline $\div 2$), then multiply by $5$.

Direct and Inverse Proportion (Higher Tier)

• Direct Proportion Definitions:     - Ratio is constant between quantities.     - If $A$ increases, $B$ increases by the same factor.

• Inverse Proportion Definitions:     - One quantity increases in proportion to the other decreasing.     - The product of the quantities is constant.

• Table Completions:     - For Direct Proportion (A:32, B:20):         - Ratio: $20 \div 32 = \frac{5}{8}$.         - To find B from A: Multiply by $\frac{5}{8}$.         - To find A from B: Divide by $\frac{5}{8}$.     - Example Inverse Proportion Calculation:         - If $A = 10, B = 14$, then product $= 140$.         - If $A = 20, B = 140 \div 20 = 7$.         - If $B = 70, A = 140 \div 70 = 2$.

Compound Measures (Higher Tier)

• Speed, Distance, and Time:     - Formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$.     - Formula: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.     - Formula: $\text{Distance} = \text{Speed} \times \text{Time}$.     - Example: Car travelling $35\,mph$ for $227.5\,miles$.         - Time $= 227.5 \div 35 = 6.5\,hours$.         - Convert to time: $6\,hours\,30\,minutes$.

• Density, Mass, and Volume:     - Formula: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$.     - Formula: $\text{Mass} = \text{Density} \times \text{Volume}$.     - Example: Box with Volume $5\,m^3$ and Density $200\,g/m^3$.         - Mass $= 200 \times 5 = 1000\,g$.

• Pressure, Force, and Area:     - Formula: $\text{Pressure} = \frac{\text{Force}}{\text{Area}}$.     - Example: $10\,N$ of force applied to area of $4\,m^2$.         - Pressure $= 10 \div 4 = 2.5\,N/m^2$.

Questions & Discussion

• Ratio Simplification Questions:     - Q1a: Simplify $45 : 63$     - Q1b: Simplify $66 : 44$     - Q1c: Simplify $320 : 440$     - ANSWERS: 1a) $5:7$; 1b) $3:2$; 1c) $8:11$.

• Form $1:n$ Questions:     - Q2a: Write $5 : 10$ in form $1:n$     - Q2b: Write $4 : 6$ in form $1:n$     - ANSWERS: 2a) $1:2$; 2b) $1:1.5$.

• Proportion and Sharing Questions:     - Q: Share $64$ in the ratio $3:5$.     - ANSWER: $24:40$.     - Q: Ann makes vanilla and chocolate cakes in ratio $2:9$. What fraction are chocolate?     - ANSWER: $\frac{9}{11}$.     - Q: Katy and Becky share money ratio $2:1$. Katy gets $\pounds 10$ more. Total each?     - ANSWER: Katy $\pounds 20$, Becky $\pounds 10$.     - Q: Claire and John share $3:2$. Claire receives $\pounds 18$. John gets?     - ANSWER: $\pounds 12$.

• Direct/Best Value Questions:     - Q: Make $24$ gingerbread men with recipe ($10$ baseline: flour $112.5\,g$, ginger $25\,g$, butter $68.75\,g$, sugar $18.75\,g$).     - ANSWER (scaling factor 2.4): $270\,g$ flour, $60\,g$ ginger, $165\,g$ butter, $45\,g$ sugar.     - Q: Packet A ($10$ rolls, $\pounds 3.50$) vs Packet B ($12$ rolls, $\pounds 3.60$).     - ANSWER: Packet B ($30\,p$ per roll).

• Compound Measure Questions:     - Q: Block exerts $120\,N$ force on grounding area of $2\,m^2$. Find pressure.     - ANSWER: $60\,N/m^2$.     - Q: Gold mass $760\,g$, volume $40\,cm^3$. Find density.     - ANSWER: $19\,g/cm^3$.     - Q: Dani drives $63\,miles$ at average speed $27\,mph$. Arrives at work (left at 08:00)?     - ANSWER: $10:20\,am$.