To divide a whole number by a decimal, the most effective strategy is to transform the division problem into one involving only whole numbers, which simplifies the calculation significantly.
Understanding the Core Principle
Dividing by a decimal can seem daunting, but it's fundamentally about making the divisor (the number you are dividing by) a whole number. This is achieved by multiplying both the dividend (the whole number being divided) and the divisor by the same power of 10 (10, 100, 1000, etc.). This operation does not change the value of the quotient because you are essentially multiplying the fraction (dividend/divisor) by 1 (e.g., 10/10 or 100/100), creating an equivalent division problem.
Step-by-Step Guide to Dividing a Whole Number by a Decimal
Follow these steps to simplify and solve the division problem:
1. Eliminate the Decimal in the Divisor
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Count Decimal Places: Determine how many decimal places are in the divisor. This number will tell you which power of 10 to use. For example, if the divisor is 0.7, there is one decimal place. If it's 0.05, there are two.
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Multiply Both Numbers: Multiply both the divisor and the dividend by the power of 10 that corresponds to the number of decimal places counted.
- If 1 decimal place, multiply by 10.
- If 2 decimal places, multiply by 100.
- If 3 decimal places, multiply by 1000, and so on.
As explained by Sal Khan, a key principle in dividing a whole number by a decimal is to transform the division into one involving only whole numbers. This is achieved by multiplying both the dividend (the whole number) and the divisor (the decimal) by the same power of 10. For instance, dividing 518 by 0.7 becomes 5,180 divided by 7.
2. Perform Whole Number Division
- Once both numbers are whole numbers, you can use standard long division techniques to find the quotient.
- Continuing with the example from Sal Khan, using long division, the result for 5,180 divided by 7 is 740, making the process easier and more manageable.
3. Place the Decimal Point in the Quotient
- Since you've effectively removed the decimal from the divisor and shifted it in the dividend, the decimal point in your final answer (the quotient) will simply be placed at the end of the whole number result.
Practical Examples
Let's look at a few examples to illustrate the process:
| Original Problem | Divisor Decimal Places | Multiplier | Transformed Problem | Result |
|---|---|---|---|---|
| 518 ÷ 0.7 | 1 | 10 | 5180 ÷ 7 | 740 |
| 240 ÷ 0.6 | 1 | 10 | 2400 ÷ 6 | 400 |
| 125 ÷ 0.05 | 2 | 100 | 12500 ÷ 5 | 2500 |
Detailed Example: Divide 240 by 0.6
- Identify the divisor and dividend:
- Dividend: 240
- Divisor: 0.6
- Make the divisor a whole number:
- 0.6 has one decimal place.
- Multiply 0.6 by 10 to get 6.
- Multiply the dividend by the same amount:
- Multiply 240 by 10 to get 2400.
- Perform the whole number division:
- Now the problem is 2400 ÷ 6.
- 2400 ÷ 6 = 400.
- Therefore, 240 ÷ 0.6 = 400.
Why This Method Works
This method is based on the principle of equivalent fractions. When you have a division like $\frac{a}{b}$ (which is $a \div b$), multiplying both the numerator and the denominator by the same non-zero number, say $c$, results in an equivalent fraction $\frac{a \times c}{b \times c}$. The value of the fraction (the quotient) remains unchanged. By multiplying both the whole number and the decimal by the same power of 10, you are essentially creating an equivalent, but easier, division problem where the divisor is a whole number.
For more visual explanations and examples, you can refer to the Dividing a whole number by a decimal (video) by Sal Khan on Khan Academy.
