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In everyday life, we constantly measure how quickly or slowly something happens. Whether it's a car moving on a highway, people completing a task, or the price of goods per kilogram, all these can be understood through the concept of rate.
Rate is simply a measure of how one quantity changes or occurs with respect to another, often time or quantity. It answers questions like: How fast? How many per hour? How much per kilogram?
Understanding rate helps us make sense of many real-world situations in travel, work, business, and science.
Rate is defined as the ratio of a quantity to the time taken or another quantity. The general formula is:
It is important to keep the units consistent when calculating rate. For example, if distance is measured in kilometers and time in hours, rate will be expressed in kilometers per hour (km/h).
This diagram shows Rate as Quantity divided by Time flowing left to right.
One of the most common examples of rate is speed. Speed tells us how far something moves in a given amount of time.
The formula for speed is:
Speed is usually measured in units like kilometers per hour (km/h) or meters per second (m/s).
Conversion between these units is important:
| From | To | Conversion Factor | Example |
|---|---|---|---|
| km/h | m/s | Divide by 3.6 | 72 km/h = 72 / 3.6 = 20 m/s |
| m/s | km/h | Multiply by 3.6 | 15 m/s = 15 x 3.6 = 54 km/h |
This conversion comes from the fact that 1 km = 1000 m and 1 hour = 3600 seconds.
Work rate measures how much of a task is completed per unit time, usually expressed as "work done per hour". This is particularly useful for problems involving people or machines working together.
If someone can finish a job in 5 hours, their work rate is:
When two or more people or machines work together, their rates add up:
graph TD A[Worker 1 Rate = 1/T₁] --> C[Total Work Rate] B[Worker 2 Rate = 1/T₂] --> C[Total Work Rate] C --> D[Time for combined work = 1 / (1/T₁ + 1/T₂)]
This means the total work rate is the sum of individual rates, and the combined time is the reciprocal of this sum.
Rate is also used to express cost per quantity, like price per kilogram or litre. It helps in comparing prices and budgeting.
If 10 kg of sugar costs INR 300, then the cost rate (price per kg) is:
Step 1: Identify distance and time:
Distance = 120 km, Time = 2 hours
Step 2: Use the speed formula:
\( Speed = \frac{Distance}{Time} = \frac{120}{2} = 60 \, \text{km/h} \)
Answer: The speed of the car is 60 km/h.
Step 1: Calculate individual work rates:
Person A's work rate = \( \frac{1}{12} \) of the job per hour
Person B's work rate = \( \frac{1}{18} \) of the job per hour
Step 2: Add the work rates:
Total work rate = \( \frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} \)
Step 3: Calculate combined time:
Time = \( \frac{1}{\text{Total work rate}} = \frac{1}{5/36} = \frac{36}{5} = 7.2 \) hours
Answer: Working together, they will finish the job in 7.2 hours (7 hours and 12 minutes).
Step 1: Identify total cost and quantity:
Total cost = INR 250, Quantity = 5 kg
Step 2: Use the cost per unit formula:
\( \text{Cost per kg} = \frac{250}{5} = 50 \, \text{INR/kg} \)
Answer: The cost per kilogram of apples is INR 50.
Step 1: Identify speeds for onward and return journeys:
Speed 1 = 40 km/h, Speed 2 = 60 km/h
Step 2: Since distances are equal, use the formula for average speed:
Calculate average speed:
\( = \frac{2 \times 40 \times 60}{40 + 60} = \frac{4800}{100} = 48 \, \text{km/h} \)
Answer: The average speed for the round trip is 48 km/h.
Step 1: Calculate individual filling rates:
Pipe 1 fills \( \frac{1}{3} \) of the tank per hour
Pipe 2 fills \( \frac{1}{6} \) of the tank per hour
Step 2: Add the rates:
Total rate = \( \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \)
Step 3: Calculate combined time:
Time = \( \frac{1}{\text{Total rate}} = \frac{1}{1/2} = 2 \) hours
Answer: Together, they will fill the tank in 2 hours.
When to use: When distances are in kilometers but time is given in minutes or seconds.
When to use: For problems involving two or more people/machines working together.
When to use: When the onward and return trips have different speeds.
When to use: To avoid mixing km with meters or hours with minutes.
When to use: To interpret and express rate questions correctly.
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