Year 8 Unit 8: Ratio, Proportion and Compound Measures

Learning Outcomes

Support Level Outcomes

• Describe the proportion of something by words, fractions, decimals, or percentages (Sparx: M267).
• To change freely between standard metric units of length or 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 (Sparx: 351).
• Express a quantity as a ratio and as a proportion (Sparx: M901).

Core Level Outcomes

• Use percentages, decimals, or fractions to calculate proportions (Sparx: M885, M267, M478; Corbett: 137, 234).
• 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 Level Outcomes

• 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 - do not include change of units (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, only $y=kx$ and $y=\frac{1}{x}$ (Sparx: M472, M665, M478; Corbett: 254, 255).

Conversion of Metric Units

Key Concepts and Words

• Metric units of length: Millimetres ($mm$), centimetres ($cm$), metres ($m$), and kilometres ($km$).
• Metric units of weight: Grams ($g$) and kilograms ($kg$).
• Metric units of capacity: Millilitres ($ml$) and litres ($l$).
• Multiple Base: All metric units utilize conversions of multiples of 10 (e.g., $10$, $100$, $1000$).

Standard Conversion Scales

• Length:   - $10\,mm = 1\,cm$   - $100\,cm = 1\,m$   - $1000\,m = 1\,km$
• Capacity:   - $1000\,ml = 1\,l$
• Weight:   - $1000\,g = 1\,kg$

Converting Areas and Volumes

• Converting Areas: To convert from $m^2$ to $cm^2$, multiply by $100^2$. To convert from $cm^2$ to $m^2$, divide by $100^2$.   - Example: $\text{Area} = 1\,m^2 = 100^2\,cm^2 = 10000\,cm^2$
• Converting Volumes: To convert from $m^3$ to $cm^3$, multiply by $100^3$. To convert from $cm^3$ to $m^3$, divide by $100^3$.   - Example: $\text{Volume} = 1\,m^3 = 100^3\,cm^3 = 1000000\,cm^3$

Examples and Practice Answers

• 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 $m^3$: $5000000 \div 1000000 = 5\,m^3$
• f) Convert $2\,m^2$ into $cm^2$: $2 \times 10000 = 20000\,cm^2$

Ratio Fundamentals

Definitions

• Ratio: The relationship between two numbers.
• Part: The numeric value that '1' of would be equivalent to.
• Simplify: Dividing both parts of a ratio by the same number to reduce it to its lowest terms.
• Equivalent: Equal in value.
• Convert: To change from one form into another form.

Key Concepts

• To simplify a ratio like $2:6$, find a common factor (in this case, 2). $2 \div 2 : 6 \div 2 = 1:3$.
• Simplifying complex ratios: For $60:40:100$, divide by 10 to get $6:4:10$, then divide by 2 to get $3:2:5$. (Alternatively, divide by 20 in one step).
• Form $1:n$: To write $2:5$ in the form $1:n$, divide both sides by the first number (2): $2 \div 2 : 5 \div 2 = 1:2.5$.

Sharing Amounts in Ratios

• Process: Add the parts to find the total number of boxes, divide the total amount by the number of boxes to find the value of one 'part', then multiply accordingly.
• Example (Share $\pounds 45$ in ratio $2:7$):   - Total parts: $2 + 7 = 9$   - Value per part: $\pounds 45 \div 9 = \pounds 5$   - Amounts: $2 \times 5 = \pounds 10$ and $7 \times 5 = \pounds 35$.
• Difference Variation (Joy and Martin): Joy and Martin share money in ratio $2:5$. Martin gets $\pounds 18$ more than Joy. How much do they each get?   - Difference in parts: $5 - 2 = 3\,parts$   - Value per part: $\pounds 18 \div 3 = \pounds 6$   - Joy gets: $2 \times 6 = \pounds 12$   - Martin gets: $5 \times 6 = \pounds 30$

Questions and Answers

• 1a) Simplify $45:63$ (divide by 9): $5:7$
• 1b) Simplify $66:44$ (divide by 22): $3:2$
• 1c) Simplify $320:440$ (divide by 40): $8:11$
• 2a) Write $5:10$ in form $1:n$: $1:2$
• 2b) Write $4:6$ in form $1:n$: $1:1.5$
• 3) Share $64$ in ratio $3:5$: Total parts = 8. $64 \div 8 = 8$. Parts are $24:40$.
• 4) Write the ratio $1:4$ as a fraction: $\frac{1}{1+4} = \frac{1}{5}$.

Dividing an Amount into Ratios

Relationship to Fractions

• A ratio can be converted into fractions by using the sum of the parts as the denominator.
• For a ratio Red : Green of $1:3$, the fraction of Red is $\frac{1}{4}$ and the fraction of Green is $\frac{3}{4}$.

Examples

• Child inheritance: A woman splits $\pounds 400$ between two children in ratio $2:3$.   - Total parts: $2 + 3 = 5$   - Value of 1 box: $400 \div 5 = 80$   - Child 1: $2 \times 80 = \pounds 160$   - Child 2: $3 \times 80 = \pounds 240$
• Party attendees: Boys and girls are in the ratio $5:2$. There are 15 more boys than girls.   - Extra boxes for boys: $5 - 2 = 3$   - Value of 1 box: $15 \div 3 = 5$   - Total people: $7\,parts \times 5 = 35\,people$.

Specific Exercises

• 1) Vanilla to chocolate cakes ratio $2:9$. Chocolate fraction? Answer: $\frac{9}{11}$.
• 2) Share $\pounds 25$ in ratio $7:3$. Answer: $\pounds 17.50$, $\pounds 7.50$.
• 3) Katy and Becky ratio $2:1$. Katy gets $\pounds 10$ more. Amounts? Answer: $\pounds 20$, $\pounds 10$.
• 4) Claire and John ratio $3:2$. Claire receives $\pounds 18$. John's amount? Answer: Claire's $3\,parts = 18$, so $1\,part = 6$. John receives $2 \times 6 = \pounds 12$.

Ratio and Direct Proportion

Key Words

• Unitary: Finding the value of a single item.
• Best Value: Comparing prices to find the cheapest option per unit.
• Proportion: A part, share, or number considered in comparative relation to a whole.
• Quantity: An amount or number of material or immaterial things.

Methods

• Unitary Method (Monetary): $\frac{\text{Cost}}{\text{Quantity}}$
• Unitary Method (Recipes): $\frac{\text{Ingredient}}{\text{Number of items}} = \text{Ingredient per item}$

Example 1: Weight calculation

• If $20\,apples$ weigh $600\,g$. How much do $28\,apples$ weigh?   - $600 \div 20 = 30\,g$ per apple.   - $28 \times 30 = 840\,g$.

Example 2: Best Value comparison

• Box A: 8 fish fingers for $\pounds 1.40$. $\pounds 1.40 \div 8 = \pounds 0.175$
• Box B: 20 fish fingers for $\pounds 3.40$. $\pounds 3.40 \div 20 = \pounds 0.17$
• Conclusion: Box B is better value as each unit costs less.

Example 3: Recipe adjustment (Making 25 Flapjacks from a 10-Flapjack Recipe)

• Recipe for 10: $80\,g$ Oats, $60\,g$ Butter, $30\,g$ Syrup, $36\,g$ Sugar.
• Unitary Method: Divide items by 10 then multiply by 25.   - Oats: $8 \times 25 = 200\,g$   - Butter: $6 \times 25 = 150\,g$   - Syrup: $3 \times 25 = 75\,g$   - Sugar: $3.6 \times 25 = 90\,g$

Direct and Inverse Proportion (Higher Tier)

Definitions

• Direct Proportion: Variables are directly proportional when the ratio between quantities is constant ($\frac{x}{y} = k$).
• Inverse Proportion: Variables are inversely proportional when one quantity increases as the other decreases ($x \times y = k$).

Calculations for Proportion Tables

• Direct Proportion Logic: If $A = 32$ and $B = 20$, the ratio is $20 \div 32 = \frac{5}{8}$. Multiply A by $\frac{5}{8}$ to get B; divide B by $\frac{5}{8}$ to get A.
• Direct Proportion Example:   - $A$ values: $5, P, 22$   - $B$ values: $9, 28.8, Q$   - Ratio $B/A = 9 \div 5 = 1.8$   - $P = 28.8 \div 1.8 = 16$   - $Q = 22 \times 1.8 = 39.6$
• Inverse Proportion Example:   - $A$ values: $10, 20, 14, R, 28$   - $B$ values: $14, P, 10, 70, 5$   - Constant ($k$): $10 \times 14 = 140$   - $P = 140 \div 20 = 7$   - $R = 140 \div 70 = 2$
• Inverse Proportion Example 2:   - $A$ values: $4, P, 18$   - $B$ values: $9, 3, Q$   - Constant: $4 \times 9 = 36$   - $P = 36 \div 3 = 12$   - $Q = 36 \div 18 = 2$

Compound Measures (Higher Tier)

Standard Formulas

• Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
• Pressure: $\text{Pressure} = \frac{\text{Force}}{\text{Area}}$
• Density: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

Calculating Time, Mass, and Pressure

• Time Calculation: A car travels $227.5\,miles$ at $35\,mph$.   - $\text{Time} = 227.5 \div 35 = 6.5\,h$   - $6.5\,h = 6\,h\,30\,mins$.
• Mass Calculation: A $5\,m^3$ box has a density of $200\,g/m^3$.   - $\text{Mass} = \text{Density} \times \text{Volume} = 200 \times 5 = 1000\,g$.
• Pressure Calculation: $10\,N$ of force applied to an area of $4\,m^2$.   - $\text{Pressure} = 10 \div 4 = 2.5\,N/m^2$.

Unit Exercises

• 1) Pressure exerted by a $120\,Newton$ block on $2\,m^2$: $120 \div 2 = 60\,N/m^2$.
• 2) Density of gold with mass $760\,g$ and volume $40\,cm^3$: $760 \div 40 = 19\,g/cm^3$.
• 3) Dani drives $63\,miles$ at $27\,mph$, leaving at 08:00.   - $\text{Duration} = 63 \div 27 = 2.\dot{3}\text{ hours} = 2\,h\,20\,mins$.   - Arrival time: $08:00 + 2\,h\,20\,mins = 10:20\,am$.