Multiplying rational numbers is a foundational skill in mathematics with various applications across different fields. Here are some practical applications and practice problems involving the multiplication of rational numbers:
Applications of Multiplying Rational Numbers
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Financial Calculations:
- When calculating discounts, taxes, or interest rates, multiplying rational numbers helps determine total costs or profits. For example, if an item costs $100 and there is a 20% discount, you can calculate the discount amount by multiplying the price by the discount rate (0.20).
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Cooking and Recipes:
- Recipes often require scaling ingredients to adjust the serving size. For instance, if a recipe that serves 4 people uses \( \frac{3}{4} \) cup of sugar, then to find out how much sugar is needed for 6 servings, you can multiply \( \frac{3}{4} \) by \( \frac{6}{4} \) (the scaling factor).
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Area and Volume:
- When calculating the area of geometric shapes with rational dimensions, like a rectangle with sides \( \frac{2}{3} \) meters and \( \frac{1}{2} \) meters, you can multiply the lengths to find the area.
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Work Rates:
- In scenarios where several people work together, multiplying rational numbers can help determine how quickly tasks can be completed. For example, if one person can complete a task in \( \frac{2}{5} \) of an hour and works with a second person who works at the same rate, their combined work rate can be found by adding their individual rates.
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Physical Science:
- In physics, calculating speed, distance, and time can involve multiplying rational numbers. For example, if an object moves at a speed of \( \frac{3}{5} \) meters per second for \( \frac{4}{3} \) seconds, the distance traveled can be calculated by multiplying the speed by the time.
Practice Problems
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Discount Calculation: A jacket costs $80. If there is a \( \frac{15}{100} \) discount, how much is the discount? \[ \text{Discount} = 80 \times \frac{15}{100} \]
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Recipe Adjustment:
If a recipe requires \( \frac{2}{3} \) cup of flour for 3 servings, how much flour is needed for 5 servings? \[ \text{Flour for 5 servings} = \frac{2}{3} \times \frac{5}{3} \] -
Area Calculation: Calculate the area of a rectangle with a length of \( \frac{5}{8} \) meters and a width of \( \frac{3}{4} \) meters. \[ \text{Area} = \frac{5}{8} \times \frac{3}{4} \]
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Work Rate: If one person can do \( \frac{1}{4} \) of a job in one hour, how much of the job can they complete in \( 2 \frac{1}{2} \) hours? \[ \text{Job done} = \frac{1}{4} \times 2.5 \]
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Distance Problem: If a cyclist rides at a speed of \( \frac{2}{3} \) miles per minute for \( \frac{5}{2} \) minutes, how far does the cyclist travel? \[ \text{Distance} = \frac{2}{3} \times \frac{5}{2} \]
Answers
- \( 80 \times \frac{15}{100} = 12 \) (discount amount)
- \( \frac{2}{3} \times \frac{5}{3} = \frac{10}{9} \) cups of flour
- \( \frac{5}{8} \times \frac{3}{4} = \frac{15}{32} \) square meters
- \( \frac{1}{4} \times 2.5 = \frac{5}{8} \) (portion of the job done)
- \( \frac{2}{3} \times \frac{5}{2} = \frac{5}{3} \) miles
Encourage practice with these scenarios to enhance understanding and application of multiplying rational numbers in everyday life!