Decimal Division And Multiplication Practical Problems
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This article delves into solving decimal division problems without resorting to long division, calculating costs based on per-unit prices, and determining distances covered by a vehicle given its fuel efficiency. We'll explore various examples and break down the steps involved in arriving at the solutions. The concepts covered here are fundamental in mathematics and have practical applications in everyday life.
(i) Dividing 324.8 by 8: Leveraging Existing Information
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When tackling decimal division, it's essential to recognize patterns and utilize existing information to simplify calculations. In this case, we're given that 32.48 ÷ 8 = 4.06. Our goal is to find the value of 324.8 ÷ 8 without performing long division. The key observation here is that 324.8 is simply 32.48 multiplied by 10. This relationship allows us to leverage the given information to arrive at the solution efficiently.
To solve 324.8 ÷ 8, we can rewrite 324.8 as 10 * 32.48. Now the problem becomes (10 * 32.48) ÷ 8. Using the associative property of multiplication and division, we can rearrange this as 10 * (32.48 ÷ 8). We already know that 32.48 ÷ 8 = 4.06, so we can substitute that in: 10 * 4.06. Multiplying 4.06 by 10 simply shifts the decimal point one place to the right, giving us 40.6. Therefore, 324.8 ÷ 8 = 40.6.
This approach highlights the power of recognizing patterns and utilizing prior knowledge. Instead of performing a complex long division, we were able to solve the problem with a simple multiplication by leveraging the given information. This technique is particularly useful in situations where you encounter similar numbers or have already solved a related problem.
This example showcases an important mathematical principle: understanding the relationship between numbers can significantly simplify calculations. By recognizing that 324.8 is a multiple of 32.48, we were able to avoid long division and arrive at the answer quickly and efficiently. This type of problem-solving skill is crucial in mathematics and various real-world scenarios.
(ii) Dividing 32.48 by 0.8: Dealing with Decimal Divisors
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Dividing by a decimal can sometimes seem daunting, but there's a simple trick to transform it into a more manageable problem. The key is to eliminate the decimal in the divisor. We are tasked with finding the value of 32.48 ÷ 0.8, again, without resorting to long division directly. To do this, we multiply both the dividend (32.48) and the divisor (0.8) by the same power of 10. This action does not change the value of the result but transforms the divisor into a whole number.
In this case, we can multiply both 32.48 and 0.8 by 10. This gives us 324.8 ÷ 8. Notice that this is the same problem we solved in part (i)! We already know that 324.8 ÷ 8 = 40.6. Therefore, 32.48 ÷ 0.8 = 40.6.
The crucial step here is understanding why multiplying both the dividend and divisor by the same number doesn't change the result. This is based on the fundamental principle that division is the inverse of multiplication. If we multiply both the numerator and denominator of a fraction by the same number, we're essentially multiplying the fraction by 1, which doesn't change its value. For example, a/b is the same as (a10) / (b10).
This technique of eliminating the decimal in the divisor is widely applicable in decimal division problems. It simplifies the process and makes the calculation much easier to perform, especially without a calculator. By converting the divisor into a whole number, we can often use mental math or other simpler division techniques to find the solution.
Mastering this skill not only helps in solving mathematical problems but also builds a strong foundation for understanding the relationship between numbers and operations. This principle is fundamental in various mathematical contexts and has practical applications in areas such as finance, science, and engineering.
6. Calculating the Cost of Cloth: Applying Decimal Multiplication
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Real-world problems often involve applying mathematical concepts to practical scenarios. Here, we need to calculate the cost of 7.25 meters of cloth given that the cost per meter is ₹175.5. This is a straightforward application of decimal multiplication. The total cost is found by multiplying the quantity of cloth (7.25 meters) by the cost per meter (₹175.5).
To find the total cost, we need to multiply 7.25 by 175.5. This can be written as 7.25 * 175.5. Performing this multiplication, we get 1272.375. Since we're dealing with currency, we typically round to two decimal places. Therefore, the cost of 7.25 meters of cloth is ₹1272.38 (rounded to the nearest paisa).
Decimal multiplication involves multiplying the numbers as if they were whole numbers and then placing the decimal point in the correct position in the product. The number of decimal places in the product is the sum of the decimal places in the multiplicand and the multiplier. In this case, 7.25 has two decimal places, and 175.5 has one decimal place, so the product has three decimal places.
This problem highlights how decimal multiplication is used in everyday financial calculations. Whether it's calculating the cost of goods, determining sale prices, or figuring out taxes, understanding decimal operations is crucial for making informed financial decisions. Furthermore, being able to perform these calculations manually or with a calculator helps in verifying the accuracy of transactions and avoiding errors.
Beyond the specific context of cloth purchasing, this type of calculation applies to a wide range of scenarios, such as calculating the cost of materials for a project, determining the total bill for services rendered, or figuring out the expenses for a trip. The ability to confidently perform decimal multiplication is a valuable life skill.
7. Calculating Distance Covered: Fuel Efficiency and Decimal Multiplication
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Another common application of mathematics is in calculating distance, speed, and fuel consumption. In this problem, we're given that a motorbike covers a distance of 78.5 kilometers in one liter of petrol. We need to determine how much distance it can cover with a certain amount of petrol (the amount is not specified in this excerpt, so we will discuss the general method). This involves using the concept of fuel efficiency and applying decimal multiplication.
To find the distance covered with a given amount of petrol, we multiply the fuel efficiency (distance per liter) by the quantity of petrol. For example, if we want to find the distance covered with 10 liters of petrol, we would multiply 78.5 km/liter by 10 liters. This gives us 785 kilometers.
The general formula for calculating the distance covered is: Distance = Fuel Efficiency * Quantity of Petrol. Fuel efficiency is the distance a vehicle can travel per unit of fuel, often expressed in kilometers per liter (km/L) or miles per gallon (mpg). The quantity of petrol is the amount of fuel used, typically measured in liters or gallons.
Let's consider another example: If the motorbike has 2.5 liters of petrol, we would calculate the distance as 78.5 km/liter * 2.5 liters. This gives us 196.25 kilometers. Again, this involves decimal multiplication, highlighting the importance of this skill in practical applications.
These types of calculations are crucial for planning trips, estimating fuel costs, and comparing the fuel efficiency of different vehicles. Understanding the relationship between distance, fuel efficiency, and fuel consumption allows for better decision-making and cost management. Moreover, these calculations are also relevant in various fields such as logistics, transportation, and environmental science.
In conclusion, by understanding and applying mathematical concepts like decimal multiplication and division, we can solve practical problems related to finance, travel, and everyday life. These skills empower us to make informed decisions and navigate the world around us more effectively.
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- Question 5: If 32.48 ÷ 8 = 4.06, find the value of: (i) 324.8 ÷ 8 (ii) 32.48 ÷ 0.8
- Question 6: Find the cost of 7.25 m of cloth at ₹175.5 per meter.
- Question 7: A motorbike covers a distance of 78.5 km in one liter of petrol. How much distance can it cover with a given amount of petrol?
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Decimal Division and Multiplication Problems Solutions
