It seems you're looking for information on unit rates and proportions, which are fundamental concepts in mathematics. Here’s a brief overview of each:
Unit Rates
A unit rate is a comparison of two different quantities where one of the quantities is expressed as a single unit. It can be used to determine how much of one item corresponds to a single unit of another item.
Example:
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If a car travels 240 miles on 8 gallons of gas, the unit rate of miles per gallon is calculated as:
\[ \text{Unit Rate} = \frac{240 \text{ miles}}{8 \text{ gallons}} = 30 \text{ miles per gallon} \]
Proportions
A proportion is an equation that states that two ratios are equivalent. For example, if \( \frac{a}{b} = \frac{c}{d} \), then the proportions are equal. Proportions can be solved using cross-multiplication.
Example: If \( \frac{3}{4} = \frac{x}{12} \), you can solve for \( x \) by cross-multiplying:
\[ 3 \times 12 = 4 \times x \]
This simplifies to:
\[ 36 = 4x \]
Dividing both sides by 4 gives:
\[ x = 9 \]
Applications
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Unit Rates are often used in real-life scenarios, such as:
- Pricing (cost per item, cost per weight)
- Speed (miles per hour)
- Efficiency (jobs per hour)
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Proportions can be used to solve real-world problems involving scale, recipes, maps, and conversion between units.
Comparing Unit Rates
When comparing different quantities, finding the unit rates allows you to make fair comparisons. For instance, if one store sells apples at $3 for 4 pounds and another at $2 for 2 pounds, finding the unit rate helps to see which is the better deal:
Store 1: \[ \frac{3}{4} = 0.75 \text{ dollars per pound} \]
Store 2: \[ \frac{2}{2} = 1.00 \text{ dollars per pound} \]
Thus, Store 1 has the better deal at $0.75 per pound.
Summary
- Unit Rates help understand per-unit quantities.
- Proportions help analyze relationships between two quantities.
Feel free to ask for specific examples or problems if you want to practice or dive deeper into these topics!
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