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\[ \text{Shop A: R}36 \text{ for } 2\text{L} \quad \text{Shop B: R}51 \text{ for } 3\text{L} \]
Shop A sells 2 litres of juice for R36. Shop B sells 3 litres for R51. Which is cheaper? Comparing only R36 and R51 is not fair. To compare fairly, we must calculate the cost per litre. This is called a unit rate.
\[ \text{Rate} = \text{comparison of different units} \]
A rate compares two quantities that are usually measured in different units. For example, 150 kilometres in 3 hours, or R240 for 8 notebooks. The units tell us exactly what is being compared.
\[ 150\text{ km}:3\text{ h} \quad \text{or} \quad \frac{150\text{ km}}{3\text{ h}} \]
A rate can be written with a colon, as a fraction, or using the word "in". All three forms mean the same thing. The fraction form is often the most useful for calculations.
\[ \text{Ratio: } 3\text{ boys}:5\text{ girls} \quad \text{Rate: } 80\text{ km/h} \]
A ratio usually compares quantities of the same type, and units may cancel out. A rate compares different units, and the units remain important. The word "per" often signals a rate, like kilometres per hour.
\[ \text{Rate} = \frac{\text{one quantity}}{\text{another quantity}} \]
A rate is a comparison of two different quantities. Always pay attention to the units, because they tell you exactly what the rate represents.
\[ \text{Divide both quantities by the same number} \]
To simplify a rate, divide both quantities by the same number. This makes the rate easier to understand and compare, just like simplifying a fraction.
\[ 180\text{ km in } 3\text{ h} = 60\text{ km/h} \]
To simplify 180 kilometres in 3 hours, divide 180 by 3 to get 60, and divide 3 by 3 to get 1. The simplified rate is 60 kilometres per hour.
\[ \frac{240\text{ bottles}}{6\text{ min}} = 40\text{ bottles/min} \]
A machine produces 240 bottles in 6 minutes. Divide 240 by 6 to get 40. The machine produces 40 bottles per minute.
\[ \text{Simplify} \rightarrow \text{divide both parts} \]
Simplifying a rate makes it cleaner and easier to work with. Always divide both the quantity and the unit of measurement by the same number.
\[ \text{Unit rate} = \text{amount for exactly one unit} \]
A unit rate gives the amount for exactly one unit of the second quantity. To find it, identify both quantities, decide which must become 1, and divide both by that number.
\[ \frac{\text{R}90}{6} = \text{R}15 \text{ per book} \]
Six exercise books cost R90. To find the cost of one book, divide R90 by 6. Each exercise book costs R15. This is the unit price.
\[ \frac{420\text{ L}}{7\text{ days}} = 60\text{ L/day} \]
A household uses 420 litres of water in 7 days. Divide 420 by 7 to find the daily rate. The household uses 60 litres per day.
\[ \frac{\text{R}137.50}{5\text{ kg}} = \text{R}27.50/\text{kg} \]
Five kilograms of rice cost R137.50. Divide 137.50 by 5. The unit price is R27.50 per kilogram. Unit rates often result in decimals, which is perfectly fine.
\[ \text{Unit rate} = \text{value for } 1 \]
Unit rates are incredibly useful because they allow us to compare different offers directly. Always include the correct units in your final answer.
\[ \text{Use compatible units first} \]
Before calculating or comparing rates, ensure the units are compatible. You cannot directly compare metres per second with kilometres per hour without converting.
\[ 1\text{ h}=60\text{ min} \quad 1\text{ km}=1000\text{ m} \]
Remember these common conversions: 1 hour equals 60 minutes, 1 minute equals 60 seconds, 1 kilometre equals 1000 metres, and 1 litre equals 1000 millilitres.
\[ \frac{600\text{ m}}{120\text{ s}} = 5\text{ m/s} \]
A runner covers 600 metres in 2 minutes. First, convert 2 minutes to 120 seconds. Then divide 600 by 120. The speed is 5 metres per second.
\[ \text{Convert} \rightarrow \text{Calculate} \]
Always check your units before calculating a rate. Converting to compatible units first prevents errors and ensures your final answer makes sense.
\[ S=\frac{D}{T} \quad D=S \times T \quad T=\frac{D}{S} \]
Speed equals distance divided by time. Distance equals speed multiplied by time. Time equals distance divided by speed. Cover the one you want to find in the triangle to see the calculation.
\[ S = \frac{270\text{ km}}{3\text{ h}} = 90\text{ km/h} \]
A bus travels 270 kilometres in 3 hours. Divide 270 by 3. The speed of the bus is 90 kilometres per hour.
\[ D = 80 \times 4 = 320\text{ km} \]
A car travels at 80 kilometres per hour for 4 hours. Multiply 80 by 4. The car travels a total distance of 320 kilometres.
\[ 1.5\text{ h} = 1\text{ h } 30\text{ min} \]
A common mistake is thinking 1.5 hours is 1 hour and 5 minutes. The 0.5 represents half an hour. Multiply 0.5 by 60 to get 30 minutes. Always convert the decimal part correctly.
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
The relationship between speed, distance, and time is fundamental. Always check that your units match before applying the formula, and convert decimal hours to minutes correctly.
\[ \text{Calculate the same unit rate for each} \]
To compare two options fairly, calculate the same unit rate for each. For prices, find the cost per kilogram or per litre. For speeds, find the distance per hour.
\[ \text{Pack A: R}72/\text{kg} \quad \text{Pack B: R}68/\text{kg} \]
Pack A is 750g for R54. Pack B is 1.2kg for R81.60. Convert to cost per kilogram. Pack A is R72/kg, Pack B is R68/kg. Pack B is the better buy.
\[ A: 0.2\text{ km/min} \quad B: 0.1875\text{ km/min} \]
Runner A covers 5 km in 25 minutes (0.2 km/min). Runner B covers 6 km in 32 minutes (0.1875 km/min). Runner A has the greater average speed.
\[ \text{Unit rate} \rightarrow \text{Fair comparison} \]
Never compare total prices or total times directly if the quantities are different. Always reduce both options to a unit rate to make a fair and accurate comparison.
\[ \text{Rate} = \frac{\text{Quantity 1}}{\text{Quantity 2}} \]
Excellent work. You can now define and simplify rates, calculate unit rates for cost and consumption, convert units correctly, use the speed-distance-time formulas, and compare rates to find the best buy. You are ready for the quiz.