Rates
Theory
A rate compares two quantities of different kinds, so it always carries units: \(\$35\) for \(5\) kg is the rate \(\$7\)/kg. A ratio compares quantities of the same kind and has no units. This Year 8 maths page shows how to simplify, change and compare rates.
Rate or ratio? \(\$7.20\) for \(3\) kg is a rate: dollars and kilograms are different kinds of quantity. \(300\) g to \(500\) g is a ratio, \(3:5\): both are masses.
A simplified rate is the amount for one unit: rate \(=\) amount \(\div\) quantity. Read the slash as per: \(\$2.40\)/kg is \(\$2.40\) per kilogram. Always write the units.
Turn the rule round. amount \(=\) rate \(\times\) quantity, and quantity \(=\) amount \(\div\) rate. The units say which way round: L/min \(\times\) min gives L.
Decimals in the division? Multiply both numbers by \(10\) or \(100\): \(64.8 \div 4.5 = 648 \div 45 = 14.4\).
Changing units. Change one unit at a time. Per minute to per hour: \(\times 60\). Per second to per hour: \(\times 3600\). L to kL: \(\div 1000\). For speed, km/h \(\div 3.6\) gives m/s and m/s \(\times 3.6\) gives km/h.
Comparing rates. Change them to the same units first: \(1.5\) km/min is \(90\) km/h.
A rate in a formula. In \(C = 30 + 2.40n\), the number multiplied by the pronumeral is the rate (\(\$2.40\) for each \(n\)); the \(\$30\) is charged once.
Fuel. \(20\) km on each litre uses \(100 \div 20 = 5\) L per \(100\) km. Net rate: a fill and a leak together give fill rate \(-\) leak rate.
Rate or ratio?
| Comparison | It is a |
|---|---|
| \(300\) g to \(500\) g | ratio \(3:5\) |
| \(180\) km in \(2\) h | rate \(90\) km/h |
| \(\$7.20\) for \(3\) kg | rate \(\$2.40\)/kg |
| \(40\) min to \(1\) h | ratio \(2:3\) |
| \(15\) mm of rain in \(3\) h | rate \(5\) mm/h |
Writing a simplified rate
- Decide which quantity goes “per” (it is the one you divide by).
- Divide: rate \(=\) amount \(\div\) quantity.
- Change units one at a time if the answer needs different units.
- Write the units, such as \(\$\)/kg, L/min or km/h.
| \(35 \div 5\) | \(=\) | \(7\) |
\(\$7\)/kg
| \(28 \times 100\) | \(=\) | \(2800 \text{ cm}\) |
| \(2800 \div 175\) | \(=\) | \(16\) |
\(16\) cm/year
(i) \(\$23.10\) for \(2.2\) kg of cherries
(ii) \(250\) mL to \(1.5\) L
| \(23.10 \div 2.2\) | \(=\) | \(231 \div 22 = 10.5\) |
| \(250:1500\) | \(=\) | \(1:6\) |
(i) A rate: \(\$10.50\)/kg (different kinds of quantity).
(ii) A ratio: \(1:6\) (both volumes, so \(1.5\) L \(= 1500\) mL first).
(i) Write this as a rate in pages per hour.
(ii) How long does a \(260\)-page job take?
| \(52 \div 8\) | \(=\) | \(6.5\) |
| \(6.5 \times 60\) | \(=\) | \(390\) |
| \(260 \div 6.5\) | \(=\) | \(40\) |
(i) \(390\) pages/h (\(6.5\) pages/min first)
(ii) \(40\) min (pages \(\div\) pages/min gives minutes)
(ii) A tap drips \(1.5\) mL every second. Write this in L/h.
| \(22.5 \times 3600\) | \(=\) | \(81\,000 \text{ m/h}\) |
| \(81\,000 \div 1000\) | \(=\) | \(81\) |
| \(1.5 \times 3600\) | \(=\) | \(5400 \text{ mL/h}\) |
| \(5400 \div 1000\) | \(=\) | \(5.4\) |
(i) \(81\) km/h (check: \(22.5 \times 3.6 = 81\))
(ii) \(5.4\) L/h
(i) At what rate does it use fuel, in L/km?
(ii) Write that rate in L/100 km.
(iii) How far can it drive on the full tank?
| \(0.08 \times 100\) | \(=\) | \(8\) |
| \(70 \div 0.08\) | \(=\) | \(7000 \div 8 = 875\) |
(i) \(0.08\) L/km (the number multiplying \(d\))
(ii) \(8\) L/100 km
(iii) \(875\) km (the tank is empty when \(F = 0\))
(ii) A car travels \(12.5\) km on each litre. Write its fuel use in L/100 km.
(iii) A tank fills at \(30\) L/min while leaking \(4\) L/min. How long does it take to gain \(520\) L?
| \(20 \times 3.6\) | \(=\) | \(72\) |
| \(1.1 \times 60\) | \(=\) | \(66\) |
| \(100 \div 12.5\) | \(=\) | \(8\) |
| \(520 \div (30 - 4)\) | \(=\) | \(20\) |
(i) \(72\), \(66\) and \(75\) km/h: \(75\) km/h is fastest.
(ii) \(8\) L/100 km
(iii) Net rate \(26\) L/min, so \(20\) min.
Common pitfalls
Frequently asked questions
What is the difference between a rate and a ratio?
A rate compares quantities of different kinds and has units, such as km/h. A ratio compares quantities of the same kind and has no units.
What is a simplified rate?
The amount for one unit, such as $7 per kilogram or 16 cm per year.
How do I change km/h to m/s?
Divide by 3.6. To change m/s to km/h, multiply by 3.6.
How do I change a rate per minute to a rate per hour?
Multiply by 60, because there are 60 minutes in an hour.
How do I compare two rates?
Change them to the same units first, then compare the numbers.