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In Context: Rates and Advanced
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Foundational Skills
Estimate result before calculating
\[ 4.8 \times 19 \approx \text{?} \]
Estimate before finding exact answer.
Identify operation for rate problems
\[ \text{Speed} = \text{?} \div \text{?} \]
Identify the correct operation.
Recognise when to convert units
\[ \text{m} \rightarrow \text{km first?} \]
Identify when conversion is needed.
Powers of 10 Multiplication
Multiply decimal by 10
\[ 3.4\text{ m} = \text{?}\text{ cm} \]
Convert using ×10.
Multiply decimal by 100
\[ 2.45\text{ kg} = \text{?}\text{ g} \]
Convert using ×100.
Multiply decimal by 1000
\[ 1.5\text{ L} = \text{?}\text{ ml} \]
Convert using ×1000.
Multiply by powers of 10
\[ 2.5 \times 100 = \text{?} \]
Decimal point movement.
Powers of 10 Division
Divide decimal by 10
\[ 340\text{ cm} = \text{?}\text{ m} \]
Convert using ÷10.
Divide decimal by 100
\[ 2450\text{ g} = \text{?}\text{ kg} \]
Convert using ÷100.
Divide decimal by 1000
\[ 3500\text{ m} = \text{?}\text{ km} \]
Convert using ÷1000.
Divide by powers of 10
\[ 56 \div 100 = \text{?} \]
Decimal point movement.
Multiplying Decimals
Multiply two decimals (tenths)
\[ 2.5 \times 1.4 = \text{?} \]
Both to 1 decimal place.
Multiply decimals (mixed places)
\[ 1.5 + 8 \times 0.25 = \text{?} \]
Different decimal places.
Calculate area with decimals
\[ 4.5 \times 3.2 = \text{?}\text{ m}^2 \]
Area with decimal dimensions.
Find volume with decimals
\[ 2.5 \times 1.2 \times 0.8 = \text{?} \]
Volume with decimal dimensions.
Dividing by Decimals
Divide whole by decimal
\[ 12 \div 0.5 = \text{?} \]
Whole number ÷ decimal.
Divide decimal by decimal (same)
\[ 8.4 \div 1.2 = \text{?} \]
Same decimal places.
Divide decimal by decimal (different)
\[ 15.6 \div 0.8 = \text{?} \]
Different decimal places.
Calculate using reciprocal
\[ 6 \div 0.25 = 6 \times 4 \]
Reciprocal method.
Speed Calculations
Calculate speed
\[ S = D \div T \]
Speed from distance and time.
Calculate distance
\[ D = S \times T \]
Distance from speed and time.
Calculate time
\[ T = D \div S \]
Time from distance and speed.
Convert speed units
\[ \text{m/s} \leftrightarrow \text{km/h} \]
Convert between speed units.
Average speed (two stages)
\[ \text{Total } D \div \text{Total } T \]
Multi-stage journey average.
Density Calculations
Calculate density
\[ \rho = M \div V \]
Density from mass and volume.
Calculate mass
\[ M = \rho \times V \]
Mass from density and volume.
Calculate volume
\[ V = M \div \rho \]
Volume from mass and density.
Unit Rates
Calculate cost per kilogram
\[ £8.50 \div 5 = £\text{?}/\text{kg} \]
Unit price calculation.
Calculate fuel consumption
\[ \text{L per 100km} \]
Fuel consumption rate.
Calculate rate per hour
\[ 135 \div 4.5 = \text{?}/\text{h} \]
Production rate per hour.
Compare unit rates
\[ £\text{?}/\text{kg vs } £\text{?}/\text{kg} \]
Which is better value?
Working Backwards
Find original after ×
\[ \text{?} \times 2.5 = 15 \]
Reverse a multiplication.
Find original after ÷
\[ \text{?} \div 0.5 = 24 \]
Reverse a division.
Find original length
\[ 5 \times 1.8 = \text{?}\text{ m} \]
Original from equal pieces.
Speed from fractional time
\[ 45 \div 0.75 = \text{?} \]
Speed with decimal time.
Averages with Decimals
Find mean of decimals
\[ \text{Mean} = \text{Sum} \div n \]
Mean of decimal measurements.
Find missing value (given mean)
\[ 3.2, 4.1, 3.8, \text{?} \; (\bar{x}=3.6) \]
Missing value from mean.
Calculate weighted mean
\[ 15.5 \times 40\% + 18.2 \times 60\% \]
Weighted average.
Multi-step Problems
Convert units then calculate rate
\[ 4500\text{ m in 3 min} \rightarrow \text{km/h} \]
Unit conversion then rate.
Calculate then multiply by rate
\[ £1.85 \times 4.5\text{ kg} = \text{?} \]
Cost using unit rate.
Multi-stage journey distance
\[ D_1 + D_2 = \text{Total} \]
Total distance for journey.
Compare rates, find difference
\[ \text{Who is faster by how much?} \]
Compare two rates.
Area with unit conversion
\[ 4.5\text{ m} \times 360\text{ cm} \]
Mixed units for area.
Percentage then decimal operation
\[ 8 \times 1.25 – 1.5 = \text{?} \]
% change then add/subtract.
Special Cases
Very large decimals in context
\[ 45.3 \text{ thousand} = \text{?} \]
Large decimals (thousands, millions).
Decimal remainders in context
\[ 15.5 \div 1.2 = \text{? full, ? left} \]
Interpret remainders.
Significant figures in context
\[ 5.4782 \rightarrow 2\text{ s.f.} \]
Round to significant figures.
Compare rates (different bases)
\[ 50\text{L}/600\text{km vs } 7.5\text{L}/100\text{km} \]
Convert to same base to compare.
Timer (Optional)
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