Division of Decimals with Whole Numbers

Certainly. To divide a decimal like 64.8 by 4 using long division, follow these steps:

• Step 1: Set up the division problem as you normally would. Place the decimal point in the quotient area directly above the decimal point in the dividend (64.8).

• Step 2: Begin dividing from left to right. Divide 6 by 4, which is 1. Write 1 above the 6.

• Step 3: Continue the long division process. Bring down the next digit (4) to make it 24. Divide 24 by 4, which is 6. Write 6 above the 4.

• Step 4: Bring down the last digit (8). Divide 8 by 4, which is 2. Write 2 above the 8.

• Step 5: The final answer, with the correctly placed decimal point, is 16.2.

If the division doesn't result in a whole number and you have a remainder, you can continue dividing by adding zeros to the right of the decimal point in the dividend. For example, to divide 7 by 2:

• Treat 7 as 7.0. Divide 7 by 2 to get 3 with a remainder of 1.

• Place the decimal point in the quotient (3.).

• Bring down the 0 to make the remainder 10. Divide 10 by 2, which is 5.

• The final answer is 3.5. You can add as many zeros as needed to find the exact answer or to round to a specific decimal place.

The shortcut for dividing a decimal by a power of 10 is to simply move the decimal point to the left. The number of places you move the decimal is equal to the number of zeros in the divisor.

• To divide by 10 (1 zero), move the decimal 1 place to the left (e.g., 78.4 ÷ 10 = 7.84).

• To divide by 100 (2 zeros), move the decimal 2 places to the left (e.g., 78.4 ÷ 100 = 0.784).

• To divide by 1000 (3 zeros), move the decimal 3 places to the left (e.g., 78.4 ÷ 1000 = 0.0784).

This method is a procedural shortcut that is based on the principles of fractions. Dividing a decimal by a whole number is the same as dividing a fraction by a whole number. For instance, 8.4 ÷ 2 is equivalent to the fraction (84/10) ÷ 2. In fraction division, this becomes 84 / (10 × 2), which can be rewritten as (84 ÷ 2) / 10. This results in 42/10, which is 4.2. Placing the decimal point directly above in the quotient is a way to ensure the place value is correctly maintained without converting to fractions each time.

When the dividend is smaller than the divisor, a common mistake is to forget the zero placeholder. In the problem 3.6 ÷ 9:

• Since 3 cannot be divided by 9, you must place a 0 in the ones place of the quotient.

• Place the decimal point immediately after the 0.

• Now, treat the dividend as 36. Divide 36 by 9 to get 4.

• The correct answer is 0.4, not 4.

The key difference lies in the first step: making the divisor a whole number.

• Decimal ÷ Whole Number (e.g., 15.5 ÷ 5): You can divide directly without any initial changes. The answer is 3.1, which is smaller than the dividend.

• Whole Number ÷ Decimal (e.g., 155 ÷ 0.5): You must first convert the divisor (0.5) into a whole number by multiplying it by a power of 10. You must also multiply the dividend by the same power of 10. The problem becomes 1550 ÷ 5, and the answer is 310, which is larger than the original dividend.

Yes, this is a very common calculation used in everyday life. Examples include:

• Splitting a Bill: If a bill of ₹850.50 is shared equally among 5 friends, you would divide 850.50 by 5 to find each person's share (₹170.10).

• Calculating Averages: To find the average rainfall over 7 days when the total was 52.5 mm, you divide 52.5 by 7.

• Finding Unit Price: If a 3 kg bag of sugar costs ₹130.50, you divide 130.50 by 3 to find the price per kg.

Estimation is a powerful tool to verify if your answer is reasonable and to catch decimal placement errors. Before calculating, round the decimal to the nearest whole number and perform a simple division. For example, in the problem 47.8 ÷ 8:

• Round 47.8 to the nearest compatible number, which is 48.

• Divide 48 by 8, which equals 6.

• Your final, precise answer should be close to 6. The actual answer is 5.975. If you had mistakenly gotten 59.75 or 0.5975, your estimation would immediately show that your decimal point is in the wrong place.