Multiplying Decimals Step By Step Solutions

by THE IDEN 44 views

In mathematics, finding the product involves multiplying numbers together. When dealing with decimals, this process requires careful attention to decimal placement. This article serves as a comprehensive guide to accurately calculating the products of various decimal combinations. We will delve into six distinct multiplication problems, providing step-by-step solutions and clear explanations to enhance your understanding of decimal multiplication.

(i) 1.3 x 0.13

When multiplying decimals like 1.3 x 0.13, the initial step is to ignore the decimal points and treat the numbers as whole numbers. Thus, we multiply 13 by 13. The product of 13 and 13 is 169. Next, we need to account for the decimal points. In the original problem, 1.3 has one decimal place (one digit after the decimal), and 0.13 has two decimal places (two digits after the decimal). Adding these together, we have a total of three decimal places. This means that in our final answer, we need to have three digits after the decimal point. Starting from the right side of 169, we count three places to the left and insert the decimal point. Therefore, the product of 1.3 and 0.13 is 0.169. This careful placement ensures the accuracy of the calculation, reflecting the true value of the product in decimal form. Understanding and applying this method will help you confidently tackle similar decimal multiplication problems.

Step-by-Step Solution

  1. Multiply the numbers as if they were whole numbers: 13 x 13 = 169.
  2. Count the total number of decimal places in the original numbers: 1.3 (1 decimal place) + 0.13 (2 decimal places) = 3 decimal places.
  3. Place the decimal point in the product so that there are three digits after the decimal point: 0.169.

Therefore, 1.3 x 0.13 = 0.169.

(ii) 2.4 x 1.5 x 2.5

Multiplying multiple decimals, such as 2.4 x 1.5 x 2.5, requires a systematic approach to ensure accuracy. First, let's multiply the first two numbers, 2.4 and 1.5. Ignore the decimal points initially and multiply 24 by 15, which equals 360. Now, count the decimal places: 2.4 has one decimal place and 1.5 has one decimal place, totaling two decimal places. Place the decimal point in 360 so that there are two digits after the decimal point, resulting in 3.60, which is equivalent to 3.6. Next, multiply 3.6 by the remaining number, 2.5. Again, ignore the decimal points and multiply 36 by 25, which equals 900. Count the decimal places: 3.6 has one decimal place and 2.5 has one decimal place, totaling two decimal places. Place the decimal point in 900 so that there are two digits after the decimal point, resulting in 9.00, which simplifies to 9. Therefore, the product of 2.4, 1.5, and 2.5 is 9. This method of multiplying in stages and carefully tracking decimal places helps break down complex calculations into manageable steps, leading to the correct final answer.

Step-by-Step Solution

  1. Multiply the first two numbers: 2.4 x 1.5. Multiply as whole numbers: 24 x 15 = 360. Count decimal places: 2.4 (1 decimal place) + 1.5 (1 decimal place) = 2 decimal places. Place the decimal point: 3.60 or 3.6.
  2. Multiply the result by the third number: 3.6 x 2.5. Multiply as whole numbers: 36 x 25 = 900. Count decimal places: 3.6 (1 decimal place) + 2.5 (1 decimal place) = 2 decimal places. Place the decimal point: 9.00 or 9.

Therefore, 2.4 x 1.5 x 2.5 = 9.

(iii) 0.8 x 3.5 x 0.05

To find the product of 0.8 x 3.5 x 0.05, we follow a similar approach to the previous example. First, multiply 0.8 and 3.5. Treat them as whole numbers and multiply 8 by 35, which results in 280. Now, count the decimal places: 0.8 has one decimal place and 3.5 has one decimal place, totaling two decimal places. Place the decimal point in 280 so that there are two digits after the decimal point, giving us 2.80, which is equivalent to 2.8. Next, multiply 2.8 by 0.05. Treat them as whole numbers and multiply 28 by 5, which equals 140. Count the decimal places: 2.8 has one decimal place and 0.05 has two decimal places, totaling three decimal places. Place the decimal point in 140 so that there are three digits after the decimal point, resulting in 0.140, which simplifies to 0.14. Therefore, the product of 0.8, 3.5, and 0.05 is 0.14. This step-by-step methodology ensures that we accurately account for all decimal places, leading to the correct answer. By breaking down the problem into manageable steps, we minimize the chances of error and gain confidence in our calculations.

Step-by-Step Solution

  1. Multiply the first two numbers: 0.8 x 3.5. Multiply as whole numbers: 8 x 35 = 280. Count decimal places: 0.8 (1 decimal place) + 3.5 (1 decimal place) = 2 decimal places. Place the decimal point: 2.80 or 2.8.
  2. Multiply the result by the third number: 2.8 x 0.05. Multiply as whole numbers: 28 x 5 = 140. Count decimal places: 2.8 (1 decimal place) + 0.05 (2 decimal places) = 3 decimal places. Place the decimal point: 0.140 or 0.14.

Therefore, 0.8 x 3.5 x 0.05 = 0.14.

(iv) 0.2 x 0.02 x 0.002

Multiplying decimals such as 0.2 x 0.02 x 0.002 can seem intricate, but the same principles apply. First, multiply the numbers as if they were whole numbers: 2 x 2 x 2 = 8. Next, count the total number of decimal places in the original numbers. 0.2 has one decimal place, 0.02 has two decimal places, and 0.002 has three decimal places. Adding these together gives us a total of six decimal places. Now, we need to place the decimal point in our result (8) so that there are six digits after the decimal point. To do this, we’ll add leading zeros: 0.000008. Thus, the product of 0.2, 0.02, and 0.002 is 0.000008. This illustrates how crucial it is to accurately count decimal places to ensure the final answer is correct. The addition of leading zeros is often necessary when the number of decimal places exceeds the digits in the product, highlighting the need for a careful and methodical approach.

Step-by-Step Solution

  1. Multiply the numbers as if they were whole numbers: 2 x 2 x 2 = 8.
  2. Count the total number of decimal places in the original numbers: 0.2 (1 decimal place) + 0.02 (2 decimal places) + 0.002 (3 decimal places) = 6 decimal places.
  3. Place the decimal point in the product so that there are six digits after the decimal point: 0.000008.

Therefore, 0.2 x 0.02 x 0.002 = 0.000008.

(v) 11.1 x 1.1 x 0.11

Calculating the product of 11.1 x 1.1 x 0.11 involves similar steps but with slightly larger numbers. First, let's multiply 11.1 by 1.1. Treat them as whole numbers: 111 x 11 = 1221. Now, count the decimal places: 11.1 has one decimal place and 1.1 has one decimal place, totaling two decimal places. Place the decimal point in 1221 so that there are two digits after the decimal point, giving us 12.21. Next, multiply 12.21 by 0.11. Treat them as whole numbers: 1221 x 11 = 13431. Count the decimal places: 12.21 has two decimal places and 0.11 has two decimal places, totaling four decimal places. Place the decimal point in 13431 so that there are four digits after the decimal point, resulting in 1.3431. Therefore, the product of 11.1, 1.1, and 0.11 is 1.3431. This methodical approach helps manage larger numbers and multiple decimal places, ensuring accuracy in the final result. Breaking the problem into smaller, manageable steps is a key strategy for success in these calculations.

Step-by-Step Solution

  1. Multiply the first two numbers: 11.1 x 1.1. Multiply as whole numbers: 111 x 11 = 1221. Count decimal places: 11.1 (1 decimal place) + 1.1 (1 decimal place) = 2 decimal places. Place the decimal point: 12.21.
  2. Multiply the result by the third number: 12.21 x 0.11. Multiply as whole numbers: 1221 x 11 = 13431. Count decimal places: 12.21 (2 decimal places) + 0.11 (2 decimal places) = 4 decimal places. Place the decimal point: 1.3431.

Therefore, 11.1 x 1.1 x 0.11 = 1.3431.

(vi) 2.1 x 0.21 x 0.021

Calculating the product of 2.1 x 0.21 x 0.021 follows the same principles as the previous examples. First, we multiply the numbers ignoring the decimal points: 21 x 21 x 21. The product of 21 x 21 is 441, and then 441 x 21 is 9261. Now, we count the decimal places. 2.1 has one decimal place, 0.21 has two decimal places, and 0.021 has three decimal places. Adding these together, we have a total of six decimal places. We need to place the decimal point in 9261 so that there are six digits after the decimal point. This requires adding leading zeros to get 0.009261. Therefore, the product of 2.1, 0.21, and 0.021 is 0.009261. This problem reinforces the importance of meticulous decimal place counting and the strategic use of leading zeros to ensure accurate results. Consistent application of these techniques builds confidence in solving similar decimal multiplication problems.

Step-by-Step Solution

  1. Multiply the numbers as if they were whole numbers: 21 x 21 x 21 = 9261.
  2. Count the total number of decimal places in the original numbers: 2.1 (1 decimal place) + 0.21 (2 decimal places) + 0.021 (3 decimal places) = 6 decimal places.
  3. Place the decimal point in the product so that there are six digits after the decimal point: 0.009261.

Therefore, 2.1 x 0.21 x 0.021 = 0.009261.

In conclusion, multiplying decimals requires a systematic approach that involves treating the numbers as whole numbers during multiplication and then carefully placing the decimal point in the final answer. By counting the total number of decimal places in the original numbers and applying this count to the product, accurate results can be achieved. This method is effective for multiplying any number of decimal values, making it a valuable skill in various mathematical contexts.