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Number - Rates and Unit Rates

Grade 7IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Rate is a ratio that compares two different quantities with different units, such as 150 kilometers3 hours\frac{150 \text{ kilometers}}{3 \text{ hours}}.

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A Unit Rate is a simplified version of a rate where the second quantity (the denominator) is exactly 11 unit. For example, 50 km/h50 \text{ km/h} is a unit rate.

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To calculate a unit rate, divide the numerator by the denominator: Rate=ab→Unit Rate=(a÷b) units per unit\text{Rate} = \frac{a}{b} \rightarrow \text{Unit Rate} = (a \div b) \text{ units per unit}.

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Unit Price is a specific type of unit rate used in commerce to find the cost of a single item or a single unit of measurement (like per gram or per liter), helping to determine the 'better buy'.

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In proportional relationships, the unit rate is often referred to as the Constant of Proportionality, represented by the variable kk in the equation y=kxy = kx.

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Rates can be converted between units. For instance, converting meters per second\text{meters per second} to kilometers per hour\text{kilometers per hour} involves using conversion factors where the value of the factor is 11.

📐Formulae

Unit Rate=Quantity 1Quantity 2\text{Unit Rate} = \frac{\text{Quantity 1}}{\text{Quantity 2}}

Average Speed=Total DistanceTotal Time\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

Unit Price=Total CostNumber of Units\text{Unit Price} = \frac{\text{Total Cost}}{\text{Number of Units}}

Quantity=Unit Rate×Time (or other base unit)\text{Quantity} = \text{Unit Rate} \times \text{Time (or other base unit)}

💡Examples

Problem 1:

A car travels 450 km450 \text{ km} in 5 hours5 \text{ hours}. What is the unit rate of speed?

Solution:

Speed=450 km5 hours=90 km/h\text{Speed} = \frac{450 \text{ km}}{5 \text{ hours}} = 90 \text{ km/h}

Explanation:

To find the unit rate (speed), we divide the total distance by the total time. The result 9090 represents the distance traveled in exactly 11 hour.

Problem 2:

A grocery store sells a 2 kg2 \text{ kg} bag of rice for Rs 160160 and a 5 kg5 \text{ kg} bag for Rs 375375. Which bag is the better buy?

Solution:

Unit Price of 2 kg bag=1602=80 per kg\text{Unit Price of 2 kg bag} = \frac{160}{2} = 80 \text{ per kg} Unit Price of 5 kg bag=3755=75 per kg\text{Unit Price of 5 kg bag} = \frac{375}{5} = 75 \text{ per kg}

Explanation:

By calculating the unit price (cost per 1 kg1 \text{ kg}), we can compare the two options directly. Since 75<8075 < 80, the 5 kg5 \text{ kg} bag is the better buy.

Problem 3:

A water pump can fill a 1200 liter1200 \text{ liter} tank in 15 minutes15 \text{ minutes}. Find the rate of the pump in liters per second.

Solution:

Rate in L/min=1200 L15 min=80 L/min\text{Rate in L/min} = \frac{1200 \text{ L}}{15 \text{ min}} = 80 \text{ L/min} Rate in L/sec=80 L60 sec=43≈1.33 L/sec\text{Rate in L/sec} = \frac{80 \text{ L}}{60 \text{ sec}} = \frac{4}{3} \approx 1.33 \text{ L/sec}

Explanation:

First, we find the rate per minute by dividing the volume by the time. Then, since 1 minute=60 seconds1 \text{ minute} = 60 \text{ seconds}, we divide the per-minute rate by 6060 to find the rate per second.