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Daily Maths

Daily Maths

Rates and unit conversion

Unit Conversion Paper 1 + 2 12 min read

On this page
  1. In this note
  2. Formula bank
  3. Metric units
  4. Area and volume units
  5. Rates: pay, flow, fuel
  6. Density, pressure and population density
  7. Converting compound units
  8. Ways they test it
  9. Mistakes to avoid
  10. Summary
  11. Practice
  12. Can I do this?
  13. Quiz

In this note

Syllabus 1.12 rates, with units of measure
  • 1.12Convert metric units of length, mass and capacity, and area and volume units.
  • 1.12Use rates: hourly pay and overtime, flow rate, fuel consumption, charges.
  • 1.12Use density, pressure and population density, and convert compound units such as km/h to m/s.

Before you start: ratio and proportion. Speed and travel graphs: the Kinematics chapter.

Formula bank

Big ↔ small unitLearn itbig → small: ×; small → big: ÷1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g, 1 litre = 1000 ml.Use it when any unit change.
Area and volume unitsLearn itarea: × (factor)\(^{2}\); volume: × (factor)\(^{3}\)\(1\text{ m}^2=10\,000\text{ cm}^2\); \(1\text{ m}^3=1\,000\,000\text{ cm}^3\).Use it when squared or cubed units.
CapacityLearn it\(1\text{ cm}^3=1\text{ ml}\); \(1000\text{ cm}^3=1\) litreUse it when volume of a container in litres.
RateLearn itrate = amount ÷ time; amount = rate × time; time = amount ÷ rate"per" means divide.Use it when flow, pay, fuel.
DensityLearn itdensity = mass ÷ volumeg/cm³ or kg/m³.Use it when mass and volume.
PressureLearn itpressure = force ÷ areaN/m².Use it when force on an area.
Population densityLearn itpeople ÷ areapeople per km².Use it when towns and countries.
km/h ↔ m/sLearn itkm/h → m/s: × 1000 ÷ 3600 (÷ 3.6)m/s → km/h: × 3.6.Use it when speeds in different units.

Metric units

MeasuresUnits, small → bigLinks
Lengthmm, cm, m, km10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km
Massmg, g, kg, tonne1000 mg = 1 g, 1000 g = 1 kg, 1000 kg = 1 tonne
Capacityml, litre1000 ml = 1 litre; \(1\text{ cm}^3=1\text{ ml}\)

General methodHow to convert any metric unit

  1. Find the link (e.g. 1 kg = 1000 g).
  2. Big unit → small unit: MULTIPLY (there are more small ones). Small → big: DIVIDE.

Example Change 850 g to kg.

Small to big: divide by 1000.

  1. \(850\div1000=0.85\)

Answer 0.85 kg

Try it yourself

1
Change 0.75 m to cm.
Show answer

Answer 75 cm

2
Change 2500 ml to litres.
Show answer

Answer 2.5 litres

3
Change 3.2 tonnes to kg.
Show answer

Answer 3200 kg

Area and volume units

A square 1 m by 1 m is also 100 cm by 100 cm, so \(1\text{ m}^2=100\times100=10\,000\text{ cm}^2\). The length factor is squared for area and cubed for volume.

1 m² = 100 cm × 100 cm1 m = 100 cm1 m = 100 cm
100 rows of 100 squares: \(10\,000\text{ cm}^2\).

General methodHow to convert area or volume units

  1. Find the LENGTH factor (m → cm is 100).
  2. Area: use the factor squared (100² = 10 000). Volume: cubed (100³ = 1 000 000).
  3. Big → small multiply; small → big divide.

Example Change \(36\,000\text{ cm}^3\) into \(\text{m}^3\).

Small to big, volume: divide by \(100^{3}\).

  1. \(36\,000\div1\,000\,000=0.036\)

Answer \(0.036\text{ m}^3\)

Try it yourself

1
Change \(2.5\text{ m}^2\) to \(\text{cm}^2\).
Show answer

Answer \(25\,000\text{ cm}^2\)

2
Change \(1.2\text{ m}^3\) to \(\text{cm}^3\).
Show answer

Answer \(1\,200\,000\text{ cm}^3\)

3
A box holds \(160\,000\text{ cm}^3\). How many litres?
Show answer

Answer 160 litres

Rates: pay, flow, fuel

A rate is how much per ONE unit of something else. "Per" means divide: 12 litres per minute = \(\dfrac{12\text{ litres}}{1\text{ minute}}\). From one rate you get three calculations: rate = amount ÷ time, amount = rate × time, time = amount ÷ rate.

General methodHow to answer any rate question

  1. Write the rate with its units (e.g. litres per minute).
  2. Make the other quantities match those units (cm³ → litres, minutes → hours).
  3. Multiply or divide using the triangle above.
  4. Time answers: change decimal parts into minutes/seconds (× 60).

Example A tank is 80 cm by 50 cm by 40 cm. A hose fills it at 12 litres per minute. How long does it take, in minutes and seconds?

Volume in litres, then ÷ the rate, then the decimal part × 60.

  1. \(80\times50\times40=160\,000\text{ cm}^3=160\) litres
  2. \(160\div12=13.33\ldots\) min
  3. \(0.33\ldots\times60=20\) s

Answer 13 minutes 20 seconds

Pay: pay = rate × hours. "Time and a half" multiplies the RATE by 1.5. Fuel: "litres per 100 km" and "km per litre" are upside down versions: 100 ÷ one gives the other.

Try it yourself

1
$14.50 an hour for 38 hours, time and a half after that. Pay for 44 hours?
Show answer

Answer $681.50

2
A plumber charges $45 plus $18 per half hour. Charge for 1 hour 30 minutes?
Show answer

Answer $99

3
A car uses 6.4 litres per 100 km. Fuel costs $1.25 a litre. Cost of 350 km?
Hint
6.4 × 3.5 litres.
Show answer

Answer $28

Density, pressure and population density

General methodHow to use any 'per' formula

  1. Learn the formula from its units: g/cm³ = grams ÷ cm³, so density = mass ÷ volume. N/m² = force ÷ area.
  2. Make the units match the formula's units.
  3. Rearrange if needed: mass = density × volume; area = force ÷ pressure.

Example A metal block is 8 cm by 5 cm by 2.5 cm and has mass 790 g. Find its density.

Density = mass ÷ volume.

  1. volume \(=8\times5\times2.5=100\text{ cm}^3\)
  2. \(790\div100=7.9\)

Answer 7.9 g/cm³

Try it yourself

1
A crate of weight 600 N rests on 0.25 m². Pressure in N/m²?
Show answer

Answer 2400

2
1 200 000 people live in 400 km². Population density?
Show answer

Answer 3000 per km²

3
Density 2.5 g/cm³, volume 40 cm³. Mass in g?
Show answer

Answer 100 g

Converting compound units

General methodHow to convert any compound unit (km/h, g/cm³, litres per minute)

  1. Write the rate as a fraction: \(\dfrac{72\text{ km}}{1\text{ h}}\).
  2. Convert the top: \(72\text{ km}=72\,000\text{ m}\).
  3. Convert the bottom: \(1\text{ h}=3600\text{ s}\).
  4. Divide: \(72\,000\div3600=20\) m/s.

Example Change 15 m/s to km/h.

Metres in an hour, then ÷ 1000.

  1. \(15\times3600=54\,000\) m per hour
  2. \(54\,000\div1000=54\)

Answer 54 km/h

Try it yourself

1
Change 54 km/h to m/s.
Show answer

Answer 15 m/s

2
Change 45 m/s to km/h.
Show answer

Answer 162 km/h

3
A tap gives 4.8 litres a minute. In ml per second?
Show answer

Answer 80 ml/s

Ways they test it

Type 1 Hidden unit change

"Maryam has 3 kg of flour and uses 120 g for each of 18 cakes. How much is left, in grams?" (2 marks)

Spot it two units for the same thing.

  1. Convert to the unit the answer wants.
  2. Then calculate.

For example 3 kg, 18 × 120 g

  1. \(3000-2160=840\)

Answer 840 g

Type 2 Area / volume units

"Change 2.5 m² to cm²." (1 mark)

Spot it a ² or ³ on the unit.

  1. Square or cube the length factor.

Type 3 Pay, overtime and charges

"Time and a half after 38 hours..."

Spot it hourly pay or a call-out charge.

  1. Normal pay + overtime pay.
  2. Overtime rate = 1.5 × normal rate.

Type 4 Fill / empty a container

"How long to fill the tank? Give minutes and seconds." (4 marks)

Spot it a volume and a flow rate.

  1. Volume (Mensuration), then in litres.
  2. Time = volume ÷ rate.
  3. Decimal minutes × 60.

Type 5 Density / pressure

"Find the mass in kg of 0.03 m³ of the metal." (2 marks)

Spot it density, pressure, people per km².

  1. Learn the formula.
  2. Match the units (g/cm³ → kg/m³ is × 1000).

For example 7.9 g/cm³ = 7900 kg/m³

  1. \(7900\times0.03=237\)

Answer 237 kg

Type 6 Compare two rates

"Which city has the greater population density?"

Spot it two rates to compare.

  1. Work out both in the same unit.
  2. Conclusion with both numbers.

Mistakes to avoid

  • ✗ \(1\text{ m}^2=100\text{ cm}^2\)✓ \(10\,000\text{ cm}^2\)Square the length factor for area.
  • ✗ 72 km/h = 72 000 ÷ 60 m/s✓ 72 000 ÷ 3600 = 20 m/sAn hour has 3600 seconds.
  • ✗ 13.33 minutes = 13 min 33 s✓ 13 min 20 s0.33 of a minute × 60 = 20 seconds.
  • ✗ Overtime pay = hours × 1.5✓ hours × (rate × 1.5)Time and a half applies to the RATE.
  • ✗ Density = volume ÷ mass✓ mass ÷ volumeg/cm³: grams on top.

Summary

  • Big → small: multiply; small → big: divide.
  • Area: factor squared; volume: factor cubed. 1000 cm³ = 1 litre.
  • "Per" means divide: rate = amount ÷ time.
  • Compound units: convert top and bottom separately.
  • Density = mass ÷ volume; pressure = force ÷ area.

If you see …

  • m², cm², m³square or cube the factor.
  • "minutes and seconds"decimal part × 60.
  • km/h with m/s× 1000 ÷ 3600.

Practice

Practice: easy, medium, hard

Easy

1
\(3\text{ m}^2\) in \(\text{cm}^2\)?
Show working
  1. \(3\times10\,000\)

Answer 30 000

2
A tank holds \(25\,000\text{ cm}^3\). How many litres?
Show working
  1. \(25\,000\div1000\)

Answer 25

3
$12 an hour, time and a half for overtime. Pay for 3 hours of overtime?
Show working
  1. overtime rate $18
  2. \(3\times18=54\)

Answer $54

Medium

1
Ravi has a 2 kg bag of sugar and uses 85 g in each of 15 cakes. How much is left, in grams?
Guide me

What is it asking? sugar left in g

What do we know? 2 kg and 15 × 85 g

Plan convert, then subtract

Show working
  1. \(15\times85=1275\)
  2. \(2000-1275=725\)

Answer 725 g

2
A car does 16 km per litre. Consumption in litres per 100 km?
Guide me

What is it asking? litres per 100 km

What do we know? 16 km per litre

Plan 100 ÷ 16

Show working
  1. \(100\div16=6.25\)

Answer 6.25

3
Pressure 2000 N/m² on an area of 0.4 m². Force in N?
Guide me

What is it asking? the force

What do we know? pressure and area

Plan force = pressure × area

Show working
  1. \(2000\times0.4=800\)

Answer 800 N

Hard

1
A paddling pool holds \(0.6\text{ m}^3\). It fills at 8 litres per minute. How long, in minutes?
Guide me

What is it asking? the time

What do we know? \(0.6\text{ m}^3\), 8 l/min

Plan m³ → litres (× 1000), then ÷ 8

Show working
  1. \(0.6\text{ m}^3=600\) litres
  2. \(600\div8=75\)

Answer 75 minutes (1 h 15 min)

PartAnswer / workingMarksGuidance
600 litresM1
\(600\div8\)M1
75 min or 1 h 15 minA1
Total: 3 marks
2
A city of 13 000 people covers 2.5 km². Another has 4200 people per km². Which is more crowded? Give the first city's density.
Guide me

What is it asking? compare densities

What do we know? 13 000 people, 2.5 km²

Plan people ÷ area, then compare

Show working
  1. \(13\,000\div2.5=5200\)
  2. 5200 > 4200

Answer 5200 per km², so the first city

PartAnswer / workingMarksGuidance
\(13\,000\div2.5\)M1
5200A1
first city, with both figuresA1
Total: 3 marks
3
A cheetah runs at 19.6 m/s. Change this to km/h.
Guide me

What is it asking? km/h

What do we know? metres per second

Plan × 3600, then ÷ 1000

Show working
  1. \(19.6\times3600=70\,560\)
  2. \(\div1000=70.56\)

Answer 70.56 km/h

PartAnswer / workingMarksGuidance
\(19.6\times3600\)M1
70.56A1
Total: 2 marks

Can I do this?

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Quiz

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