Chapter 1 Rates and Ratios
1A Rates and Concentrations
Converting a rate
1. Write the rate as a fraction. First quantity is the numerator and 1 is the denominator.
2. Convert the first amount to the required unit.
3. Convert the second amount to the required unit.
4. Simplify the fraction
E.g. 55200 m/h to m/min
55200/60
920m/min
Unitary Method
1. Find one unit quantity by dividing the amount
2. Multiply the result to solve the problem
E.g. A car travels 240 km on 30L of petrol. How far does it travel on 7L
30L = 240km
= 240 ÷ 30 x 7
= 56km
Speed
Distance = speed X time
Speed= Distance / Time
Time= Distance / speed
1B Heart Rate
Maximum Heart Rate MHR= 220 - Age
1C Energy Rate
,1kW = 1000W
E.g find the cost of a 200W television for 20 hours at $0.21/kWh
0.2kW × 20hrs = 4 kWh
Cost = 4 × 0.21 = $0.84
1D Fuel Consumption Rate
Fuel consumption = Amount of fuel (L) × 100
Distance travelled
E.g. A car travelled 850 km using 78.2 L of petrol, what was the fuel consumption
78.2 x 100 = 9.2L/100km
850
1E Calculations with Ratios
Equivalent ratios are found by multiplying or dividing by the same number
1F Dividing a Quantity into a Given Ratio
Divides each amount in the Ratio by the total number of parts.
E.g. Mikhail and Ilya were given $450 to share in the ratio 4:5. How much did each get?
1. Calculate the total number of parts by adding each amount in the ratio (4 + 5) = 9
2. Divide the quantity ($450) by the total number of parts (9 parts) to determine the value of
one part. (450 ÷ 9 = $50)
3. Multiply each amount of the ratio by $50 (50X4 -- mikhail, 50X5 -- illya)
4. Mikhail got $200, Ilya got $250
,1G Scale Drawings
Scale of a drawing = Drawing length : Actual length
A scale drawing has a scale of 1:50.
Find the actual length if the drawing length is 30mm. Answer to the nearest centimetre
Find the drawing length if the actual length is 4.5m. Answer to the nearest millimetre.
1. Multiply the drawing length by 50 to determine the actual length.
2. Divide by 10 to change millimetres to centimetres.
3. Divide the actual length by 50 to determine the drawing length.
4. Multiply by 1000 to change metres to millimetres.
Actual length = 30 × 50 mm = 1500 mm
= 150 cm
Drawing length = 4.5 ÷ 50 m = 0.09 m
= 90 mm
1H Plans & Elevations
A plan is a view of an object from the top.
An elevation is a view of an object from one side, such as a front elevation or side elevation.
1I Perimeter, Area & Volume
Perimeter:to calculate the perimeter of land, multiply the length on the aerial photograph by the
scale.
Area:to estimate the area of land, divide it into square grids and count the number of squares.
Volume:If water covers an area (such as the bottom of a pool), to a constant depth, its
volume can be calculated using V = Ah
Trapezoidal rule:
, 2D Shapes – Area Formulas
Shape Formula Variables
Square A = s² s = side length
Rectangle A=l×w l = length, w = width
Triangle A=½×b×h b = base, h = height
Circle A = π × r² r = radius
Semicircle A = ½ × π × r² r = radius
Trapezoid A = ½ × (a + b) × h a, b = bases, h =
height
Parallelogram A=b×h b = base, h = height
3D Shapes – Volume and Surface Area Formulas
Shape Volume Surface Area Variables
Formula Formula
Cube V = s³ SA = 6 × s² s = side length
Rectangular V=l×w×h SA = 2(lw + lh + l = length, w = width, h = height
Prism wh)
Cylinder V = π × r² × h SA = 2πr²+2πrh r = radius, h = height
Sphere V = ⁴⁄₃ × π × r³ SA = 4π × r² r = radius
Hemisphere V = ²⁄₃ × π × r³ SA = 3π × r² r = radius
1A Rates and Concentrations
Converting a rate
1. Write the rate as a fraction. First quantity is the numerator and 1 is the denominator.
2. Convert the first amount to the required unit.
3. Convert the second amount to the required unit.
4. Simplify the fraction
E.g. 55200 m/h to m/min
55200/60
920m/min
Unitary Method
1. Find one unit quantity by dividing the amount
2. Multiply the result to solve the problem
E.g. A car travels 240 km on 30L of petrol. How far does it travel on 7L
30L = 240km
= 240 ÷ 30 x 7
= 56km
Speed
Distance = speed X time
Speed= Distance / Time
Time= Distance / speed
1B Heart Rate
Maximum Heart Rate MHR= 220 - Age
1C Energy Rate
,1kW = 1000W
E.g find the cost of a 200W television for 20 hours at $0.21/kWh
0.2kW × 20hrs = 4 kWh
Cost = 4 × 0.21 = $0.84
1D Fuel Consumption Rate
Fuel consumption = Amount of fuel (L) × 100
Distance travelled
E.g. A car travelled 850 km using 78.2 L of petrol, what was the fuel consumption
78.2 x 100 = 9.2L/100km
850
1E Calculations with Ratios
Equivalent ratios are found by multiplying or dividing by the same number
1F Dividing a Quantity into a Given Ratio
Divides each amount in the Ratio by the total number of parts.
E.g. Mikhail and Ilya were given $450 to share in the ratio 4:5. How much did each get?
1. Calculate the total number of parts by adding each amount in the ratio (4 + 5) = 9
2. Divide the quantity ($450) by the total number of parts (9 parts) to determine the value of
one part. (450 ÷ 9 = $50)
3. Multiply each amount of the ratio by $50 (50X4 -- mikhail, 50X5 -- illya)
4. Mikhail got $200, Ilya got $250
,1G Scale Drawings
Scale of a drawing = Drawing length : Actual length
A scale drawing has a scale of 1:50.
Find the actual length if the drawing length is 30mm. Answer to the nearest centimetre
Find the drawing length if the actual length is 4.5m. Answer to the nearest millimetre.
1. Multiply the drawing length by 50 to determine the actual length.
2. Divide by 10 to change millimetres to centimetres.
3. Divide the actual length by 50 to determine the drawing length.
4. Multiply by 1000 to change metres to millimetres.
Actual length = 30 × 50 mm = 1500 mm
= 150 cm
Drawing length = 4.5 ÷ 50 m = 0.09 m
= 90 mm
1H Plans & Elevations
A plan is a view of an object from the top.
An elevation is a view of an object from one side, such as a front elevation or side elevation.
1I Perimeter, Area & Volume
Perimeter:to calculate the perimeter of land, multiply the length on the aerial photograph by the
scale.
Area:to estimate the area of land, divide it into square grids and count the number of squares.
Volume:If water covers an area (such as the bottom of a pool), to a constant depth, its
volume can be calculated using V = Ah
Trapezoidal rule:
, 2D Shapes – Area Formulas
Shape Formula Variables
Square A = s² s = side length
Rectangle A=l×w l = length, w = width
Triangle A=½×b×h b = base, h = height
Circle A = π × r² r = radius
Semicircle A = ½ × π × r² r = radius
Trapezoid A = ½ × (a + b) × h a, b = bases, h =
height
Parallelogram A=b×h b = base, h = height
3D Shapes – Volume and Surface Area Formulas
Shape Volume Surface Area Variables
Formula Formula
Cube V = s³ SA = 6 × s² s = side length
Rectangular V=l×w×h SA = 2(lw + lh + l = length, w = width, h = height
Prism wh)
Cylinder V = π × r² × h SA = 2πr²+2πrh r = radius, h = height
Sphere V = ⁴⁄₃ × π × r³ SA = 4π × r² r = radius
Hemisphere V = ²⁄₃ × π × r³ SA = 3π × r² r = radius
